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Network Coding: An Introduction
Tracey Ho
Desmond S. Lun
Contents
Preface page vi
1 Introduction 1
1.1 What is network coding? 1
1.2 What is network coding good for? 3
1.2.1 Throughput 3
1.2.2 Robustness 6
1.2.3 Complexity 9
1.2.4 Security 10
1.3 Network model 10
1.4 Outline of book 13
1.5 Notes and further reading 15
2 Lossless Multicast Network Coding 16
2.0.1 Notational conventions 16
2.1 Basic network model and multicast network coding
problem formulation 16
2.2 Delay-free scalar linear network coding 17
2.3 Solvability and throughput 20
2.3.1 The unicast case 20
2.3.2 The multicast case 21
2.3.3 Multicasting from multiple source nodes 22
2.3.4 Maximum throughput advantage 23
2.4 Multicast network code construction 25
2.4.1 Centralized polynomial-time construction 25
2.4.2 Random linear network coding 28
2.5 Packet networks 32
2.5.1 Distributed random linear coding for packet
networks 33
iii
iv Contents
2.6 Networks with cycles and convolutional network coding 36
2.6.1 Algebraic representation of convolutional
network coding 37
2.7 Correlated source processes 40
2.7.1 Joint source-network coding 41
2.7.2 Separation of source coding and network coding 43
2.8 Notes and further reading 44
2.A Appendix: Random network coding 46
3 Inter-Session Network Coding 56
3.1 Scalar and vector linear network coding 57
3.2 Fractional coding problem formulation 58
3.3 Insufficiency of linear network coding 60
3.4 Information theoretic approaches 62
3.4.1 Multiple unicast networks 67
3.5 Constructive approaches 68
3.5.1 Pairwise XOR coding in wireline networks 68
3.5.2 XOR coding in wireless networks 70
3.6 Notes and further reading 74
4 Network Coding in Lossy Networks 76
4.1 Random linear network coding 79
4.2 Coding theorems 81
4.2.1 Unicast connections 81
4.2.2 Multicast connections 94
4.3 Error exponents for Poisson traffic with i.i.d. losses 96
4.4 Notes and further reading 98
5 Subgraph Selection 99
5.1 Flow-based approaches 100
5.1.1 Intra-session coding 101
5.1.2 Computation-constrained coding 127
5.1.3 Inter-session coding 128
5.2 Queue-length-based approaches 132
5.2.1 Intra-session network coding for multiple
multicast sessions 133
5.2.2 Inter-session coding 147
5.3 Notes and further reading 148
6 Security Against Adversarial Errors 149
6.1 Introduction 149
6.1.1 Notational conventions 150
6.2 Error correction 150
Contents v
6.2.1 Error correction bounds for centralized
network coding 150
6.2.2 Distributed random network coding and
polynomial-complexity error correction 162
6.3 Detection of adversarial errors 168
6.3.1 Model and problem formulation 169
6.3.2 Detection probability 171
6.4 Notes and further reading 173
6.A Appendix: Proof of results for adversarial error
detection 173
Bibliography 179
Index 188
Preface
The basic idea behind network coding is extraordinarily simple. As it is
defined in this book, network coding amounts to no more than perform-
ing coding operations on the contents of packets—performing arbitrary
mappings on the contents of packets rather than the restricted functions
of replication and forwarding that are typically allowed in conventional,
store-and-forward architectures. But, although simple, network coding
has had little place in the history of networking. This is for good reason:
in the traditional wireline technologies that have dominated networking
history, network coding is not very practical or advantageous.
Today, we see the emergence, not only of new technologies, but also
of new services, and, in designing new network protocols for these new
technologies and services, we must be careful not to simply transplant old
protocols because we are familiar with them. Instead, we must consider
whether other, hitherto unused, ideas may lead to better solutions for
the new situation.
Network coding shows much promise as such an idea. In particular,
various theoretical and empirical studies suggest that significant gains
can be obtained by using network coding in multi-hop wireless networks
and for serving multicast sessions, which are certainly examples of fast-
emerging technologies or services. These studies have, for example, en-
couraged Microsoft to adopt network coding as a core technology of
its Avalanche project—a research project that aims to develop a peer-
to-peer file distribution system—exploiting the advantages offered by
network coding for multicast services. Thus, the time may finally be
ripe for network coding.
Hence the motivation for this book: we feel that network coding may
have a great deal to offer to the future design of packet networks, and
we would like to help this potential be realized. We would like also to
vi
Preface vii
encourage more research in this burgeoning field. Thus, we have aimed
the book at two (not necessarily distinct) audiences: first, the practi-
tioner, whose main interest is applications; and, second, the theoretician,
whose main interest is developing further understanding of the proper-
ties of network coding. Of these two audiences, we have tended to favor
the first, though the content of the book is nevertheless theoretical. We
have aimed to expound the theory in such a way that it is accessible to
those who would like to implement network coding, serving an important
purpose that was, in our opinion, inadequately served. The theoretician,
in contrast to the practitioner, is spoiled. Besides this book, a survey
of important theoretical results in network coding is provided in Yeung
et al.’s excellent review, Network Coding Theory [149, 150]. Because
of our inclination toward applications, however, our presentation differs
substantially from that of Yeung et al.
Our presentation draws substantially from our doctoral dissertations
[60, 93], and a bias toward work with which we have personally been
involved is shown. We endeavor, however, to ensure that most of the
significant work in network coding is either covered in the text or men-
tioned in summary—a goal aided by the notes and further reading in the
final section of each chapter. There will inevitably be some unintended
omissions, for which we apologize in advance.
Broadly, we intend for the book to be accessible to any reader who
has a background in electrical engineering or computer science. Some
mathematical methods that we use, such as some algebraic methods and
some optimization techniques, may be unfamiliar to readers in this cat-
egory and, though we do not cover these methods, we provide references
to suitable textbooks where relevant.
We have many to thank for their help and support during the develop-
ment of this book. Foremost among them are Muriel M´edard and Ralf
Koetter, doctoral and postdoctoral advisers, respectively, to us both,
who have been exemplary personal and professional mentors. We would
also like to thank the wonderful group of collaborators that worked
with us on network coding: Ebad Ahmed, Yu-Han Chang, Supratim
Deb, Michelle Effros, Atilla Eryilmaz, Christina Fragouli, Mario Gerla,
Keesook J. Han, Nick Harvey, Sidharth (Sid) Jaggi, David R. Karger,
Dina Katabi, Sachin Katti, J¨org Kliewer, Michael Langberg, Hyunjoo
Lee, Ben Leong, Petar Maymounkov, Payam Pakzad, Joon-Sang Park,
Niranjan Ratnakar, Siddharth Ray, Jun Shi, Danail Traskov, Sriram
Vishwanath, Harish Viswanathan, and Fang Zhao. In general, the en-
tire network coding community has been a very delightful, friendly, and
viii Preface
intellectually-stimulating community in which to work, and we would
like to thank all its members for making it so. We would also like to
thank Tao Cui, Theodoros Dikaliotis and Elona Erez for their helpful
suggestions and comments on drafts of this book. There are two further
groups that we would like to thank—without them this book certainly
could not have been produced. The first is the great group of profes-
sionals at Cambridge University Press who saw the book to publication:
we thank, in particular, Phil Meyler, Anna Littlewood, and Daisy Bar-
ton. The second is our families, for their love and support during our
graduate studies and the writing of this book.
1
Introduction
Network coding, as a field of study, is young. It was only in 2000 that the
seminal paper by Ahlswede, Cai, Li, and Yeung [4], which is generally
attributed with the “birth” of network coding, was published. As such,
network coding, like many young fields, is characterized by some degree
of confusion, of both excitement about its possibilities and skepticism
about its potential. Clarifying this confusion is one of the principal aims
of this book. Thus, we begin soberly, with a definition of network coding.
1.1 What is network coding?
Defining network coding is not straightforward. There are several defi-
nitions that can be and have been used.
In their seminal paper [4], Ahlswede, Cai, Li, and Yeung say that they
“refer to coding at a node in a network as network coding”, where, by
coding, they mean an arbitrary, causal mapping from inputs to outputs.
This is the most general definition of network coding. But it does not
distinguish the study of network coding from network, or multitermi-
nal, information theory—a much older field with a wealth of difficult
open problems. Since we do not wish to devote this book to network in-
formation theory (good coverage of network information theory already
exists, for example, in [28, Chapter 14]), we seek to go further with our
definition.
A feature of Ahlswede et al.’s paper that distinguishes it from most
network information theory papers is that, rather than looking at general
networks where essentially every node has an arbitrary, probabilistic
effect on every other node, they look specifically at networks consisting
of nodes interconnected by error-free point-to-point links. Thus the
network model of Ahlswede et al. is a special case of those ordinarily
1
2Introduction
studied in network information theory, albeit one that is very pertinent
to present-day networks—essentially all wireline networks can be cast
into their model once the physical layer has been abstracted into error-
free conduits for carrying bits.
Another possible definition of network coding, then, is coding at a
node in a network with error-free links. This distinguishes the function
of network coding from that of channel coding for noisy links; we can
similarly distinguish the function of network coding from that of source
coding by considering the former in the context of independent incom-
pressible source processes. This definition is frequently used and, under
it, the study of network coding reduces to a special case of network
information theory. This special case has in fact been studied well be-
fore 2000 (see, for example, [52, 133]), which detracts from some of the
novelty of network coding, but we can still go further with our definition.
Much work in network coding has concentrated around a particu-
lar form of network coding: random linear network coding. Random
linear network coding was introduced in [62] as a simple, randomized
coding method that maintains “a vector of coefficients for each of the
source processes,” which is “updated by each coding node”. In other
words, random linear network coding requires messages being commu-
nicated through the network to be accompanied by some degree of extra
information—in this case a vector of coefficients. In today’s communica-
tions networks, there is a type of network that is widely-used, that easily
accommodates such extra information, and that, moreover, consists of
error-free links: packet networks. With packets, such extra information,
or side information, can be placed in packet headers and, certainly, plac-
ing side information in packet headers is common practice today (e.g.,
sequence numbers are often placed in packet headers to keep track of
order).
A third definition of network coding, then, is coding at a node in a
packet network (where data is divided into packets and network coding
is applied to the contents of packets), or more generally, coding above
the physical layer. This is unlike network information theory, which is
generally concerned with coding at the physical layer. We use this defini-
tion in this book. Restricting attention to packet networks does, in some
cases, limit our scope unnecessarily, and some results with implications
beyond packet networks may not be reported as such. Nevertheless,
this definition is useful because it grounds our discussion in a concrete
setting relevant to practice.
1.2 What is network coding good for? 3
t2
s
t1
1
3 4
2
b1
b2
b1
b1⊕b2
b2
b1
b2
b1⊕b2
b1⊕b2
Fig. 1.1. The butterfly network. In this network, every arc represents a di-
rected link that is capable of carrying a single packet reliably. There are two
packets, b1and b2, present at the source node s, and we wish to communicate
the contents of these two packets to both of the sink nodes, t1and t2.
1.2 What is network coding good for?
Equipped with a definition, we now proceed to discuss the utility of
network coding. Network coding can improve throughput, robustness,
complexity, and security. We discuss each of these performance factors
in turn.
1.2.1 Throughput
The most well-known utility of network coding—and the easiest to illustrate—
is increase of throughput. This throughput benefit is achieved by using
packet transmissions more efficiently, i.e., by communicating more infor-
mation with fewer packet transmissions. The most famous example of
this benefit was given by Ahlswede et al. [4], who considered the problem
of multicast in a wireline network. Their example, which is commonly
referred to as the butterfly network (see Figure 1.1). features a multi-
cast from a single source to two sinks, or destinations. Both sinks wish
to know, in full, the message at the source node. In the capacitated
network that they consider, the desired multicast connection can be es-
tablished only if one of the intermediate nodes (i.e., a node that is neither
source nor sink) breaks from the traditional routing paradigm of packet
networks, where intermediate nodes are allowed only to make copies of
received packets for output, and performs a coding operation—it takes
4Introduction
s2
t2
t1
s1
1 2
b1⊕b2
b1
b1⊕b2
b1⊕b2
b1
b2
b2
Fig. 1.2. The modified butterfly network. In this network, every arc represents
a directed link that is capable of carrying a single packet reliably. There is one
packet b1present at source node s1that we wish to communicate to sink node
t1and one packet b2present at source node s2that we wish to communicate
to sink node t2.
two received packets, forms a new packet by taking the binary sum,
or xor, of the two packets, and outputs the resulting packet. Thus, if
the contents of the two received packets are the vectors b1and b2, each
comprised of bits, then the packet that is output is b1⊕b2, formed from
the bitwise xor of b1and b2. The sinks decode by performing further
coding operations on the packets that they each receive. Sink t1recovers
b2by taking the xor of b1and b1⊕b2, and likewise sink t2recovers b1
by taking the xor of b2and b1⊕b2. Under routing, we could commu-
nicate, for example, b1and b2to t1, but we would then only be able to
communicate one of b1or b2to t2.
The butterfly network, while contrived, illustrates an important point:
that network coding can increase throughput for multicast in a wireline
network. The nine packet transmissions that are used in the butter-
fly network communicate the contents of two packets. Without coding,
these nine transmissions cannot used to communicate as much informa-
tion, and they must be supplemented with additional transmissions (for
example, an additional transmission from node 3 to node 4).
While network coding can increase throughput for multicast in a wire-
line network, its throughput benefits are not limited to multicast or to
wireline networks. A simple modification of the butterfly network leads
to an example that involves two unicast connections that, with coding,
can be established and, without coding, cannot (see Figure 1.2). This
1.2 What is network coding good for? 5
2
3
1
s
t1
t2
b1
b2
b1⊕b2
b1
b2
Fig. 1.3. The wireless butterfly network. In this network, every hyperarc
represents a directed link that is capable of carrying a single packet reliably
to one or more nodes. There are two packets, b1and b2, present at the source
node s, and we wish to communicate the contents of these two packets to both
of the sink nodes, t1and t2.
example involves two unicast connections. For unicast in the lossless
wireline networks that have been considered so far, a minimum of two
unicast connections is necessary for there to be a throughput gain from
network coding. As we establish more concretely in Section 2.3, network
coding yields no throughput advantage over routing for a single unicast
connection in a lossless wireline network.
Network coding can also be extended to wireless networks and, in
wireless networks, it becomes even easier to find examples where net-
work coding yields a throughput advantage over routing. Indeed, the
wireless counterparts of the butterfly network (Figure 1.3) and the mod-
ified butterfly network (Figure 1.4) involve fewer nodes—six and three
nodes, respectively, as opposed to seven and six. As before, these exam-
ples show instances where the desired communication objective is not
achievable using routing, but is achievable using coding. These wireless
examples differ in that, rather than assuming that packet transmissions
are from a single node to a single other node, they allow for packet
transmissions to originate at a single node and end at more than one
node. Thus, rather than representing transmissions with arcs, we use
hyperarcs—generalizations of arcs that may have more than one end
node.
The examples that we have discussed so far demonstrate that, even
in the absence of losses and errors, network coding can yield a through-
6Introduction
s1
s2
1
b1
b2
b1⊕b2
Fig. 1.4. The modified wireless butterfly network. In this network, every
hyperarc represents a directed link that is capable of carrying a single packet
reliably to one or more nodes. There is one packet b1present at source node s1
that we wish to communicate to node s2and one packet b2present at source
node s2that we wish to communicate to node s1.
put advantage when it is applied either to one or more simultaneous
multicast connections or two or more simultaneous unicast connections.
This is true both when packets are transmitted only from a single node
to a single other node (wireline networks) and when they are transmit-
ted from a single node to one or more other nodes (wireless networks).
These examples are, however, mere contrived, toy examples, and it is
natural to wonder whether network coding can be generalized and, if
so, to what end. Much of the remainder of this book will be devoted to
generalizing the observations made thus far to network coding in more
general settings.
1.2.2 Robustness
1.2.2.1 Robustness to packet losses
But before we proceed, we address an important issue in packet net-
works, particularly wireless packet networks, that we have thus far ne-
glected: packet loss. Packet loss arises for various reasons in networks,
which include buffer overflow, link outage, and collision. There are a
number of ways to deal with such losses. Perhaps the most straightfor-
ward, which is the mechanism used by the transmission control protocol
(tcp), is to set up a system of acknowledgments, where packets received
by the sink are acknowledged by a message sent back to the source and, if
1.2 What is network coding good for? 7
21 3
Fig. 1.5. Two-link tandem network. Nodes 1 and 2 are each capable of inject-
ing a single packet per unit time on their respective outgoing links.
the source does not receive the acknowledgment for a particular packet,
it retransmits the packet. An alternative method that is sometimes used
is channel coding or, more specifically, erasure coding. An erasure code,
applied by the source node, introduces a degree of redundancy to the
packets so that the message can be recovered even if only a subset of the
packets sent by the source are received by the sink.
Erasure coding is coding applied by the source node. What about
coding applied by intermediate nodes? That is, what about network
coding? Is network coding useful in combating against packet losses? It
is; and the reason it is can be seen with a very simple example. Con-
sider the simple, two-link tandem network shown in Figure 1.5. In this
network, packets are lost on the link joining nodes 1 and 2 with prob-
ability ε12 and on the link joining nodes 2 and 3 with probability ε23.
An erasure code, applied at node 1, allows us to communicate informa-
tion at a rate of (1 −ε12)(1 −ε23) packets per unit time. Essentially we
have, between nodes 1 and 3, an erasure channel with erasure probability
1−(1−ε12 )(1−ε23 ), whose capacity, of (1−ε12 )(1−ε23 ), can be achieved
(or approached) with a suitably-designed code. But the true capacity of
the system is greater. If we apply an erasure code over the link joining
nodes 1 and 2 and another over the link joining nodes 2 and 3, i.e., if
we use two stages of erasure coding with full decoding and re-encoding
at node 2, then we can communicate information between nodes 1 and
2 at a rate of 1 −ε12 packets per unit time and between nodes 2 and 3
at a rate of 1 −ε23 packets per unit time. Thus, we can communicate
information between nodes 1 and 3 at a rate of min(1 −ε12 ,1−ε23),
which is in general greater than (1 −ε12)(1 −ε23).
So why isn’t this solution used in packet networks? A key reason is
delay. Each stage of erasure coding, whether it uses a block code or a
convolutional code, incurs some degree of delay because the decoder of
each stage needs to receive some number of packets before decoding can
begin. Thus, if erasure coding is applied over every link a connection,
the total delay would be large. But applying extra stages of erasure
coding is simply a special form of network coding—it is coding applied
8Introduction
2
1 3
Fig. 1.6. The packet relay channel. Nodes 1 and 2 are each capable of injecting
a single packet per unit time on their respective outgoing links.
at intermediate nodes. Thus, network coding can be used to provide
robustness against packet losses, which can be translated into through-
put gains. What we want from a network coding solution is not only
increased throughput, however, we want a solution that goes beyond
merely applying additional stages of erasure coding—we want a net-
work coding scheme that applies additional coding at intermediate code
without decoding. In Chapter 4, we discuss how random linear network
coding satisfies the requirements of such a coding scheme.
Losses add an additional dimension to network coding problems and,
when losses are present, even a single unicast connection suffices for
gains to be observed. Losses are very pertinent to wireless networks,
and considering losses makes network coding more relevant to wireless
applications. Another characteristic of wireless networks that we have
discussed is the presence of broadcast links—links that reach more than
one end node—and we have yet to combine losses and broadcast links.
In Figure 1.6, we show a modification of the two-link tandem net-
work that we call the packet relay channel. Here, the link coming out
of node 1 doesn’t only reach node 2, but also reaches node 3. Be-
cause of packet loss, however, whether a packet transmitted by node 1
is received by neither node 2 nor node 3, by node 2 only, by node 3
only, or by both nodes 2 and 3 is determined probabilistically. Let’s
say packets transmitted by node 1 are received by node 2 only with
probability p1(23)2, by node 3 only with probability p1(23)3, and by both
nodes 2 and 3 with probability p1(23)(23)) (they are lost entirely with
probability 1 −p1(23)2 −p1(23)3 −p1(23)(23)). As for packets transmit-
ted by node 2, let’s say packets transmitted by node 2 are received
by node 3 with probability p233 (they are lost entirely with probability
1−p233). Network coding, in particular random linear network cod-
ing, allows for the maximum achievable throughput in such a set-up,
1.2 What is network coding good for? 9
known as the min-cut capacity, to be reached, which in this case is
min(p1(23)2 +p1(23)3 +p1(23)(23), p1(23)3 +p1(23)(23) +p233 ).
This is no mean feat: first, from the standpoint of network infor-
mation theory, it is not even clear that there would exist a simple,
capacity-achieving network code and, second, it represents a significant
shift from the prevailing approach to wireless packet networks. The pre-
vailing, routing approach advocates treating wireless packet networks as
an extension of wireline packet networks. Thus, it advocates sending
information along routes; in this case, either sending information from
node 1 to node 2, then to node 3, or directly from node 1 to node 3, or,
in more sophisticated schemes, using a combination of the two. With
network coding, there are no paths as such—nodes contribute transmis-
sions to a particular connection, but these nodes do not necessarily fall
along a path. Hence a rethink of routing is necessary. This rethink
results in subgraph selection, which we examine in Chapter 5.
1.2.2.2 Robustness to link failures
Besides robustness against random packet losses, network coding is also
useful for protection from non-ergodic link failures. Live path protection,
where a primary and a backup flow are transmitted for each connection,
allows very fast recovery from link failures, since rerouting is not re-
quired. By allowing sharing of network resources among different flows,
network coding can improve resource usage. For a single multicast ses-
sion, there exists, for any set of failure patterns from which recovery is
possible with arbitrary rerouting, a static network coding solution that
allows recovery from any failure pattern in the set.
1.2.3 Complexity
In some cases, although optimal routing may be able to achieve similar
performance to that of network coding, the optimal routing solution
is difficult to obtain. For instance, minimum-cost subgraph selection
for multicast routing involves Steiner trees, which is complex even in
a centralized setting, while the corresponding problem with network
coding is a linear optimization that admits low-complexity distributed
solutions. This is discussed further in Section 5.1.1.
Network coding has also been shown to substantially improve per-
formance in settings where practical limitations necessitate suboptimal
solutions, e.g., gossip-based data dissemination [33] and 802.11 wireless
ad hoc networking [76].
10 Introduction
1.2.4 Security
From a security standpoint, network coding can offer both benefits and
drawbacks. Consider again the butterfly network (Figure 1.1). Suppose
an adversary manages to obtain only the packet b1⊕b2. With the packet
b1⊕b2alone, the adversary cannot obtain either b1or b2; thus we have a
possible mechanism for secure communication. In this instance, network
coding offers a security benefit.
Alternatively, suppose that node 3 is a malicious node that does not
send out b1⊕b2, but rather a packet masquerading as b1⊕b2. Because
packets are coded rather than routed, such tampering of packets is more
difficult to detect. In this instance, network coding results in a poten-
tial security drawback. We discuss the security implications of network
coding in Chapter 6.
We have now given a number of toy examples illustrating some benefits
of network coding. That these examples bear some relevance to packet
networks should be evident; exactly how the principles they illustrate
can be exploited in actual settings is perhaps not. We address more
general cases using the model that we put forth in the following section.
1.3 Network model
Packet networks, especially wireless packet networks, are immensely
complex and, as such, difficult to accurately model. Moreover, network
coding is used in such a wide variety of contexts that it is not sensible
to always use the same model. Nevertheless, there are common aspects
to all the models that we employ, which we now discuss. The specific
aspects of the various models we use are discussed as we encounter them.
As a starting point for our model, we assume that there are a number
of connections, or sessions, that we wish to establish. These connections
may be unicast (with a single source node and a single sink node) or
multicast (with a single source node and more than one sink node). In a
multicast connection, all the sink nodes wish to know the same message
originating from the source node. These connections are accompanied
with packets that we wish to communicate at rates that may or may
not be known. Thus, our model ignores congestion control, i.e., our
model does not consider having to regulate the rates of connections. We
consider congestion control to be separate problem that is beyond the
scope of this book.
We represent the topology of the network with a directed hypergraph
1.3 Network model 11
H= (N,A), where Nis the set of nodes and Ais the set of hyperarcs.
A hypergraph is a generalization of a graph, where, rather than arcs,
we have hyperarcs. A hyperarc is a pair (i, J ), where i, the start node,
is an element of Nand J, the set of end nodes, is a non-empty subset
of N. Each hyperarc (i, J ) represents a broadcast link from node ito
nodes in the non-empty set J. In the special case where Jconsists of
a single element j, we have a point-to-point link. The hyperarc is now
a simple arc and we sometimes write (i, j) instead of (i, {j}). If the
network consists only of point-to-point links (as in wireline network),
then His a graph, denoted alternatively as Grather than H. The link
represented by hyperarc (i, J ) may be lossless or lossy, i.e., it may or
may not be subject to packet erasures.
To establish the desired connection or connections, packets are in-
jected on hyperarcs. Let ziJ be the average rate at which packets are
injected on hyperarc (i, J ). The vector z, consisting of ziJ , (i, J )∈ A,
defines the rate at which packets are injected on all hyperarcs in the
network. In this abstraction, we have not explicitly accounted for any
queues. We assume that queueing occurs at a level that is hidden from
our abstraction and, provided that zlies within some constraint set Z,
all queues in the network are stable. In wireline networks, links are
usually independent, and the constraint set Zdecomposes as the Carte-
sian product of |A| constraints. In wireless networks, links are generally
dependent and the form of Zmay be complicated (see, for example,
[29, 74, 75, 82, 140, 144]). For the time being, we make no assumptions
about Zexcept that it is a convex subset of the positive orthant and
that it contains the origin.
The pair (H, Z) defines a capacitated graph that represents the net-
work at our disposal, which may be a full, physical network or a sub-
network of a physical network. The vector z, then, can be thought of
as a subset of this capacitated graph—it is the portion actually under
use—and we call it the coding subgraph for the desired connection or
connections. We assume that the coding subgraph defines not only the
rates of packet injections on hyperarcs, but also the specific times at
which these injections take place. Thus, the classical networking prob-
lems of routing and scheduling are special subproblems of the problem
of selecting a coding subgraph.
The examples discussed in the previous section give instances of coding
subgraphs—instances where packet injections have been chosen, and the
task that remains is to use them as effectively as possible. Perhaps the
simplest way of representing a coding subgraph in a lossless network
12 Introduction
b1
b3=f(b1, b2)
b2
Fig. 1.7. Coding at a node in a static subgraph. Two packets, b1and b2, are
each carried by one of the incoming arcs. The outgoing arc carries a packet
that is a function of b1and b2.
is to represent each packet transmission over some time period as a
separate hyperarc, as we have done in Figures 1.1–1.4. We may have
parallel hyperarcs as we do in Figure 1.3 (where there are two hyperarcs
(s, {1,2})), representing multiple packets transmitted and received by
the same nodes in a single time period. Coding at a node is shown in
Figure 1.7. We call this representation of a subgraph a static subgraph.
In a static subgraph, time is not represented explicitly, and it appears
as though events occur instantaneously. Presumably in reality, there is
some delay involved in transmitting packets along links, so the output
of packets on a link are delayed from their input. Thus, static subgraphs
hide some timing details that, though not difficult to resolve, must be
kept in mind. Moreover, we must restrict our attention to acyclic graphs,
because cyclic graphs lead to the instantaneous feedback of packets.
Despite its limitations, static subgraphs suffice for much that we wish
to discuss, and they will be used more or less exclusively in Chapter 2,
where we deal with lossless networks.
For lossy networks, the issue of time becomes much more important.
The network codes that are used for lossy networks are generally much
longer than those for lossless networks, i.e., one coding block involves
many more source packets. Looked at another way, the time period that
must be considered for a network code in a lossy network is much longer
than that in a lossless network. Hence it becomes imperative to examine
the interplay of coding and time at a coding node. To do this, we extend
static subgraphs to time-expanded subgraphs.
A time-expanded subgraph represents not only the injection and re-
ception points of packets, but also the times at which these injections
and receptions take place. We draw only successful receptions, hence,
1.4 Outline of book 13
b3=f(b1, b2)
b1b2
time1 2 3
Fig. 1.8. Coding at a node in a time-expanded subgraph. Packet b1is received
at time 1, and packet b2is received at time 2. The thick, horizontal arcs have
infinite capacity, and represent data stored at a node. Thus, at time 3, packets
b1and b2can be used to form packet b3.
in a lossy network, a time-expanded subgraph in fact represents a par-
ticular element in the random ensemble of a coding subgraph. Sup-
pose for example that, in Figure 1.7, packet b1is received at time 1,
packet b2is received at time 2, and packet b3is injected at time 3. In
a time-expanded subgraph, we represent these injections and receptions
as shown in Figure 1.8. In this example, we have used integral values of
time, but real values of time can just as easily be used. We now have
multiple instances of the same node, with each instance representing the
node at a different time. Joining these instances are infinite capacity
links that go forward in time, representing the ability of nodes to store
packets. We use time-expanded subgraphs in Chapter 4, where we deal
with lossy networks.
1.4 Outline of book
We begin, in Chapter 2, with the setting for which network coding theory
was originally developed: multicast in a lossless wireline network. The
prime example illustrating the utility of network coding in this setting
is the butterfly network (Figure 1.1), and, in Chapter 2, we extend the
insight developed from this example to general topologies. We character-
ize the capacity of network coding for the multicast problem and discuss
both deterministic and random code constructions that allow this ca-
pacity to be achieved. In the theoretical discussion, we deal with static
14 Introduction
subgraphs, which, as we have noted, hide the details of time. Thus,
it is not immediately apparent how the theory applies to real packet
networks, where dynamical behavior is usually important. We discuss
packet networks explicitly in Section 2.5 and show how random network
coding can be applied naturally to dynamic settings. The strategy dis-
cussed here is revisited in Chapter 4. We conclude the chapter with two
other extensions of the basic theory for lossless multicast: the case of
networks with cycles (recall the issue of instantaneous feedback in static
subgraphs) and that of correlated source processes.
In Chapter 3, we extend our discussion to non-multicast problems in
lossless wireline networks, i.e., we consider the situation where we wish
to establish multiple connections, and we potentially code packets from
separate connections together, as in the case of the modified butterfly
network (Figure 1.2). We refer to this type of coding as inter-session
coding. This is as opposed to the situation where each connection is kept
separate (as in Chapter 2, where we only consider a single connection),
which we call intra-session coding. We show that linear network coding
is not in general sufficient to achieve capacity and that non-linear codes
may be required. Given the difficulty of constructing codes without a
linear structure, work on constructing inter-session codes has generally
focused on suboptimal approaches that give non-trivial performance im-
provements. We discuss some of these approaches in Chapter 3.
In Chapter 4, we consider lossy networks. We show how random linear
network coding can be applied in lossy networks, such as the packet relay
channel (Figure 1.6), and that it yields a capacity-achieving strategy for
single unicast or multicast connections in such networks. We also derive
an error exponent that quantifies the rate at which the probability of
error decays with coding delay.
Chapters 2–4 all assume that the coding subgraph is already defined.
Chapter 5 considers the problem of choosing an appropriate coding sub-
graph. Two types of approaches are considered: flow-based approaches,
where we assume that the communication objective is to establish con-
nections at certain, given flow rates, and queue-length-based approaches,
where the flow rates, though existent, are not known. We deal primar-
ily with subgraph selection for intra-session coding, though we do also
discuss approaches to subgraph selection for inter-session coding.
In Chapter 6, we consider the problem of security against adversarial
errors. This problem is motivated by the application of network coding
to overlay networks, where not all nodes can necessarily be trusted.
1.5 Notes and further reading 15
Thus, mechanisms are necessary to allow errors introduced by malicious
nodes to be either corrected or detected.
1.5 Notes and further reading
Error-free networks have been considered for some time, and work on
error-free networks includes that of Han [52] and Tsitsiklis [133]. Ahlswede
et al. [4] were the first to consider the problem multicast of an error-free
network. In their work, which had its precursor in earlier work relating
to specific network topologies [146, 120, 152, 151], they showed that cod-
ing at intermediate nodes is in general necessary to achieve the capacity
of a multicast connection in an error-free network and characterized that
capacity. This result generated renewed interest in error-free networks,
and it was quickly strengthened by Li et al. [88] and Koetter and M´edard
[85], who independently showed that linear codes (i.e., codes where nodes
are restricted to performing operations that are linear over some base
finite field) suffice to achieve the capacity of a multicast connection in
an error-free network.
Ho et al. [62] introduced random linear network coding as a method for
multicast in lossless packet networks and analyzed its properties. Ran-
dom linear network coding for multicast in lossless packet networks was
further studied in [27, 66]; it was also studied as a method for data dis-
semination in [34] and as a method for data storage in [1]; in [94, 98], its
application to lossy packet networks was examined. Protocols employ-
ing random linear network coding in peer-to-peer networks and mobile
ad-hoc networks (manets) are described in [50] and [112], respectively.
The butterfly network first appears in [4]; its modified form first ap-
pears in [84, 119]. The wireless butterfly network first appears in [97];
its modified form first appears in [139].
The basic static subgraph model that we use derives principally from
[85]. The use of time-expanded subgraphs first appears in [136].
2
Lossless Multicast Network Coding
Multicast refers to the case where the same information is transmitted
to multiple sink nodes. The first application of network coding to be
discovered was that network coding allows the maximum possible multi-
cast rate to be achieved in a noise-free, or lossless, network. Clearly, this
rate can be no more than the capacity between the source and each sink
individually. As we will see, network coding allows joint use of network
resources by multiple sink nodes, so that any rate possible for all sinks
individually is simultaneously achievable for all sinks together.
2.0.1 Notational conventions
We denote matrices with bold uppercase letters and vectors with bold
lowercase letters. All vectors are row vectors unless indicated otherwise
with a subscript T. We denote by [x,y] the concatenation of two row vec-
tors xand y. For any vector (or matrix) whose entries (rows/columns)
are indexed by the arcs of a network, we assume a consistent ordering of
the vector entries (matrix rows/columns) corresponding to a topological
ordering of the arcs.
2.1 Basic network model and multicast network coding
problem formulation
Leaving aside for a start the complexities of real packet networks, we
consider a very simple network model and problem formulation, widely
used in the network coding literature, to gain some fundamental insights
and develop some basic results. We will later apply these insights and
results to more complex network models.
The basic problem we consider is a single-source multicast on an
16
2.2 Delay-free scalar linear network coding 17
acyclic network. The network is represented by a graph G= (N,A)
where Nis the set of nodes and Ais the set of arcs. There are rsource
processes X1,...,Xroriginating at a given source node s∈ N . Each
source process Xiis a stream of independent random bits of rate one bit
per unit time.
The arcs are directed and lossless, i.e. completely reliable. Each arc
l∈ A can transmit one bit per time unit from its start (origin) node o(l)
to its end (destination) node d(l); there may be multiple arcs connecting
the same pair of nodes. Arc lis called an input arc of d(l) and an
output arc of o(l). We denote by I(v),O(v) the set of input links and
output links respectively of a node v. We refer to the random bitstream
transmitted on an arc las an arc process and denote it by Yl. For each
node v, the input processes of vare the arc processes Ykof input arcs
kof vand, if v=s, the source processes X1,...,Xr. The arc process
on each of v’s output arcs lis a function of one or more of v’s input
processes, which are called the input processes of l; we assume for now
that all of v’s input processes are input processes of l.†
All the source processes must be communicated to each of a given set
T ⊂ N \sof sink nodes. We assume without loss of generality that the
sink nodes do not have any output arcs‡. Each sink node t∈ T forms
output processes Zt,1,...,Zt,r as a function of its input processes. Spec-
ifying a graph G, a source node s∈ N , source rate r, and a set of sink
nodes T ⊂ N \sdefines a multicast network coding problem. A solution
to such a problem defines coding operations at network nodes and decod-
ing operations at sink nodes such that each sink node reproduces the val-
ues of all source processes perfectly, i.e. Zt,i =Xi∀t∈ T , i = 1,...,r,.
A network coding problem for which a solution exists is said to be solv-
able.
2.2 Delay-free scalar linear network coding
We first consider the simplest type of network code, delay-free scalar
linear network coding over a finite field. As we will see, this type of
network code is sufficient to achieve optimal throughput for the acyclic
multicast problem described above, but is not sufficient in general.
Each bitstream corresponding to a source process Xior arc process
†We will drop this assumption in Section 2.5 when we extend the model to packet
networks. In the case without network coding, each arc has only one input process.
‡Given any network, we can construct an equivalent network coding problem sat-
isfying this condition by adding, for each sink node t, a virtual sink node t′and r
links from tto t′.
18 Lossless Multicast Network Coding
Ylis divided into vectors of mbits. Each m-bit vector corresponds to
an element of the finite field Fqof size q= 2m. We can accordingly view
each source or arc process as a vector of finite field symbols instead of
bits.
Since we are considering an acyclic network, we do not have to ex-
plicitly consider arc transmission delays – we simply assume that the
nth symbol for each arc lis transmitted only after o(l) has received
the nth symbol of each of its input processes. This is equivalent, as
far as the analysis is concerned, to assuming that all transmissions hap-
pen instantaneously and simultaneously, hence the term delay-free. This
assumption would lead to stability problems if there is a cycle of depen-
dent arcs, i.e. a directed cycle of arcs each transmitting data that is a
function of data on its predecessor in the cycle, possibly coded together
with other inputs.
In scalar linear network coding, the nth symbol transmitted on an
arc lis a scalar linear function, in Fq, of the nth symbol of each input
process of node o(l), and this function is the same for all n. Thus,
instead of working with processes that are streams of symbols, it suffices
to consider just one symbol for each process. For notational convenience,
in the remainder of this section, Xi,Yland Zt,i refer to a single symbol
of the corresponding source, arc and output processes respectively.
Scalar linear network coding for an arc lcan be represented by the
equation
Yl=X
k∈I(o(l))
fk,lYk+Piai,l Xiif o(l) = s
0 otherwise (2.1)
where ai,l, fk,l are scalar elements from Fq, called (local) coding coeffi-
cients, specifying the coding operation. In the case of multicast with
independent source processes, we will see that it suffices for each sink
node tto form its output symbols as a scalar linear combination of its
input symbols
Zt,i =X
k∈I(t)
bt,i,kYk(2.2)
We denote by aand fthe vectors of coding variables (ai,l : 1 ≤i≤r, l ∈
A) and (fk,l :, l, k ∈ A) respectively, and by bthe vector of decoding
variables (bt,i,k :t∈ T ,1≤i≤r, k ∈ A).
Since all coding operations in the network are scalar linear operations
of the form (2.1), it follows inductively that for each arc l,Ylis a scalar
2.2 Delay-free scalar linear network coding 19
linear function of the source symbols Xi. In equation form,
Yl=
r
X
i=1
ci,lXi(2.3)
where coefficients ci,l ∈Fqare functions of the coding variables (a,f).
The vector
cl= [c1,l . . . cr,l]∈Fr
q,
is called the (global) coding vector of arc l. It specifies the overall map-
ping from the source symbols to Ylresulting from the aggregate effect of
local coding operations at network nodes. For the source’s output arcs
l,cl= [a1,l . . . ar,l]. Since the network is acyclic, we can index the arcs
topologically, i.e. such that at each node all incoming arcs have lower
indexes than all outgoing arcs†, and inductively determine the coding
vectors clin ascending order of lusing (2.1).
Outputs Zt,i are likewise scalar linear operations of the source symbols
Xi. Thus, the mapping from the vector of source symbols x= [X1...Xr]
to the vector of output symbols zt= [Zt,1...Zt,r] at each sink tis
specified by a linear matrix equation
zt=xMt.
Mtis a function of (a,f,b) and can be calculated as the matrix product
Mt=A(I−F)−1BT
t
where
•A= (ai,l) is an r× |A| matrix whose nonzero entries ai,l are the coef-
ficients with which source symbols Xiare linearly combined to form
symbols on the source’s output arcs l(ref Equation (2.1)). Columns
corresponding to all other arcs are all zero. Matrix Acan be viewed
as a transfer matrix from the source symbols to the source’s output
arcs.
•F= (fk,l) is a |A| × |A| matrix whose nonzero entries fk,l are the
coefficients with which node d(k) linearly combines arc symbols Yk
to form symbols on output arcs l(ref Equation (2.1)). fk,l = 0 if
d(k)6=o(l). For n= 1,2,..., the (k, l)th entry of Fngives the
mapping from Ykto Yldue to (n+ 1)-hop (or arc) paths. Since we are
†The topological order is an extension, generally non-unique, of the partial order
defined by the graph.
20 Lossless Multicast Network Coding
considering an acyclic network, Fis nilpotent, i.e. F˜n=0for some ˜n,
and the (k, l)th entry of
(I−F)−1=I+F+F2+...
gives the overall mapping from Ykto Yldue to all possible paths
between arcs kand l. (I−F)−1can thus be considered as a transfer
matrix from each arc to every other arc.
•Bt= (bt,i,k) is an r× |A| matrix. Its nonzero entries bt,i,k are the
coefficients with which sink tlinearly combines symbols Ykon its input
arcs kto form output symbols Zt,i (ref Equation (2.2)). Columns
corresponding to all other arcs are all zero. Btis the transfer matrix
representing the decoding operation at sink t.
The value of (a,f,b), or equivalently, the value of (A,F,Bt:t∈ T ),
specifies a scalar linear network code. Finding a scalar linear solution
is equivalent to finding a value for (a,f,b) such that A(I−F)−1BT
t=
I∀t∈ T , i.e. the source symbols are reproduced exactly at each sink
node.
Defining the matrix C:= A(I−F)−1, we have that the lth column
of Cgives the transpose of the coding vector clfor arc l.
Having developed a mathematical framework for scalar linear network
coding, we proceed in the following to address some basic questions:
•Given a multicast network coding problem, how do we determine the
maximum multicast rate for which the problem is solvable?
•What is the maximum throughput advantage for multicast network
coding over routing?
•Given a solvable multicast network coding problem, how do we con-
struct a solution?
2.3 Solvability and throughput
2.3.1 The unicast case
As a useful step towards characterizing solvability of a multicast prob-
lem, we consider the special case of unicast: communicating at rate r
between a source node sand a single sink node tin our basic network
model. This can be viewed as a degenerate multicast network coding
problem with rsource processes originating at sand a single sink t.
The famous max-flow/min-cut theorem for a point-to-point connec-
tion tells us that the following two conditions are equivalent:
2.3 Solvability and throughput 21
(C1) There exists a flow of rate rbetween the sand t.
(C2) The value of the minimum cut between sand t†is at least r.
The network coding framework provides another equivalent condition
which is of interest to us because it generalizes readily to the multicast
case:
(C3) The determinant of the transfer matrix Mtis nonzero over the
ring of polynomials F2[a,f,b].
Theorem 2.1 Conditions (C1) and (C3) are equivalent.
The proof of this theorem and the next uses the following lemma:
Lemma 2.1 Let fbe a nonzero polynomial in variables x1, x2,...,xn
over F2, and let dbe the maximum degree of fwith respect to any
variable. Then there exist values for x1, x2,...,xnin Fn
2msuch that
f(x1, x2,...,xn)6= 0, for any msuch that 2m> d.
Proof Consider fas a polynomial in x2,...,xnwith coefficients from
F2[x1]. Since the coefficients of fare polynomials of degree at most d
they are not divisible by x2m
1−x1(the roots of which are the elements
of F2m). Thus, there exists an element α∈F2msuch that fis nonzero
when x1=α. The proof is completed by induction on the variables.
Proof of Theorem 2.1: If (C1) holds, we can use the Ford-Fulkerson
algorithm to find rarc-disjoint paths from sto t. This corresponds to
a solution where Mt=I, so (C3) holds. Conversely, if (C3) holds, by
Lemma 2.1, there exists a value for (a,f,b) over a sufficiently large finite
field, such that det(Mt)6= 0. Then ˜
Bt= (MT
t)−1Btsatisfies C˜
BT
t=I,
and (A,F,˜
B) is a solution to the network coding problem, implying
(C1).
2.3.2 The multicast case
The central theorem of multicast network coding states that if a com-
munication rate ris possible between a source node and each sink node
individually, then with network coding it is possible to multicast at rate
†A cut between sand tis a partition of Ninto two sets Q, N \Q, such that s∈Q
and t∈ N \Q. Its value is the number of arcs whose start node is in Qand whose
end node is in N \Q.
22 Lossless Multicast Network Coding
rto all sink nodes simultaneously. An intuitive proof is obtained by
extending the preceding results for the unicast case.
Theorem 2.2 Consider an acyclic delay-free multicast problem where r
source processes originating at source node sare demanded by a set T
of sink nodes. There exists a solution if and only if for each sink node
t∈ T there exists a flow of rate rbetween sand t.
Proof We have the following sequence of equivalent conditions:
∀t∈ T there exists a flow of rate rbetween sand t
⇔ ∀t∈ T the transfer matrix determinant det Mtis nonzero over
the ring of polynomials F2[a,f,b]
⇔Qt∈T det Mtis nonzero over the ring of polynomials F2[a,f,b]
⇔there exists a value for (a,f,b) in a large enough finite field such
that Qt∈T det Mtevaluates to a nonzero value. From this we can
obtain a solution (a,f,b′), since each sink tcan multiply the corre-
sponding vector of output values ztby M−1
tto recover the source
values x.
where the first step follows from applying Theorem 2.1 to each sink and
the last step follows from Lemma 2.1.
Corollary 2.1 The maximum multicast rate is the minimum, over all
sink nodes, of the minimum cut between the source node and each sink
node.
2.3.3 Multicasting from multiple source nodes
The analysis developed for the case of multicasting from one source
node readily generalizes to the case of multicasting from multiple source
nodes to the same set of sink nodes. Consider a multiple-source multicast
problem on a graph (N,A) where each source process Xi, i = 1,...,r,
instead of originating at a common source node, originates at a (possibly
different) source node si∈ N .
One approach is to allow ai,l to take a nonzero value only if Xiorigi-
nates at o(l), i.e. we replace Equation 2.1 with
Yl=X
i:o(l)=si
ai,lXi+X
k∈I(o(l))
fk,lYk(2.4)
An alternative is to convert the multiple-source multicast problem
2.3 Solvability and throughput 23
into an equivalent single-source multicast problem, by adding to Na
virtual source node sfrom which rsource processes originate, and to A
one virtual arc (s, si) for each source process Xi. We can then apply
similar analysis as in the single source node case to obtain the following
multiple source counterpart to Theorem 2.2.
Theorem 2.3 Consider an acyclic delay-free multicast problem on a
graph G= (N,A)with rsource processes Xi, i = 1,...,r originating at
source nodes si∈ N respectively, demanded by a set Tof sink nodes.
There exists a solution if and only if for each sink node t∈ T and each
subset S ⊂ {si:i= 1,...,r}of source nodes, the max flow/min cut
between Sand tis greater than or equal to |S|.
Proof Let G′be the graph obtained by adding to Na virtual source node
sand to Aone virtual arc (s, si) for each source process Xi. We apply
Theorem 2.2 to the equivalent single-source multicast problem on G′.
For any cut Qsuch that s∈Q, t ∈ N \Q, let S(Q) = Q∩ {si: 1,...,r}
be the subset of actual source nodes in Q. The condition that the value
of the cut Qin G′is at least ris equivalent to the condition that the
value of the cut S(Q) in Gis at least |S(Q)|, since there are r− |S (Q)|
virtual arcs crossing the cut Qfrom sto the actual source nodes not in
Q.
2.3.4 Maximum throughput advantage
The multicast throughput advantage of network coding over routing for
a given network graph (N,A) with arc capacities z= (zl:l∈ A),
source node s∈ N and sink nodes T ⊂ N \sis defined as the ratio of
the multicast capacity with network coding to the multicast capacity
without network coding. The capacity with network coding is given
by the max flow min cut condition, from Corollary 2.1. The capacity
without network coding is equal to the fractional Steiner tree packing
number, which is given by the following linear program:
max
uX
k∈K
uk
subject to X
k∈K:l∈k
uk≤zl∀l∈ A
uk≥0∀k∈ K
(2.5)
24 Lossless Multicast Network Coding
where Kis the set of all possible Steiner trees in the network, and ukis
the flow rate on tree k∈ K.
It is shown in [3] that for a given directed network, the maximum mul-
ticast throughput advantage of network coding over routing is equal to
the integrality gap of a linear programming relaxation for the minimum
weight directed Steiner tree. Specifically, consider a network (N,A, s, T)
with arc weights w= (wl:l∈ A). The minimum weight Steiner tree
problem can be formulated as an integer program
min
aX
l∈A
wlal
subject to X
l∈Γ+(Q)
al≥1∀Q∈ C
al∈ {0,1} ∀ l∈ A
(2.6)
where C:= {Q⊂ N :s∈Q, T 6⊂ Q}denotes the set of all cuts between
the source and at least one sink, Γ+(Q) := {(i, j ) : i∈Q, j /∈Q}
denotes the set of forward arcs of a cut Q, and alis the indicator function
specifying whether arc lis in the Steiner tree. This integer program
has an integer relaxation obtained by replacing the integer constraint
al∈ {0,1}with the linear constraint 0 ≤al≤1.
Theorem 2.4 For a given network (N,A, s, T),
max
z≥0
Mc(N,A, s, T,z)
Mr(N,A, s, T,z)= max
w≥0
WIP (N,A, s, T,w)
WLP (N,A, s, T,w)
where Mc(N,A, s, T,z)and Mr(N,A, s, T,z)denote the multicast ca-
pacity with and without network coding respectively, under arc capacities
z, and WIP (N,A, s, T,w)and WLP (N,A, s, T,w)denote the optimum
of the integer program (2.6) and its linear relaxation respectively.
Determining the maximum value of the integrality gap, maxw≥0WIP (N,A,s,T,w)
WLP (N,A,s,T,w),
is a long-standing open problem in computer science. From a known
lower bound on this integrality gap, we know that the multicast through-
put advantage for coding can be Ω((log n/ log log n)2) for a network with
nsink nodes. For undirected networks, there is a similar correspondence
between the maximum multicast throughput advantage for coding and
the integrality gap of the bidirected cut relaxation for the undirected
Steiner tree problem. Interested readers can refer to [3] for details.
2.4 Multicast network code construction 25
2.4 Multicast network code construction
Next, we address the question: given a solvable multicast network cod-
ing problem, how do we construct a solution? Note that here we are
considering a single multicast session (i.e. all the sink nodes demand the
same information) on a given graph whose arcs have their entire capacity
dedicated to supporting that multicast session. When there are multiple
sessions sharing a network, one possible approach for intra-session net-
work coding is to first allocate to each session a subset of the network,
called a subgraph, and then apply the techniques described in this sec-
tion to construct a code for each session on its allocated subgraph. The
issue of subgraph selection is covered in Chapter 5.
2.4.1 Centralized polynomial-time construction
Consider a solvable multicast network coding problem on an acyclic net-
work with rsource processes and dsink nodes. The following centralized
algorithm constructs, in polynomial time, a solution over a finite field
Fq, where q≥d.
The algorithm’s main components and ideas are as follows:
•The algorithm first finds rarc-disjoint paths Pt,1,...,Pt,r from the
source sto each sink t∈ T . Let A′⊂ A be the set of arcs in the union
of these paths. By Theorem 2.2, the subgraph consisting of arcs in A′
suffices to support the desired multicast communication, so the coding
coefficients for all other arcs can be set to zero.
•The algorithm sets the coding coefficients of arcs in A′in topological
order, maintaining the following invariant: for each sink t, the coding
vectors of the arcs in the set Stform a basis for Fr
q, where Stcomprises
the arc from each of the paths Pt,1,...,Pt,r whose coding coefficients
were set most recently. The invariant is initially established by adding
a virtual source node s′and rvirtual arcs from s′to sthat have
linearly independent coding vectors [0i−1,1,0r−i], i = 1,...,r, where
0jdenotes the length-jall zeros row vector. The invariant ensures
that at termination, each sink has rlinearly independent inputs.
•To facilitate efficient selection of the coding coefficients, the algorithm
maintains, for each sink tand arc l∈ St, a vector dt(l) satisfying the
condition
dt(l)·ck=δl,k ∀l, k ∈ St
26 Lossless Multicast Network Coding
where
δl,k := 1l=k
0l6=k.
By this condition, dt(l) is orthogonal to the subspace spanned by
the coding vectors of the arcs in Stexcluding l. Note that a vector
v∈Fr
qis linearly independent of vectors {ck:k6=l, k ∈ St}if and
only if v·dt(l)6= 0. This can be seen by expressing vin the basis
corresponding to {ck:k∈ St}as v=Pk∈Stbkck, and noting that
v·dt(l) = X
k∈St
bkck·dt(l) = bl.
•For an arc lon a path Pt,i , the arc immediately preceding lon Pt,i is
denoted pt(l), and the set of sinks tfor which an arc lis in some path
Pt,i is denoted T(l). To satisfy the invariant, the coding coefficients
of each arc lare chosen such that the resulting coding vector clis
linearly independent of the coding vectors of all arcs in St\{pt(l)}for
all t∈ T (l), or equivalently, such that
cl·dt(l)6= 0 ∀t∈ T (l).(2.7)
This can be done by repeatedly choosing random coding coefficients
until the condition (2.7) is met. Alternatively, this can be done deter-
ministically by applying Lemma 2.2 below to the set of vector pairs
{(cpt(l),dt(pt(l))) : t∈ T (l)}.
Lemma 2.2 Let n≤q. For a set of pairs {(xi,yi)∈Fr
q×Fr
q: 1 ≤i≤
n}such that x·yi6= 0 ∀i, we can, in O(n2r)time, find a vector un
that is a linear combination of x1,...,xn, such that un·yi6= 0 ∀i.
Proof This is done by the following inductive procedure, which con-
structs vectors u1,...,unsuch that ui·yl6= 0 ∀1≤l≤i≤n. Set
u1:= x1. Let Hbe a set of ndistinct elements of Fq. For i= 1,...,n−1,
if ui·yi+1 6= 0, set ui+1 := ui; otherwise set ui+1 := αui+xi+1 where
αis any element in
H\ {−(xi+1 ·yl)/(ui·yl) : l≤i},
which is nonempty since |H| > i. This ensures that
ui+1 ·yl=αui·yl+xi+1 ·yl6= 0 ∀l≤i.
Each dot product involves length-rvectors and is found in O(r) time,
2.4 Multicast network code construction 27
each uiis found in O(nr) time, and u1,...,unis found in O(n2r) time.
The full network code construction algorithm is given in Algorithm 1.
It is straightforward to verify that ck·dt(l) = δk,l ∀k, l ∈ S′
tat the end
Algorithm 1: Centralized polynomial-time algorithm for multicast
linear network code construction
Input:N,A, s, T, r
N:= N ∪ {s′}
A:= A ∪ {l1,...,lr}where o(li) = s′, d(li) = sfor i= 1,...,r
Find rarc-disjoint paths Pt,1,...,Pt,r from s′to each sink t∈ T ;
Choose field size q= 2m≥ |T |
foreach i= 1,...,r do cli:= [0i−1,1,0r−i]
foreach t∈ T do
St:= {l1,...,lr}
foreach l∈ Stdo dt(l) := cl
foreach k∈ A\{l1,...,lr}in topological order do
choose, by repeated random trials or by the procedure of
Lemma 2.2, ck=Pk′∈P(k)fk′,kck′such that ckis linearly
independent of {ck′:k′∈ St, k′6=pt(k)}for each t∈ T (k)
foreach t∈ T (k)do
S′
t:= {k} ∪ St\pt(k)
d′
t(k) := (ck·dt(pt(k)))−1dt(pt(k))
foreach k′∈ St\pt(k)do
Ad′
t(k′) := dt(k′)−(ck·dt(k′))d′
t(k)
(St,dt) := (S′
t,d′
t)
return f
of step A.
Theorem 2.5 For a solvable multicast network coding problem on an
acyclic network with rsource processes and dsink nodes, Algorithm 1
deterministically constructs a solution in O(|A| dr(r+d)) time.
Proof The full proof is given in [72].
Corollary 2.2 A finite field of size q≥dis sufficient for a multicast
network coding problem with dsink nodes on an acyclic network.
28 Lossless Multicast Network Coding
4
t2
s
t1
1
3
2
X1, X2
ξ1X1+ξ2X2
ξ5(ξ1X1+ξ2X2)
+ξ6(ξ3X1+ξ4X2)
ξ3X1+ξ4X2
Fig. 2.1. An example of random linear network coding. X1and X2are the
source processes being multicast to the receivers, and the coefficients ξiare
randomly chosen elements of a finite field. The label on each arc represents
the process being transmitted on the arc. Reprinted with permission from
[56].
For the case of two source processes, a tighter bound of q≥p2d−7/4+
1/2 is shown in [46] using a coloring approach.
2.4.2 Random linear network coding
A simple approach that finds a solution with high probability is to choose
coding coefficients (a,f) independently at random from a sufficiently
large finite field. The value of (a,f) determines the value of the coding
vector Ylfor each network arc l(which equals the lth column of C=
A(I−F)−1).
It is not always necessary to do random coding on every arc. For
instance, in our lossless network model, a node with a single input can
employ simple forwarding, as in Figure 2.1. Or if we have found r
disjoint paths from the source to each sink as in the algorithm of the
previous section, we can restrict coding to occur only on arcs where two
or more paths to different sinks merge. We will bound the probability
2.4 Multicast network code construction 29
that random network coding on ηarcs yields a solution to a feasible
multicast problem.
Recall from the proof of Theorem 2.2 that for a solvable multicast
problem, the product of transfer matrix determinants Qt∈T det(A(I−
F)−1BT
t) is nonzero over the ring of polynomials F2[a,f,b]. Since the
only nonzero rows of BT
tare those corresponding to input arcs of sink t,
A(I−F)−1BT
tis nonsingular only if thas a set It⊂ I (t) of rinput arcs
with linearly independent coding vectors, or equivalently, the submatrix
CItformed by the rcolumns of Ccorresponding to Itis nonsingular.
Then each sink tcan decode by setting the corresponding submatrix
of Bt(whose columns correspond to arcs in It) to C−1
It, which gives
Mt=I.
To obtain a lower bound on the probability that random coding yields
a solution, we assume that for each sink tthe set Itis fixed in advance
and other inputs, if any, are not used for decoding. A solution then
corresponds to a value for (a,f) such that
ψ(a,f) = Y
t∈T
det CIt(2.8)
is nonzero. The Schwartz-Zippel theorem (e.g., [105]) states that for any
nonzero polynomial in F2[x1,...,xn], choosing the values of variables
x1,...,xnindependently and uniformly at random from F2mresults in
a nonzero value for the polynomial with probability at least 1 −d/2m,
where dis the total degree of the polynomial. To apply this theorem
to the polynomial ψ(a,f), we need a bound on its total degree; we
can obtain a tighter bound by additionally bounding the degree of each
variable.
These degree bounds can be obtained from the next lemma, which
expresses the determinant of the transfer matrix Mt=A(I−F)−1BT
t
in a more transparent form, in terms of the related matrix
Nt=A 0
I−F BT
t.(2.9)
Lemma 2.3 For an acyclic delay-free network, the determinant of the
transfer matrix Mt=A(I−F)−1BT
tfor receiver tis equal to
det Mt= (−1)r(|A|+1) det Nt.
30 Lossless Multicast Network Coding
Proof Note that
I−A(I−F)−1
0 I A 0
I−F BT
t=0−A(I−F)−1BT
t
I−F BT
t
Since I−A(I−F)−1
0 I has determinant 1,
det A 0
I−F BT
t = det 0−A(I−F)−1BT
t
I−F BT
t
= (−1)r|A|det −A(I−F)−1BT
t0
BT
tI−F
= (−1)r|A|det(−A(I−F)−1BT
t)det(I−F)
= (−1)r(|A|+1)det(A(I−F)−1BT
t)det(I−F)
The result follows from observing that det(I−F) = 1 since Fis upper-
triangular with zeros along the main diagonal.
This lemma can be viewed as a generalization of a classical result
linking (uncoded) network flow and bipartite matching. The problem of
checking the feasibility of an s−tflow of size ron graph G= (N,A)
can be reduced to a bipartite matching problem by constructing the
following bipartite graph: one node set of the bipartite graph has r
nodes u1,...,ur, and a node vl,1corresponding to each arc l∈ A; the
other node set of the bipartite graph has rnodes w1,...,wr, and a node
vl,2corresponding to each arc l∈ A. The bipartite graph has
•an arc joining each node uito each node vl,1such that o(l) = s
(corresponding to an output link of source s)
•an arc joining node vl,1to the corresponding node vl,2for all l∈ A,
•an arc joining node vl,2to vj,1for each pair (l, j )∈ A × A such that
d(l) = o(j) (corresponding to incident links), and
•an arc joining each node wito each node vl,2such that d(l) = t
(corresponding to input links of sink t.
The s−tflow is feasible if and only if the bipartite graph has a per-
fect matching. The matrix Ntdefined in Equation (equation:N) can
be viewed as a network coding generalization of the Edmonds matrix
(see e.g., [105]) used for checking if the bipartite graph has a perfect
matching.
Since each coding coefficient appears in only one entry of Nt, we can
2.4 Multicast network code construction 31
easily obtain degree bounds for det Ntusing the complete expansion of
the determinant, as shown in the following lemma.
Lemma 2.4 Consider a random network code in which ηarcs lhave
associated coding coefficients ai,l and/or fk,l that are randomly chosen.
The determinant of Nthas maximum degree ηin the random variables
{ai,l, fk,l }, and is linear in each of these variables.
Proof Note that the variables ai,l, fk,l corresponding to an arc leach
appear once, in column lof Nt. Thus, only the ηcolumns corresponding
to arcs with associated random coefficients contain variable terms. The
determinant of Ntcan be written as the sum of products of r+|A|
entries, each from a different column (and row). Each such product
is linear in each variable ai,l, fk,l, and has degree at most ηin these
variables.
Noting that det CItequals det Mtfor some bt†, and using Lemmas 2.3
and 2.4, we have that ψ(a,f) (defined in (2.8)) has total degree at most
dη in the randomly chosen coding coefficients and each coding coefficient
has degree at most d, where ηis the number of arcs with randomly chosen
coding coefficients and dis the number of sink nodes.
Theorem 2.6 Consider a multicast problem with dsink nodes, and a
network code in which some or all of the coding coefficients (a,f)are
chosen uniformly at random from a finite field Fqwhere q > d, and the
remaining coding coefficients, if any, are fixed. If there exists a solution
to the multicast problem with these fixed coding coefficients, then the
probability that the random network code yields a solution is at least
(1 −d/q)η, where ηis the number of arcs lwith associated random
coding coefficients ai,l, fk,l .
Proof See Appendix 2.A.
The bound of Theorem 2.6 is a worst-case bound applying across all
networks with dsink nodes and ηlinks with associated random coding
coefficients. For many networks, the actual probability of obtaining a
solution is much higher. Tighter bounds can be obtained by considering
additional aspects of network structure. For example, having more re-
dundant capacity in the network increases the probability that a random
linear code will be valid.
†when Btis the identity mapping from arcs in Itto outputs at t
32 Lossless Multicast Network Coding
In the rest of this chapter, we will extend our basic network model
and lossless multicast problem in a few ways: from static source and arc
processes to time-varying packet networks, from acyclic networks to net-
works with cycles, and from independent to correlated source processes.
2.5 Packet networks
The algebraic description of scalar linear network coding in Section 2.2,
developed for the idealized static network model of Section 2.1, is readily
adapted to the case of transmission of a finite batch (generation) of
packets over a time-varying network where each packet can potentially
undergo different routing and coding operations.
Let the source message be composed of a batch of rexogenous source
packets. A packet transmitted by a node vis formed as a linear combi-
nation of one or more constituent packets, which may be source packets
originating at vor packets received previously by v. For a multicast
problem, the objective is to reproduce, at each sink, the rsource pack-
ets.
For scalar linear network coding in a field Fq, the bits in each packet
are grouped into vectors of length mwhich are viewed as symbols from
Fq,q= 2m. We thus consider each packet as a vector of symbols from
Fq; we refer to such a vector as a packet vector.
We can think of source packets and transmitted packets as analogous
to source processes and arc processes respectively in the static network
model. The kth symbol of a transmitted packet is a scalar linear function
of the kth symbol of each of its constituent packets, and this function is
the same for all k. This is analogous to the formation of an arc process
Ylas a linear combination of one or more of the input processes of node
o(l) in the static model.
For a given sequence Sof packet transmissions, we can consider a
corresponding static network Gwith the same node set and with arcs
corresponding to transmissions in S, where for each packet ptransmitted
from node vto win S,Ghas one unit-capacity arc ˜pfrom vto w. The
causality condition that each packet ptransmitted by a node vis a
linear combination of only those packets received by vearlier in the
sequence translates into a corresponding restriction in Gon the subset
of v’s inputs that can be inputs of ˜p. This departs from our previous
assumption in Section 2.2 that each of node v’s inputs is an input of each
of v’s outgoing arcs. The restriction that arc kis not an input of arc lis
equivalent to setting the coding coefficient fk,l to zero. Such restrictions
2.5 Packet networks 33
are conveniently specified using a line graph: the line graph G′of Ghas
one node wlfor every arc lof G, and contains the arc (wk, wl) if wkis
an input of wl.
2.5.1 Distributed random linear coding for packet networks
2.5.1.1 Coding vector approach
The random linear network coding approach of Section 2.4.2 can form
the basis of a practical, distributed multicast technique for time-varying
packet networks. Applying this approach to the packet network model,
each packet transmitted by a node vis an independent random linear
combination of previously received packets and source packets gener-
ated at v. The coefficients of these linear combinations are chosen with
the uniform distribution from the finite field Fq, and the same linear
operation is applied to each symbol in a packet.
In a distributed setting, network nodes independently choose random
coding coefficients, which determine the network code and the corre-
sponding decoding functions at the sinks. Fortunately, a sink node does
not need to know all these coefficients in order to know what decoding
function to use. It suffices for the sink to know the overall linear trans-
formation from the source packets to the packets it has received. As
in the static model, the overall linear transformation from the source
packets to a packet pis called the (global) coding vector of p.
There is a convenient way to convey this information to the sinks,
which is analogous to using a pilot tone or finding an impulse response.
For a batch of source packets with indexes i= 1,2,...,r, we add to the
header of the ith source packet its coding vector, which is the length
runit vector [0 ...010...0] with a single nonzero entry in the ith
position. For each packet formed subsequently by a coding operation,
the same coding operation is applied to each symbol of the coding vector
as to the data symbols of the packet. Thus, each packet’s header contains
the coding vector of that packet.
A sink node can decode the whole batch when it has received rlinearly
independent packets. Their coding vectors form the rows of the transfer
matrix from the source packets to the received packets. The transfor-
mation corresponding to the inverse of this matrix can be applied to the
received packets to recover the original source packets. Decoding can
alternatively be done incrementally using Gaussian elimination.
Note that each coding vector is rlog qbits long, where qis the coding
34 Lossless Multicast Network Coding
field size. The proportional overhead of including the coding vector in
each packet decreases with the amount of data in each packet, so for
large packets this overhead is relatively small. For small packets, this
overhead can be reduced by decreasing the field size qor batch size r(by
dividing a large batch of source packets into smaller batches, and only
allowing coding among packets of the same batch). Decreasing qand
ralso reduces decoding complexity. However, the smaller the field size
the higher the probability that more transmissions are required, since
there is higher probability of randomly picking linearly dependent trans-
missions. Also, reducing batch size reduces our ability to code across
bursty variations in source rates or arc capacities, resulting in reduced
throughput if packets near the batch boundaries have to be transmitted
without coding. An illustration is given in Figure 2.2. In this example,
a source node sis multicasting to sink nodes yand z. All the arcs have
average capacity 1, except for the four labeled arcs which have average
capacity 2. In the optimal solution, arc (w, x) should transmit coded
information for both receivers at every opportunity. However, variabil-
ity in the instantaneous capacities of arcs (u, w), (u, y), (t, w) and (t, z)
can cause the number of sink ypackets in a batch arriving at node w
to differ from the number of sink zpackets of that batch arriving at w,
resulting in some throughput loss.
Because of such trade-offs, appropriate values for qand rwill depend
on the type of network and application. The effect of these parameter
choices, and performance in general, are also dependent on the choice
of subgraph (transmission opportunities) for each batch.†The effects
of such parameters are investigated by Chou et al. [26] under a particu-
lar distributed policy for determining when a node switches to sending
packets of the next batch.
2.5.1.2 Vector space approach
A more general approach for distributed random linear coding encodes
a batch of information in the choice of the vector space spanned by the
source packet vectors.
Specifically, let x1,...,xrdenote the source packet vectors, which are
length-nrow vectors of symbols from Fq. We denote by Xthe r×n
matrix whose ith row is xi. Consider a sink node t, and let Ytbe the
matrix whose rows are given by t’s received packet vectors. Xand Yt
†Subgraph selection is the topic of Chapter 5.
2.5 Packet networks 35
s
u
t
z
y
w x
2
2
2
2
Fig. 2.2. An example illustrating throughput loss caused by restricting coding
to occur only among packets of a batch. Reprinted with permission from [58].
are linearly related by a matrix equation
Y=GtX‡
In a random network code, Gtis determined by the random coding
coefficients of network nodes. The vector space approach is based on the
observation that for any value of Gt, the row space of Yis a subspace of
the row space of X. If the sink receives rlinearly independent packets,
it recovers the row space of X.
Let P(Fn
q) denote the set of all subspaces of Fn
q, i.e. the projective
geometry of Fn
q. In this approach, a code corresponds to a nonempty
subset of P(Fn
q), and each codeword is a subspace of Fn
q. A codeword is
transmitted as a batch of packets; the packet vectors of the source pack-
ets in the batch form a generating set for the corresponding subspace
or its orthogonal complement. It is natural to consider codes whose
codewords all have the same dimension r.†Note that the coding vector
approach of the previous section is a special case where the code con-
sists of all subspaces with generator matrices of the form [U|I], where
U∈Fr×(n−r)
qand Iis the r×ridentity matrix (corresponding to the
coding vectors). Since only a subset of all r-dimensional subspaces of Fn
q
correspond to codewords in the coding vector approach, the number of
‡Gtis analogous to the transpose of matrix CI(t)defined in Section 2.4.2 for the
static network model.
†Such codes can be described as particular vertices of a Grassmann graph/q-
Johnson scheme. Details are given in [83].
36 Lossless Multicast Network Coding
codewords and hence the code rate is lower than in the vector space ap-
proach, though the difference becomes asymptotically negligible as the
packet length ngrows relative to the batch size r.
An important motivation for the vector space approach is its appli-
cation to correction of errors and erasures in networks. We discuss this
briefly in Section 6.2.2.2.
2.6 Networks with cycles and convolutional network coding
In our basic network model, which is acyclic, a simple delay-free net-
work coding approach (ref Section 2.2) can be used. Many networks of
interest contain cycles, but can be operated in a delay-free fashion by
imposing restrictions on the network coding subgraph to prevent cyclic
dependencies among arcs. For instance, we can restrict network coding
to occur over an acyclic subgraph of the network line graph (defined in
Section 2.5). Another type of restriction is temporal, as in the finite
batch packet model of the previous section: if we index the transmitted
packets according to their creation time, each transmitted packet has a
higher index than the constituent packets that formed it, so there are no
cyclic dependencies among packets. This can be viewed conceptually as
expanding the network in time. In general, a cyclic graph with vnodes
and rate rcan be converted to a time-expanded acyclic graph with κv
nodes and rate at least (κ−v)r; communication on this expanded graph
can be emulated in κtime steps on the original cyclic graph.
For some network problems, such as those in Figure 2.3, the opti-
mal rate cannot be achieved over any acyclic subgraph of the network
line graph. In this example, to multicast both sources simultaneously
to both sinks, information must be continuously injected into the cycle
(the square in the middle of the network) from both sources. Convert-
ing the network to a time-expanded acyclic graph gives a time-varying
solution that asymptotically achieves the optimal rate, but at the ex-
pense of increasing delay and decoding complexity. An alternative for
such as networks is to take an approach akin to convolutional coding,
where delays are explicitly considered, and information from different
time steps is linearly combined. This approach, termed convolutional
network coding, enables the optimal rate to be achieved with a time-
invariant solution.
2.6 Networks with cycles and convolutional network coding 37
s1
t1
s2
t2
Fig. 2.3. An example of a multicast problem in which the optimal rate cannot
be achieved over any acyclic subgraph of the network line graph. Each arc has
a constant rate of one packet per unit time. Reprinted with permission from
[64].
2.6.1 Algebraic representation of convolutional network
coding
Convolutional network codes can be cast in a mathematical framework
similar to that of Section 2.2 for delay-free scalar network codes, by rep-
resenting the random processes algebraically in terms of a delay operator
variable Dwhich represents a unit time delay or shift: if
Xi(D) =
∞
X
τ=0
Xi(τ)Dτ
Yl(D) =
∞
X
τ=0
Yτ(l)Dτ, Yl(0) = 0
Zt,i(D) =
∞
X
τ=0
Zt,i(τ)Dτ, Zt,i(0) = 0.
38 Lossless Multicast Network Coding
The results for delay-free scalar linear network coding carry over to this
model by replacing the finite field Fqwith the field Fq(D) of rational
functions in the delay variable D. Analogously to the delay-free case,
the transfer matrix from source processes to sink output processes can
be calculated as the matrix product
Mt=A(D)(I−F(D))−1Bt(D)T
where A(D) = (ai,l(D)),F(D) = (fk,l (D)),Bt(D) = (bt,i,k (D)) are ma-
trices whose entries are elements of Fq(D).
A simple type of convolutional network code uses a network model
where each arc has fixed unit delay; arcs with longer delay can be mod-
eled as arcs in series. At time τ+ 1, each non-sink node vreceives
symbols Yk(τ) on its input arcs kand/or source symbols Xi(τ) if v=si,
and linearly combines them to form symbols Yl(τ+ 1) on its output arcs
l. The corresponding coding operation at an arc lat time τis similar
to Equation 2.4 but with time delay:
Yl(τ+ 1) = X
{i:si=o(l)}
ai,lXi(τ)
+X
{k:d(k)=o(l)}
fk,lYk(τ)
which can be expressed in terms of Das
Yl(D) = X
{i:si=o(l)}
Dai,lXi(D)
+X
{k:d(k)=o(l)}
Dfk,l Yk(D)
In this case, the coding coefficients at non-sink nodes are given by
ai,l(D) = Dai,l , fk,l (D) = Dfk,l. By considering D= 0, we can see
that the matrix I−F(D) is invertible. In a synchronous setting, this
does not require memory at non-sink nodes (though in practical settings
where arc delays are variable, some buffering is needed since the (τ+1)st
symbol of each output arc is transmitted only after reception of the τth
symbol of each input). The sink nodes, on the other hand, require mem-
ory: the decoding coefficients bt,i,k(D) are, in general, rational functions
of D, which corresponds to the use of past received and decoded symbols
2.6 Networks with cycles and convolutional network coding 39
for decoding. The corresponding equations are
Zt,i(τ+ 1) =
µ
X
u=0
b′
t,i(u)Zt,i(τ−u)
+X
{k:d(k)=t}
µ
X
u=0
b′′
t,i,k(u)Yk(τ−u)
and
Zt,i(D) = X
{k:d(k)=t}
bt,i,k(D)Yk(D),
where
bt,i,k(D) = Pµ
u=0 Du+1b′′
t,i,k(u)
1−Pµ
u=0 Du+1b′
t,i(u).(2.10)
The amount of memory required, µ, depends on the structure of the
network. A rational function is realizable if it is defined when D= 0†,
and a matrix of rational entries is realizable if all its entries are realizable.
By similar arguments as in the acyclic delay-free case, we can extend
Theorem 2.2 to the case with cycles.
Theorem 2.7 Consider a multicast problem where rsource processes
originating at source node sare demanded by a set Tof sink nodes.
There exists a solution if and only if for each sink node t∈ T there
exists a flow of rate rbetween sand t.
Proof The proof is similar to that of Theorem 2.2, but with a change of
field. Consider the simple unit arc delay model and network code. We
have the following equivalent conditions:
∀t∈ T there exists a flow of rate rbetween sand t
⇔ ∀t∈ T the transfer matrix determinant det Mtis a nonzero ratio
of polynomials from the ring F2(D)[a,f,b′,b′′]
⇔Qt∈T det Mtis a nonzero ratio of polynomials from the ring
F2(D)[a,f,b′,b′′]
⇔there exists a value for (a,f,b′,b′′) over F2m, for sufficiently large
m, such that Qt∈T det Mtis nonzero in F2m(D)
⇔there exist realizable matrices Bt(D) such that Mt=DuI∀t∈ T
for some sufficiently large decoding delay u.
†i.e. For a realizable rational function, the denominator polynomial in lowest terms
has a nonzero constant coefficient
40 Lossless Multicast Network Coding
More generally, it is not necessary to consider delays on every arc. To
ensure stability and causality of information flow, we only need every
directed cycle in the network to contain at least one delay element. Fur-
thermore, the delays can be associated with nodes instead of links: an
alternative model for convolutional network coding considers delay-free
links and associates all delays with coding coefficients at network nodes.
With this model, we can work in the binary field F2; having delay or
memory at nodes corresponds to coding coefficients that are polynomi-
als in F2[D]. For acyclic networks, such codes can be constructed in
polynomial time using an approach analogous to that in Section 2.4.1,
where the invariant becomes: for each sink t, the coding vectors of the
arcs in the set Stspan F2[D]r. For each arc, the coding coefficients
can be chosen from a set of d+ 1 values, where dis the number of
sinks. The block network codes we have considered in previous sections
achieve capacity for acyclic networks, but in some cases convolutional
network codes can have lower delay and memory requirements. One rea-
son is that for block codes each coding coefficient is from the same field,
whereas for convolutional network codes the amount of delay/memory
can be different across coding coefficients. The case of cyclic networks
is more complicated since there is no well-defined topological order in
which to set the coding coefficients. An algorithm described in [?] up-
dates the coding coefficients associated with each sink’s subgraph in turn
(each sink’s subgraph consists of rarc-disjoint paths and the associated
coding coefficients can be updated in topological order).
2.7 Correlated source processes
In our basic network model, the source processes are independent. In this
section we consider correlated, or jointly distributed, source processes.
Such correlation can be exploited to improve transmission efficiency.
The problem of lossless multicasting from correlated sources is a gen-
eralization of the classical distributed source coding problem of Slepian
and Wolf, where correlated sources are separately encoded but jointly
decoded. The classical Slepian-Wolf problem corresponds to the special
case of a network consisting of one direct arc from each of two source
nodes to a common receiver. In the network version of the problem, the
sources are multicast over a network of intermediate nodes that can per-
form network coding. It turns out that a form of random linear network
2.7 Correlated source processes 41
coding with nonlinear decoding is asymptotically rate-optimal. This can
be viewed as a network coding generalization of a classical random lin-
ear coding approach for the Slepian-Wolf problem, and as an instance
of joint source-network coding.
2.7.1 Joint source-network coding
For simplicity, we consider two sources†X1, X2with respective rates r1
and r2bits per unit time. The source bits at Xiare grouped into vectors
of ribits which we refer to as symbols. The two sources’ consecutive
pairs of output symbols are drawn i.i.d. from the same joint distribution
Q.
We employ a vector linear network code that operates on blocks of bits
corresponding to nsymbols from each source. Specifically, linear coding
is done in F2over blocks consisting of nribits from each source Xi.‡Let
ckbe the capacity of arc k. For each block, each node vtransmits, on
each of its output arcs k,nckbits formed as random linear combinations
of input bits (source bits originating at vand bits received on input arcs).
This is illustrated in Figure 2.4. x1∈Fnr1
2and x2∈Fnr2
2are vectors
of source bits being multicast to the receivers, and the matrices Υiare
matrices of random bits. Suppose the capacity of each arc is c. Matrices
Υ1and Υ3are nr1×nc, Υ2and Υ4are nr2×nc, and Υ5and Υ6are
nc ×nc.
To decode, sink maps its block of received bits to a block of decoded
values that has minimum entropy or maximum Q-probability among all
possible source values consistent with the received values.
We give a lower bound on the probability of decoding error at a sink
for Let m1and m2be the minimum cut capacities between the receiver
and each of the sources respectively, and let m3be the minimum cut
capacity between the receiver and both sources. We denote by Lthe
maximum source-receiver path length. The type Pxof a vector x∈F˜n
2is
the distribution on F2defined by the relative frequencies of the elements
of F2in x, and joint types Pxy are analogously defined.
Theorem 2.8 The error probability of the random linear network code
†We use the term “source” in place of “source process” for brevity.
‡The approach can be extended to coding over larger finite fields.
42 Lossless Multicast Network Coding
4
t2
s
t1
1
3
2
x1, x2
Υ1x1+ Υ2x2
Υ5(Υ1x1+ Υ2x2)
+Υ6(Υ3x1+ Υ4x2)
Υ3x1+ Υ4x2
Fig. 2.4. An example illustrating vector linear coding. The label on each arc
represents the process being transmitted on the arc. Reprinted with permis-
sion from [56].
is at most P3
i=1 pi
e, where
p1
e≤exp (−nmin
X,Y D(PXY ||Q)
+
m1(1 −1
nlog L)−H(X|Y)
+!
+22r1+r2log(n+ 1))
p2
e≤exp (−nmin
X,Y D(PXY ||Q)
+
m2(1 −1
nlog L)−H(Y|X)
+!
+2r1+2r2log(n+ 1))
p3
e≤exp (−nmin
X,Y D(PXY ||Q)
+
m3(1 −1
nlog L)−H(XY )
+!
+22r1+2r2log(n+ 1))
2.7 Correlated source processes 43
and X, Y are dummy random variables with joint distribution PXY .
Proof See Appendix 2.A.
The error exponents
e1= min
X,Y D(PXY ||Q)
+
m1(1 −1
nlog L)−H(X|Y)
+!
e2= min
X,Y D(PXY ||Q)
+
m2(1 −1
nlog L)−H(Y|X)
+!
e3= min
X,Y D(PXY ||Q)
+
m3(1 −1
nlog L)−H(XY )
+!,
for general networks reduce to those for the Slepian-Wolf network where
L= 1, m1=R1, m2=R2, m3=R1+R2:
e1= min
X,Y D(PXY ||Q) + |R1−H(X|Y)|+
e2= min
X,Y D(PXY ||Q) + |R2−H(Y|X)|+
e3= min
X,Y D(PXY ||Q) + |R1+R2−H(X Y )|+.
2.7.2 Separation of source coding and network coding
A separate source coding and network coding scheme first performs
source coding to describe each source as a compressed collection of bits,
and then uses network coding to losslessly transmit to each sink a subset
of the resulting bits. Each sink first decodes the network code to recover
the transmitted bits, and then decodes the source code to recover the
original sources. Such an approach allows use of existing low complexity
source codes. However, separate source and network coding is in general
not optimal. This is shown by an example in Figure 2.5 which is for-
mally presented in [114]; we give here an informal description providing
44 Lossless Multicast Network Coding
t1
t2
s1
s2
s3
ǫ
Fig. 2.5. An example in which separate source coding and network coding is
suboptimal. Each arc has capacity 1 + ǫ, except the arc from s3to t1which
has capacity ǫ, where 0 < ǫ < 1. Reprinted with permission from [114].
intuition for the result. Suppose source s1is independent of sources s2
and s3, while the latter two are highly correlated, and all three sources
have entropy 1. In the limit as ǫgoes to 0 and sources s2and s3are
invertible functions of each other, we have essentially the equivalent of
the modified butterfly network in Figure 2.6, where sources s2and s3in
Figure 2.5 together play the role of source s2in Figure 2.6. The problem
is thus solvable with joint source-network coding. It is not however solv-
able with separate source and network coding– sink t2needs to do joint
source and network decoding based on the correlation between sources
s2and s3.
2.8 Notes and further reading
The field of network coding has its origins in the work of Yeung et
al. [151], Ahlswede et al. [4] and Li et al. [88]. The famous butterfly
network and the max flow min cut bound for network coded multicast
were given in Ahlswede et al. [4]. Li et al. [88] showed that linear coding
with finite symbol size is sufficient for multicast connections. Koetter
2.8 Notes and further reading 45
s2
t2
t1
s1
1 2
b1⊕b2
b1
b1⊕b2
b1⊕b2
b1
b2
b2
Fig. 2.6. The modified butterfly network. In this network, every arc has ca-
pacity 1.
and M´edard [85] developed the algebraic framework for linear network
coding used in this chapter.
The connection between multicast throughput advantage and inte-
grality gap of the minimum weight Steiner tree problem was given by
Agarwal and Charikar [3], and extended to the case of average through-
put by Chekuri et al. [23].
Concurrent independent work by Sanders et al. [123] and Jaggi et
al. [68] developed the centralized polynomial-time algorithm for acyclic
networks presented in this chapter. The coding vector approach for dis-
tributed random linear network coding was introduced in Ho et al. [62].
The network coding generalization of the Edmonds matrix was given in
Ho et al. [55, 57] and used in Harvey et al. [54], which presented a deter-
ministic construction for multicast codes based on matrix completion.
A practical batch network coding protocol based on random network
coding was presented in Chou et al. [27]. The vector space approach for
distributed random network coding was proposed by Koetter and Kschis-
chang [83]. A gossip protocol using random linear network coding was
presented in Deb and M´edard [34]. Fragouli and Soljanin [47] developed
a code construction technique based on information flow decomposition.
A number of works have considered the characteristics of network
codes needed for achieving capacity on different types of multicast net-
work problems. Lower bounds on coding field size were presented by
Rasala Lehman and Lehman [116] and Feder et al. [43]. Upper bounds
were given by Jaggi et al. [73] (acyclic networks), Ho et al. [66] (general
46 Lossless Multicast Network Coding
networks), Feder et al. [43] (graph-specific) and Fragouli et al. [47] (two
sources).
Convolutional network coding was first discussed in Ahlswede et al. [4].
The algebraic convolutional network coding approach presented in this
chapter is from Koetter and M´edard [85]. Various aspects of convo-
lutional network coding, including constructive coding and decoding
techniques, are addressed in Erez and Feder [40, 41], Fragouli and Sol-
janin [45], and Li and Yeung [87]. The linear network coding approach
for correlated sources is an extension by Ho et al. [65] of the linear cod-
ing approach in Csisz´ar [30] for the Slepian-Wolf problem. Separation
of source coding and network coding is addressed in [114].
2.A Appendix: Random network coding
Lemma 2.5 Let Pbe a nonzero polynomial in F[ξ1, ξ2,...]of degree less
than or equal to dη, in which the largest exponent of any variable ξiis
at most d. Values for ξ1, ξ2, . . . are chosen independently and uniformly
at random from Fq⊆F. The probability that Pequals zero is at most
1−(1 −d/q)ηfor d < q.
Proof For any variable ξ1in P, let d1be the largest exponent of ξ1in
P. Express Pin the form P=ξd1
1P1+R1, where P1is a polynomial of
degree at most dη −d1that does not contain variable ξ1, and R1is a
polynomial in which the largest exponent of ξ1is less than d1. By the
Principle of Deferred Decisions (e.g., [105]), the probability Pr[P= 0]
is unaffected if we set the value of ξ1last after all the other coefficients
have been set. If, for some choice of the other coefficients, P16= 0, then
Pbecomes a polynomial in F[ξ1] of degree d1. By the Schwartz-Zippel
Theoremhim, this probability Pr[P= 0|P16= 0] is upper bounded by
d1/q. So
Pr[P= 0] ≤Pr[P16= 0] d1
q+ Pr[P1= 0]
= Pr[P1= 0] 1−d1
q+d1
q.(2.11)
Next we consider Pr[P1= 0], choosing any variable ξ2in P1and letting
d2be the largest exponent of ξ2in P1. We express P1in the form
P1=ξd2
2P2+R2, where P2is a polynomial of degree at most dη −d1−d2
that does not contain variables ξ1or ξ2, and R2is a polynomial in which
2.A Appendix: Random network coding 47
the largest exponent of ξ2is less than d2. Proceeding similarly, we assign
variables ξiand define diand Pifor i= 3,4,... until we reach i=k
where Pkis a constant and Pr[Pk= 0] = 0. Note that 1 ≤di≤d < q ∀i
and Pk
i=1 di≤dη, so k≤dη. Applying Schwartz-Zippel as before, we
have for k′= 1,2,...,k
P r[Pk′= 0] ≤P r[Pk′+1 = 0] 1−dk′+1
q+dk′+1
q.(2.12)
Combining all the inequalities recursively, we can show by induction that
Pr[P= 0] ≤Pk
i=1 di
q−Pi6=ldidl
q2
+···+ (−1)k−1Qk
i=1 di
qk.
Now consider the integer optimization problem
Maximize f=Pdη
i=1 di
q−Pi6=ldidl
q2+
···+ (−1)dη−1Qdη
i=1 di
qdη
subject to 0 ≤di≤d < q ∀i∈[1, dη],
dη
X
i=1
di≤dη, and diinteger (2.13)
whose maximum is an upper bound on Pr[P= 0].
We first consider the problem obtained by relaxing the integer condition
on the variables di. Let d∗={d∗
1,...,d∗
dη}be an optimal solution.
For any set Shof hdistinct integers from [1, dη], let fSh= 1 −Pi∈Shdi
q+
Pi,l∈Sh,i6=ldidl
q2− · · · + (−1)hQi∈Shdi
qh. We can show by induction on h
that 0 < fSh<1 for any set Shof hdistinct integers in [1, dη]. If
Pdη
i=1 d∗
i< dη, then there is some d∗
i< d, and there exists a feasible
solution d={d1,...,ddη }such that di=d∗
i+ǫ,ǫ > 0, and dh=d∗
hfor
h6=i, which satisfies
f(d)−f(d∗)
=ǫ
q1−Ph6=id∗
h
q+···+ (−1)dη−1Qh6=id∗
h
qdη−1.
This is positive, contradicting the optimality of d∗, so Pdη
i=1 d∗
i=dη.
Next suppose 0 < d∗
i< d for some d∗
i. Then there exists some d∗
lsuch
48 Lossless Multicast Network Coding
that 0 < d∗
l< d, since if d∗
l= 0 or dfor all other l, then Pdη
i=1 d∗
i6=dη.
Assume without loss of generality that 0 < d∗
i≤d∗
l< d. Then there
exists a feasible vector d={d1,...,ddη }such that di=d∗
i−ǫ,dl=d∗
l+ǫ,
ǫ > 0, and dh=d∗
h∀h6=i, l, which satisfies
f(d)−f(d∗) = −(d∗
i−d∗
l)ǫ−ǫ2
q2
1−Ph6=i,l d∗
h
q− · · · + (−1)dη−2Qh6=i,land d∗
h
qdη−2.
This is again positive, contradicting the optimality of d∗.
Thus, Pdη
i=1 d∗
i=dη, and d∗
i= 0 or d. So exactly ηof the variables d∗
i
are equal to d. Since the optimal solution is an integer solution, it is
also optimal for the integer program (2.13). The corresponding optimal
f=ηd
q−η
2d2
q2+···+ (−1)η−1dη
qη= 1 −1−d
qη.
Proof of Theorem 2.8: We consider transmission, by random lin-
ear network coding, of one block of source bits, represented by vector
[x1,x2]∈Fn(r1+r2)
2. The transfer matrix CI(t)specifies the map-
ping from the vector of source bits [ x1,x2] to the vector zof bits on
the set I(t) of terminal arcs incident to the receiver.
The decoder maps a vector zof received bits onto a vector [ ˜
x1,˜
x2]∈
Fn(r1+r2)
2minimizing α(Px1x2) subject to [ x1,x2]CI(t)=z. For a
minimum entropy decoder, α(Px1x2)≡H(Px1x2), while for a maximum
Q-probability decoder, α(Px1x2)≡ − log Qn(x1x2). We consider three
types of errors: in the first type, the decoder has the correct value for x2
but outputs the wrong value for x2; in the second, the decoder has the
correct value for x1but outputs the wrong value for x2; in the third, the
decoder outputs wrong values for both x1and x2. The error probability
is upper bounded by the sum of the probabilities of the three types of
errors, P3
i=1 pi
e.
(Joint) types of sequences are considered as (joint) distributions PX
(PX,Y , etc.) of dummy variables X, Y , etc. The set of different types of
sequences in Fk
2is denoted by P(Fk
2). Defining the sets of types
Pi
n=
{PX˜
XY ˜
Y∈P(Fnr1
2×Fnr1
2×Fnr2
2×Fnr2
2)|
˜
X6=X, ˜
Y=Y}i= 1
{PX˜
XY ˜
Y∈P(Fnr1
2×Fnr1
2×Fnr2
2×Fnr2
2)|
˜
X=X, ˜
Y6=Y}i= 2
{PX˜
XY ˜
Y∈P(Fnr1
2×Fnr1
2×Fnr2
2×Fnr2
2)|
˜
X6=X, ˜
Y6=Y}i= 3
2.A Appendix: Random network coding 49
sequences
TXY ={[x1,x2]∈Fn(r1+r2)
2|
Px1x2=PXY }
T˜
X˜
Y|XY (x1x2) = {[ ˜
x1,˜
x2]∈Fn(r1+r2)
2|
P˜
x1˜
x2x1x2=P˜
X˜
Y XY }
we have
p1
e≤X
PX˜
XY ˜
Y∈P1
n:
α(P˜
XY )≤α(PXY )X
(x1,x2)∈
TXY
Qn(x1x2) Pr ∃(˜
x1,˜
x2)∈
T˜
X˜
Y|XY (x1x2) s.t.[ x1−˜
x1,0]CI(t)=0!
≤X
PX˜
XY ˜
Y∈P1
n:
α(P˜
XY)≤α(PXY )X
(x1,x2)∈TXY
Qn(x1x2)
min (X
(˜
x1,˜
x2)∈
T˜
X˜
Y|XY (x1x2)
Pr [x1−˜
x1,0]CI(t)=0,1)
Similarly,
p2
e≤X
PX˜
XY ˜
Y∈P2
n:
α(PX˜
Y)≤α(PXY )X
(x1,x2)∈TXY
Qn(x1x2)
min (X
(˜
x1,˜
x2)∈
T˜
X˜
Y|XY (x1x2)
Pr [0,x2−˜
x2]CI(t)=0,1)
p3
e≤X
PX˜
XY ˜
Y∈P3
n:
α(P˜
X˜
Y)≤α(PXY )X
(x1,x2)∈TXY
Qn(x1x2)
min (X
(˜
x1,˜
x2)∈
T˜
X˜
Y|XY (x1x2)
Pr [x1−˜
x1,x2−˜
x2]CI(t)=0,1)
where the probabilities are taken over realizations of the network transfer
matrix CI(t)corresponding to the random network code. The probabil-
50 Lossless Multicast Network Coding
ities
P1= Pr [x1−˜
x1,0]CI(t)=0
P2= Pr [0,x2−˜
x2]CI(t)=0
P3= Pr [x1−˜
x1,x2−˜
x2]CI(t)=0
for nonzero x1−˜
x1,x2−˜
x2can be calculated for a given network, or
bounded in terms of nand parameters of the network as we will show
later.
We can apply some simple cardinality bounds
|P1
n|<(n+ 1)22r1+r2
|P2
n|<(n+ 1)2r1+2r2
|P3
n|<(n+ 1)22r1+2r2
|TXY | ≤ exp{nH (X Y )}
|T˜
X˜
Y|XY (x1x2)| ≤ exp{nH (˜
X˜
Y|XY )}
and the identity
Qn(x1x2) = exp{−n(D(PXY ||Q) + H(X Y ))},
(x1,x2)∈TXY (2.14)
to obtain
p1
e≤exp (−nmin
PX˜
XY ˜
Y∈P1
n
:
α(P˜
XY)≤α(PXY )D(PXY ||Q) +
−1
nlog P1−H(˜
X|XY )
++ 22r1+r2log(n+ 1))
p2
e≤exp (−nmin
PX˜
XY ˜
Y
∈P2
n
:
α(PX˜
Y)≤
α(PXY )
D(PXY ||Q) +
−1
nlog P2−H(˜
Y|XY )
++ 2r1+2r2log(n+ 1))
p3
e≤exp (−nmin
PX˜
XY ˜
Y
∈P3
n
:
α(P˜
X˜
Y)≤
α(PXY )
D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y|XY )
++ 22r1+2r2log(n+ 1)),
2.A Appendix: Random network coding 51
where the exponents and logs are taken with respect to base 2.
For the minimum entropy decoder, we have
α(P˜
X˜
Y)≤α(PXY )⇒
H(˜
X|XY )≤H(˜
X|Y)≤H(X|Y) for Y=˜
Y
H(˜
Y|XY )≤H(˜
Y|X)≤H(Y|X) for X=˜
X
H(˜
X˜
Y|XY )≤H(˜
X˜
Y)≤H(XY )
which gives
p1
e≤exp (−nmin
XY D(PX Y ||Q) +
−1
nlog P1−H(X|Y)
+!
+22r1+r2log(n+ 1))(2.15)
p2
e≤exp (−nmin
XY D(PX Y ||Q) +
−1
nlog P2−H(Y|X)
+!
+2r1+2r2log(n+ 1))(2.16)
p3
e≤exp (−nmin
XY D(PX Y ||Q) +
−1
nlog P3−H(XY )
+!
+22r1+2r2log(n+ 1)).(2.17)
We next show that these bounds also hold for the maximum Q-
probability decoder, for which, from (2.14), we have
α(P˜
X˜
Y)≤α(PXY )
⇒D(P˜
X˜
Y||Q) + H(˜
X˜
Y)≤D(PXY ||Q) + H(X Y ).(2.18)
For i= 1, ˜
Y=Y, and (2.18) gives
D(P˜
XY ||Q) + H(˜
X|Y)≤D(PXY ||Q) + H(X|Y).(2.19)
52 Lossless Multicast Network Coding
We show that
min
PX˜
XY ˜
Y∈P1
n:
α(P˜
X˜
Y)≤
α(PXY )
D(PXY ||Q) +
−1
nlog P1−H(˜
X|XY )
+!
≥min
PX˜
XY ˜
Y∈P1
n:
α(P˜
X˜
Y)≤
α(PXY )
D(PXY ||Q) +
−1
nlog P1−H(˜
X|Y)
+!
≥min
XY D(PX Y ||Q) +
−1
nlog P1−H(X|Y)
+!
by considering two possible cases for any X, ˜
X, Y satisfying (2.19):
Case 1: −1
nlog P1−H(X|Y)<0. Then
D(PXY ||Q) +
−1
nlog P1−H(˜
X|Y)
+
≥D(PXY ||Q) +
−1
nlog P1−H(X|Y)
+
≥min
XY D(PX Y ||Q) +
−1
nlog P1−H(X|Y)
+!
Case 2: −1
nlog P1−H(X|Y)≥0. Then
D(PXY ||Q) +
−1
nlog P1−H(˜
X|Y)
+
≥D(PXY ||Q) + −1
nlog P1−H(˜
X|Y)
≥D(P˜
XY||Q) + −1
nlog P1−H(X|Y)by (2.19)
=D(P˜
XY ||Q) +
−1
nlog P1−H(X|Y)
+
2.A Appendix: Random network coding 53
which gives
D(PXY ||Q) +
−1
nlog P1−H(˜
X|Y)
+
≥1
2"D(PXY ||Q) +
−1
nlog P1−H(˜
X|Y)
+
+D(P˜
XY ||Q) +
−1
nlog P1−H(X|Y)
+#
≥1
2"D(PXY ||Q) +
−1
nlog P1−H(X|Y)
+
+D(P˜
XY ||Q) +
−1
nlog P1−H(˜
X|Y)
+#
≥min
XY D(PX Y ||Q) +
−1
nlog P1−H(X|Y)
+!.
A similar proof holds for i= 2.
For i= 3, we show that
min
PX˜
XY ˜
Y∈P3
n:
α(P˜
X˜
Y)≤
α(PXY )
D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y|XY )
+!
≥min
PX˜
XY ˜
Y∈P3
n:
α(P˜
X˜
Y)≤
α(PXY )
D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y)
+!
≥min
XY D(PX Y ||Q) +
−1
nlog P3−H(XY )
+!
by considering two possible cases for any X, ˜
X, Y , ˜
Ysatisfying (2.18):
Case 1: −1
nlog P3−H(XY )<0. Then
D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y)
+
≥D(PXY ||Q) +
−1
nlog P3−H(XY )
+
≥min
XY D(PX Y ||Q) +
−1
nlog P3−H(XY )
+!
54 Lossless Multicast Network Coding
Case 2: −1
nlog P3−H(XY )≥0. Then
D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y)
+
≥D(PXY ||Q) + −1
nlog P3−H(˜
X˜
Y)
≥D(P˜
X˜
Y||Q) + −1
nlog P3−H(XY )by (2.18)
=D(P˜
X˜
Y||Q) +
−1
nlog P3−H(XY )
+
which gives
D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y)
+
≥1
2"D(PXY ||Q) +
−1
nlog P3−H(˜
X˜
Y)
+
+D(P˜
X˜
Y||Q) +
−1
nlog P3−H(XY )
+#
≥min
XY D(PX Y ||Q) +
−1
nlog P3−H(XY )
+!.
We bound the probabilities Piin terms of nand the network param-
eters mi, i = 1,2,the minimum cut capacity between the receiver and
source Xi,m3, the minimum cut capacity between the receiver and both
sources, and L, the maximum source-receiver path length.
Let G1and G2be subgraphs of graph Gconsisting of all arcs down-
stream of sources 1 and 2 respectively, where an arc kis considered
downstream of a source Xiif si=o(k) or if there is a directed path
from the source to o(k). Let G3be equal to G.
Note that in a random linear network code, any arc kwhich has at
least one nonzero input transmits the zero process with probability 1
2nck,
where ckis the capacity of k. Since the code is linear, this probability is
the same as the probability that a pair of distinct values for the inputs
of kare mapped to the same output value on k.
For a given pair of distinct source values, let Ekbe the event that
the corresponding inputs to arc kare distinct, but the corresponding
values on kare the same. Let E(˜
G) be the event that Ekoccurs for
some arc kon every source-receiver path in graph ˜
G.Piis then equal to
the probability of event E(Gi).
2.A Appendix: Random network coding 55
Let G′
i, i = 1,2,3 be the graph consisting of minode-disjoint paths,
each consisting of Larcs each of unit capacity. We show by induction
on mithat Piis upper bounded by the probability of event E(G′
i).
We let ˜
Gbe the graphs Gi,G′
i, i = 1,2,3 in turn, and consider any
particular source-receiver path P˜
Gin ˜
G. We distinguish two cases:
Case 1: Ekdoes not occur for any of the arcs kon the path P˜
G. In
this case the event E(˜
G) occurs with probability 0.
Case 2: There exists some arc ˆ
kon the path P˜
Gfor which Ekoccurs.
Thus, we have Pr(E(˜
G)) = Pr(case 2) Pr(E(˜
G)|case 2). Since PG′
i
has at least as many arcs as PGi, Pr(case 2 for G′
i)≥Pr(case 2 for Gi).
Therefore, if we can show that Pr(E(G′
i)|case 2) ≥Pr(E(Gi)|case 2), the
induction hypothesis Pr(E(G′
i)) ≥Pr(E(Gi)) follows.
For mi= 1, the hypothesis is true since Pr(E(G′
i)|case 2) = 1. For
mi>1, in case 2, removing the arc ˆ
kleaves, for G′
i, the effective equiv-
alent of a graph consisting of mi−1 node-disjoint length-Lpaths, and,
for Gi, a graph of minimum cut at least mi−1. The result follows from
applying the induction hypothesis to the resulting graphs.
Thus, Pr(E(G′
i)) gives an upper bound on probability Pi:
Pi≤1−(1 −1
2n)Lmi
≤L
2nmi
.
Substituting this into the error bounds (2.15)-(2.17) gives the desired
result.
3
Inter-Session Network Coding
So far, we have considered network coding for a single communication
session, i.e. unicast communication to one sink node or multicast of com-
mon information to multiple sink nodes. This type of coding is called
intra-session network coding, since we only code together information
symbols that will be decoded by the same set of sink nodes. For intra-
session network coding, it suffices for each node to form its outputs
as random linear combinations of its inputs. Each sink node can de-
code once it has received enough independent linear combinations of the
source processes.
When there are multiple sessions sharing the network, a simple prac-
tical approach is to allocate disjoint subsets of the network capacity
to each session. If each session’s allocated subgraph satisfies the max-
flow/min-cut condition for each sink (Theorems 2.2 and 2.7), we can
obtain a solution with intra-session network coding among information
symbols of each session separately. Sections 5.1.1 and 5.2.1 discuss such
an approach.
In general, however, achieving optimal rates may require inter-session
network coding, i.e. coding among information symbols of different ses-
sions. Inter-session network coding is more complicated than intra-
session network coding. Coding must be done strategically in order
to ensure that each sink can decode its desired source processes – nodes
cannot simply combine all their inputs randomly, since the sink nodes
may not have sufficient incoming capacity to decode all the randomly
combined source processes. Unlike intra-session network coding, decod-
ing may have to be done at non-sink nodes. We will see an example
further on, in Section 3.5.1.
At present, for general multi-session network problems, it is not yet
known how to determine feasibility or construct optimal network codes.
56
3.1 Scalar and vector linear network coding 57
In this chapter, we first discuss some theoretical approaches and results.
We then describe constructions and implementations of suboptimal but
practical inter-session network codes.
3.1 Scalar and vector linear network coding
Scalar linear network coding was described in Section 2.2 for a single
multicast session. In the general case with multiple sessions, each sink
t∈ T can demand an arbitrary subset
Dt⊂ {1,2,...,r}(3.1)
of the information sources. Scalar linear network coding for the general
case is defined similarly, the only difference being that each sink needs
only decode its demanded subset of information sources. We can gener-
alize the scalar linear solvability criterion as follows. In the single session
case, the criterion is that the transfer matrix determinant det Mtfor
each sink node t, as a function of coding coefficients (a,f,b), is nonzero
– this corresponds to each sink node being able to decode all the source
processes. In the inter-session case, the criterion is that there exists a
value for coefficients (a,f,b) such that
(i) the submatrix of Mtconsisting of rows whose indexes are in Dt
is nonsingular
(ii) the remaining rows of Mtare all zero.
This corresponds to each sink node being able to extract its demanded
source processes while removing the effect of other interfering (unwanted)
source processes.
The problem of determining whether a general network problem has a
scalar linear solution has been shown (by reduction from 3-CNF) to be
NP-hard. It can be reduced to the problem of determining whether a re-
lated algebraic variety is empty, as follows. Let m1(a,f,b),...,mK(a,f,b)
denote all the entries of Mt, t ∈ T that must evaluate to zero accord-
ing to condition (ii). Let d1(a,f,b),...,dL(a,f,b) denote the deter-
minants of the submatrices that must be nonzero according to condi-
tion (i). Let ξbe a new variable, and let Ibe the ideal generated
by m1(a,f,b),...,mK(a,f,b),1−ξQL
i=1 di(a,f,b). Then the decod-
ing conditions are equivalent to the condition that the variety associ-
ated with the ideal Iis nonempty. This can be decided by computing
a Gr¨obner basis for I, which is not polynomial in complexity but for
which standard mathematical software exists.
58 Inter-Session Network Coding
Unlike the single multicast session case, scalar linear network coding
is not optimal in the general case. Scalar coding is time-invariant. The
approach outlined above for determining scalar linear solvability does not
cover time-sharing among scalar solutions. Figure 3.1 gives an example
network problem which can be solved by time-sharing among different
routing strategies, but not by scalar linear network coding.
The class of vector linear network codes includes both scalar linear
network coding and time-sharing as special cases. In vector linear net-
work coding, the bitstream corresponding to each source and arc process
is divided into vectors of finite field symbols; the vector associated with
an arc is a linear function, specified by a matrix, of the vectors associated
with its inputs. Vector linear network coding was used in Section 2.7 for
multicasting from correlated sources.
3.2 Fractional coding problem formulation
In the basic model and problem formulation of Section 2.1, a solution is
defined with respect to fixed source rates and arc capacities (all assumed
to be equal); for a given network problem and class of coding/routing
strategies, a solution either exists or does not. A more flexible approach,
useful for comparing various different classes of strategies in the multiple
session case, is to define a network problem in terms of source/sink
locations and demands, and to ask what rates are possible relative to
the arc capacities.
The most general characterization of the rates achievable with some
class of strategies is a rate region which gives the trade-offs between the
rates of different sources. A simpler characterization, which suffices for
the purpose of comparing different classes of network codes, assumes that
the source rates are equal to each other and that the arc rates are equal
to each other, and asks what is the maximum ratio between the rate of a
source and the rate of an arc. Specifically, we consider sending a vector
of ksymbols from each source with a vector of nsymbols transmitted
on each arc. The symbols are from some alphabet (in the codes we have
considered so far, the alphabet is a finite field, but we can consider more
general alphabets such as rings). Such a network problem is defined
by a graph (N,A), source nodes si∈ N , i = 1,...,r, a set T ⊂ N of
sink nodes, and the sets Dt⊂ {1,...,r}of source processes demanded
by each sink t∈ T . For brevity, we will in the following refer to such
a network problem simply as a network. A (k, n)fractional solution
defines coding operations at network nodes and decoding operations at
3.2 Fractional coding problem formulation 59
A′, B′
3 4
2
5
7 86 9
A, A′B, B′
A, B A, B′A′, B
1
(a) An example network problem which can be solved by
time-sharing among different routing strategies, but not by
scalar linear network coding.
B′
1, B2
3 4
2
5
7 86 9
A1, A2, B1, B2A′
1, A′
2, B1, B2
A2, B1A2, B′
2A′
1, B1A′
1, B′
2
A1, A2, B′
1, B′
2A′
1, A′
2, B′
1, B′
2
B1, B2, B′
1, B′
2
A1, A2, A′
1, A′
2
A′
1, A2
B1, B′
2
A1, A′
2
1
(b) A time-sharing routing solution.
Fig. 3.1.
sink nodes such that each sink perfectly reproduces the values of its
demanded source processes. A solution, as in Section 2.1, corresponds
to the case where k=n. A scalar solution is a special case where
k=n= 1. The coding capacity of a network with respect to an alphabet
60 Inter-Session Network Coding
Aand a class Cof network codes is
sup{k/n :∃a (k, n) fractional coding solution in Cover A}
3.3 Insufficiency of linear network coding
Linear network coding is not sufficient in general for the case of multiple
sessions. This was shown by an example network Pwhich has a nonlin-
ear coding capacity of 1, while there is no linear solution. The network
Pand its nonlinear solution are shown in Figure 3.2. The class of lin-
ear codes for which Phas no solution is more general than the class of
vector linear codes over finite fields described in Section 3.1. It includes
linear network codes where the source and arc processes are associated
with elements of any finite R-module Gwith more than one element,
for any ring R. (A ring is a generalization of a field, the difference being
that elements of a ring do not need to have multiplicative inverses. An
R-module is a generalization of a vector space using a ring Rin place of
a field.)
The construction of Pis based on connections with matroid theory.
By identifying matroid elements with source and arc processes in a net-
work, a (non-unique) matroidal network can be constructed from a given
matroid, such that dependencies and independencies of the matroid are
reflected in the network. Circuits (minimal dependent sets) of the ma-
troid are reflected in the dependence of output arc processes of a node
(or decoded output processes of a sink node) on the input processes of
the node. Bases (maximal independent sets) of the matroid correspond
to the set of all source processes, or the input processes of sink nodes
that demand all the source processes. Properties of the matroid thus
carry over to its associated matroidal network(s).
The network Pis based on two matroidal networks P1and P2(shown
in Figures 3.4 and 3.6) associated with the well-known Fano and non-
Fano matroids respectively (shown in Figures 3.3 and 3.5. In the case of
vector linear coding, P1has no vector linear solution of any dimension
over a finite field with odd characteristic, while P2has no vector linear
solution of any dimension over a finite field with characteristic 2.†This
†In the more general case of R-linear coding over G, vector linear solvability corre-
sponds to scalar linear solvability over a larger module. It can be shown that if P
has a scalar linear solution, then it has a scalar linear solution for the case where
Racts faithfully on Gand is a ring with an identity element I. In this case, P1
has no scalar R-linear solution over Gif I+I6= 0, while P2has no scalar R-linear
solution over Gif I+I= 0.
3.3 Insufficiency of linear network coding 61
d+e
d
1211104 5 6 7 8 9
123
a b c
13 14 15 16
17 18 19 20
21 22 23 24 25 26 27 28
29 30 31 32 33 34 35 36
43
37 38 39 40 41 42 45 4644
c c
c
cba a e db
a⊕b
a⊕b⊕c
a⊕c
b⊕c
a+b+c
a+b
a+c
b+c
t(c) + d+e
t(c) + d
t(c) + e
e
Fig. 3.2. An example network problem whose nonlinear coding capacity is
greater than its linear coding capacity. The arc labels give a nonlinear solution
over an alphabet of size 4. Symbols + and −denote addition and subtraction
in the ring Z4of integers modulo 4, ⊕denotes bitwise xor, and tdenotes the
operation of reversing the order of the bits in a 2-bit binary string.
incompatibility is exploited to construct the example network which does
not have a linear solution, but has a nonlinear solution over an alphabet
of size 4, shown in Figure 3.2.
For the case of vector linear codes over finite fields, it can be shown,
using information theoretic arguments described in the next section, that
the linear coding capacity of this example is 10/11. The proof is given
in [36]. Thus, vector linear network coding over finite fields is not even
asymptotically optimal.
62 Inter-Session Network Coding
ˆz
ˆy
ˆxˆw
ˆa
ˆc
ˆ
b
Fig. 3.3. A geometric representation of the Fano matroid. The labeled dots
correspond to the elements of the underlying set, any three elements of which
are dependent if and only if in the diagram they are collinear or on the circle.
3.4 Information theoretic approaches
Determining the coding capacity of an arbitrary network is an open prob-
lem, but progress in characterizing/bounding capacity in various cases
has been made using information theoretic entropy arguments. The in-
formation theoretic approach represents the source and arc processes as
random variables (or sequences of random variables). The entropies of
the source random variables can be identified with (or lower bounded
by) the source rates, while the entropies of the arc random variables
can be upper bounded by the arc capacities. Other constraint relations
involving the joint/conditional entropies of various subsets of these ran-
dom variables can be derived from the coding dependencies and decoding
requirements.
In the following we denote by S={1,...,r}the set of information
sources, and by I(v),O(v)∈ A,S(v)⊂ S the set of input arcs, output
arcs and information sources respectively of a node v. As before, Dt⊂ S
denotes the set of information sources demanded by a sink t.
In the case of an acyclic network, suppose there exists a (k , n) frac-
tional network coding solution over an alphabet B. Let the ith source
process, 1 ≤i≤r, be represented by a random vector Xiconsisting of
kindependent random variables distributed uniformly over B. Denote
by Yjthe corresponding random vector transmitted on each arc j∈ A
under the fractional network code. We use the abbreviated notation
YA′={Yj:j∈ A′}for a set of arcs A′⊂ A, and XS′={Xi:i∈ S ′}
for a set of sources S′⊂ S. Then we have the following entropy condi-
3.4 Information theoretic approaches 63
z
2 3
4 5
6 7
8 9
10 11
12 13 14
c ab
a b c
w y
x
1
Fig. 3.4. A network associated with the Fano matroid. Source processes a, b, c
originate at nodes 1,2,3 respectively, and are demanded by the sink nodes 14,
13, 12 respectively. The source and edge labels indicate the correspondence
with the elements of the Fano matroid as shown in Figure 3.3. The network
has no vector linear solution of any dimension over a finite field with odd
characteristic.
tions:
H(Xi) = k(3.2)
H(XS) = rk (3.3)
H(Yj)≤n∀j∈ A (3.4)
H(YO(v)|XS(v), YI(v)) = 0 ∀v∈ N (3.5)
H(XDt|YI(t)) = 0 ∀t∈ T (3.6)
Equations (3.2)-(3.3) state that the source vectors Xieach have entropy
k(in units scaled by the log of the alphabet size) and are indepen-
dent. Inequality (3.4) upper bounds the entropy of each arc vector Yj
by the arc capacity n. Equation (3.5) expresses the condition that each
64 Inter-Session Network Coding
ˆz
ˆc
ˆ
b
ˆa
ˆy
ˆxˆw
Fig. 3.5. A geometric representation of the non-Fano matroid. The labeled
dots correspond to the elements of the underlying set, any three elements of
which are dependent if and only if in the diagram they are collinear.
node’s outputs are a deterministic function of its inputs. Equation (3.6)
expresses the requirement that each sink can reproduce its demanded
sources as a deterministic function of its inputs.
Given equations (3.2)-(3.6) for a particular network problem, informa-
tion inequalities can be applied with the aim of simplifying the equations
and obtaining a bound on the ratio k/n. Information inequalities are in-
equalities involving information measures (entropy, conditional entropy,
mutual information and conditional mutual information) of subsets of a
set of random variables, that hold under any joint distribution for the
random variables. Intuitively, information inequalities are constraints
which must be satisfied by the values of these information measures in
order for the values to be consistent with some joint distribution. For
any set Nof discrete random variables, and any subsets U,U′,U′′ of N,
we have the following basic inequalities:
H(U)≥0
H(U|U′)≥0
I(U;U′)≥0
I(U;U′|U′′)≥0
The basic inequalities and all inequalities implied by the basic inequali-
ties are called Shannon-type information inequalities. For four or more
random variables, there exist inequalities not implied by the basic in-
equalities, called non-Shannon-type information inequalities. An exam-
3.4 Information theoretic approaches 65
a, b, c
2 3
4
5
678
9 10 11
15
12 13 14
c b a
a b c
z
w x y
1
Fig. 3.6. A network associated with the non-Fano matroid. Source processes
a, b, c originate at nodes 1,2,3 respectively, and are demanded by the sets
of sink nodes {14,15},{13,15},{12,15}respectively. The source and edge
labels indicate the correspondence with the elements of the non-Fano matroid
as shown in Figure 3.5. The network has no vector linear solution of any
dimension over a finite field with characteristic 2.
ple involving four random variables X1, X2, X3, X4is the inequality
2I(X3;X4)≤I(X1;X2)+I(X1;X3, X4)+3I(X3;X4|X1)+I(X3;X4|X2),
(3.7)
Shannon-type inequalities are insufficient in general for computing cod-
ing capacity; this was shown in [37] by a network problem constructed
based on the V´amos matroid (e.g., [110]), for which the use of the non-
Shannon-type inequality (3.7) yields a tighter bound than any bound
derived only with Shannon-type inequalities.
The information theoretic approach yields an implicit characterization
of the rate region for deterministic network coding on a general acyclic
network with arc capacities ck, k ∈ A. We first introduce some defini-
66 Inter-Session Network Coding
tions. Let Nbe a set of random variables denoted {Xi:i∈ S} ∪ {Yj:
j∈ A}. Let HNbe the (2|N | −1)-dimensional Euclidian space whose
coordinates correspond to the 2|N | −1 nonempty subsets of N. A vector
g∈ HNis entropic if there exists some joint distribution for the random
variables in Nsuch that each component of gequals the entropy of the
corresponding subset of random variables. Define the region
Γ∗
N={g∈ HN:gis entropic}
This region essentially summarizes the effect of all possible information
inequalities involving variables in N(we do not yet have a complete
characterization of this region or, equivalently, of all possible information
inequalities).
We define the following regions in HN:
C1=(h∈ HN:H(XS) = X
i∈S
H(Xi))
C2=h∈ HN:H(YO(v)|XS(v), YI(v)) = 0 ∀v∈ N
C3={h∈ HN:H(Yj)< cj∀j∈ A}
C4=h∈ HN:H(XDt|YI(t)) = 0 ∀t∈ T
Theorem 3.1 The capacity region for an arbitrary acyclic network with
multiple sessions is given by
R= Λ projXSconv(Γ∗
N∩ C12)∩ C3∩ C4,
where for any region C ⊂ HN, projXS(C) = {hXS:h∈ C} is the
projection of Con the coordinates hXia corresponding to the source
entropies, conv(·)denotes the convex hull operator, the overbar denotes
the closure, and C12 =C1∩ C2.
Proof The proof is given in [145].
For networks with cycles, we need to also consider delay and causality
constraints. This can be done by considering, for each arc j, a se-
quence Y(1)
j,...,Y(T)
jof random variables corresponding to time steps
τ= 1,...,T. Then we have, in place of Equation (3.5),
H(Y(1)
O(v),...,Y(τ)
O(v)|XS(v), Y (1)
I(v),...,Y(τ−1)
O(v)) = 0 ∀v∈ N .
3.4 Information theoretic approaches 67
2 3
1 4
5t1
t2
Fig. 3.7. Gadget for converting any general multiple session network coding
problem into a multiple unicast problem. Reprinted with permission from [37].
3.4.1 Multiple unicast networks
A special case of the multiple session network coding problem is the
multiple unicast case, i.e. multiple communication sessions each consist-
ing of one source node and one sink node. For directed wired networks,
any general multiple session network coding problem can be converted
into a multiple unicast problem without changing the solvability or lin-
ear solvability of the problem. Thus, it suffices to study the multiple
unicast case as far as capacity is concerned. The conversion procedure
uses the gadget shown in Figure 3.7. Suppose t1and t2are sink nodes
in an arbitrary directed network that both demand source X. We then
add five additional nodes as shown, where node 1 is an additional source
demanded by node 5, and node 4 demands source X. In the resulting
network, t1and t2are not sink nodes but must decode Xin order to
satisfy the demands of the new sinks 4 and 5.
For an alternative undirected wired network model where each arc’s
capacity can be arbitrarily partitioned between the two directions of
information flow, it was conjectured in [90] that network coding does not
increase throughput for multiple unicast sessions. This is not however
68 Inter-Session Network Coding
s2
t2
t1
s1
1 2
b1⊕b2
b1
b1⊕b2
b1⊕b2
b1
b2
b2
Fig. 3.8. The two-unicast butterfly network. Each arc represents a directed
arc that is capable of carrying a single packet reliably. There is one packet b1
present at source node s1that we wish to communicate to sink node t1and
one packet b2present at source node s2that we wish to communicate to sink
node t2.b1⊕b2is the packet formed from the bitwise xor of the packets b1
and b2.
the case for undirected wireless networks, as we will see in the following
section.
3.5 Constructive approaches
The complexity of inter-session network coding motivates consideration
of suboptimal but useful classes of network codes that are feasible to con-
struct and implement in practice. A number of constructive approaches
are based on generalizations of simple example networks where bitwise
xor coding is useful.
3.5.1 Pairwise XOR coding in wireline networks
Figure 3.8, which we have seen before in Chapter 1, gives a simple exam-
ple of a wireline network where bitwise xor coding across two unicast
sessions doubles the common throughput that can be simultaneously
achieved for each session. This example has a “poison-antidote” inter-
pretation: the coded b1⊕b2packet is called a poison packet, since by itself
it is not useful to either sink; each sink needs an antidote packet from
the other session’s source in order to recover its own session’s packet.
To generalize this example, each arc can be replaced by a path seg-
3.5 Constructive approaches 69
remedy
poisoned
poisoned
remedy
s1t2
t1
s2
remedy request
remedy request
poisoned
Fig. 3.9. The canonical poison-antidote scenario. This is an example of inter-
session xor coding between two unicast sessions where decoding must take
place at intermediate non-sink nodes.
ment. The antidote segment can be between intermediate nodes rather
than directly from source to sink node, as shown in Figure 3.9. We can
also consider a stream of packets rather than a single packet from each
source.
For more than two unicast sessions, a tractable approach is to restrict
consideration to xor coding across pairs of unicasts. The throughput-
optimization problem then becomes the problem of optimally packing,
in the network, poison-antidote butterfly structures of the form shown
in Figure 3.9. This can be formulated as a linear optimization problem,
which we discuss in Section 5.1.3.
70 Inter-Session Network Coding
b1+b2
123
b1b2
Fig. 3.10. Information exchange via a single relay node.
3.5.2 XOR coding in wireless networks
3.5.2.1 Canonical scenarios
The broadcast nature of the wireless medium gives rise to more situa-
tions where network coding is beneficial. The simplest example, where
two sources exchange packets via a common relay node, is given in Fig-
ure 3.10 (it is the same as Figure 1.4 of Chapter 1, but drawn without
the hyperarcs explicitly shown).
For the case where the two sources exchange a stream of packets, this
example can be generalized by replacing the single relay node with a
multiple-relay path, as illustrated in Figure 3.11. Each coded broadcast
transmission by a relay node can transfer useful information in both
directions, replacing two uncoded point-to-point transmissions. This
canonical scenario is called the information exchange scenario.
The information exchange scenario involves wireless broadcast of coded
packets by relay nodes. The path intersection scenario, illustrated in Fig-
ure 3.12, involves wireless broadcast of coded packets as well as uncoded
packets which are used for decoding the coded packets. The information
exchange and path intersection scenarios are called wireless one-hop xor
coding scenarios, since each coded packet travels one hop before being
decoded.
The poison-antidote scenario described earlier for wireline networks
can also be adapted to wireless networks. The information exchange
scenario and the path intersection scenario with two unicast sessions
can be viewed as special cases where the coded packets travel exactly
one hop, and the antidote packets travel zero and one hop respectively.
3.5.2.2 Opportunistic coding
The Completely Opportunistic Coding (cope) protocol, proposed by
Katti et al., is based on the canonical wireless one-hop xor coding sce-
3.5 Constructive approaches 71
a1b1
12345
(a) The initial packets are sent uncoded from each source . . .
b1
a2a1
12345
b2
(b) . . . until a packet from each source reaches an intermediate relay node.
a1+b1
a3a2b2
12345
b3
(c) This node can take the bitwise xor of the two packets to form a coded
packet. It then broadcasts the coded packet to its two neighbor nodes with
a single wireless transmission. Each neighbor node has already seen (and
stored) one of the two original packets, and so is able to decode the other.
Thus, the coded broadcast is equivalent to forwarding the two original pack-
ets individually.
a1+b3
b4
a4a2+b2
a3+b1
12345
(d) Each neighbor node now has a packet from each source, and can similarly
xor them to form a coded packet, which is then broadcast.
Fig. 3.11. The multiple-relay information exchange scenario. After [138].
narios described in the previous section, and operates over any under-
lying routing protocol (e.g., shortest path or geographic routing) and
medium access protocol (e.g., 802.11 MAC).
There is an opportunistic listening (overhearing) component: the broad-
cast nature of the wireless medium allows nodes to overhear transmis-
sions intended for other nodes; all overheard packets are stored for some
period of time. Nodes periodically send reception reports, which can
be annotations to their own transmissions, informing their neighbors of
which packets they have overheard.
In the opportunistic coding component, at each transmission opportu-
nity, a node makes a decision on which of its queued packets to code and
transmit, based on the packets’ next-hop nodes and which of these pack-
ets have been overheard by the next-hop nodes. A node with packets to
72 Inter-Session Network Coding
t3
t2s2
s3
s1
t1
Fig. 3.12. The path intersection scenario with three unicast sessions. Each
source communicates with its sink via a common relay node, and each source’s
transmission is received by the relay node as well as the other two sessions’ sink
nodes. The relay node broadcasts the bitwise xor of the three sources’ packets.
Each sink can decode as it has the other two sources’ packets. Reprinted with
permission from [39].
forward to different next hop neighbors looks for the largest subset Sof
packets such that
•each packet u∈ S has a different next hop node vu
•each of these next hop nodes vualready has all the packets in Sexcept
u.
The packets in Sare xored together to form a coded packet which is
broadcast to nodes vu, u ∈ S, each of which has enough information
to decode its intended packet u. This policy maximizes the number of
packets that are communicated by its transmission, assuming that the
nodes that receive the packet will attempt to decode immediately upon
reception. For example, consider the situation illustrated in Figure 3.13.
In this situation, the coding decision made by node 1 is to send out the
packet b1⊕b2⊕b3, because this allows three packets, b1,b2, and b3,
to be communicated to their next hops, while allowing the nodes that
receive the packet to decode upon reception and recover the packets that
they each desire. Node 1 does not, for example, send out the packet
b1⊕b2⊕b4, because only node 2 is able to immediately decode and
recover the packet that it desires, b4. The ability to make the coding
decision requires each node to know the contents of the queues of the
neighboring nodes.
3.5 Constructive approaches 73
b2
b1b2b3b4
b1
b1
3
4
2 1
b3
b3
b2
Fig. 3.13. An example of the Katti et al. queue-length-based approach. Sup-
pose the next hop for packet b1is node 4, the next hop for packet b2is node
3, and the next hop for packets b3and b4is node 2. Node 1 has a transmis-
sion opportunity on its lossless broadcast arc reaching nodes 2, 3, and 4. The
decision it makes is to send out the packet b1⊕b2⊕b3, because this allows
three packets, b1,b2, and b3, to be communicated to their next hops. After
[78].
Experiments have shown that opportunistic coding can significantly
improve throughput in ad hoc wireless networks using 802.11 and geo-
graphic routing, particularly under congested network conditions. We
discuss cope further in Section 5.2.2, in relation to subgraph selection.
3.5.2.3 Coding-influenced routing
There can be greater scope for network coding gains when routing is
influenced by potential coding opportunities. For instance, consider the
74 Inter-Session Network Coding
s2
s1
t2
t1
Fig. 3.14. The reverse carpooling scenario.
information exchange scenario, which involves two unicast flows such
that the source of each unicast is co-located with the sink of the other.
This can be generalized to the case of two unicast sessions whose source
and sink nodes are not co-located, by selecting routes for the two sessions
which overlap in opposite directions. In analogy to carpooling, each
session’s route may involve a detour which is compensated for by the
sharing of transmissions on the common portion of the route, hence the
name reverse carpooling. An illustration is given in Figure 3.14.
The problem of finding the best solution within a class of network
codes is a subgraph selection problem. Tractable optimization problems
can be obtained by limiting consideration to strategies involving reverse
carpooling, poison-antidote and/or path intersection scenarios with a
limit on the number of constituent sessions of a coded packet. This is
discussed in Sections 5.1.3 and 5.2.2.
3.6 Notes and further reading
The algebraic characterization and construction of scalar linear non-
multicast network codes presented in this chapter is from Koetter and
M´edard [85]. Rasala Lehman and Lehman [116] determined the com-
plexity classes of different scalar linear network coding problems. Ex-
ample networks requiring vector linear rather than scalar linear coding
solutions were given in Rasala Lehman and Lehman [116], M´edard et
al. [104] and Riis [119]. Dougherty et al. showed in [38] that linear
3.6 Notes and further reading 75
coding is insufficient in general for non-multicast networks, and in [37]
that Shannon-type inequalities are insufficient in general for analyzing
network coding capacity, using connections between matroid theory and
network coding described in [37]. Group network codes have been con-
sidered by Chan [21].
The entropy function-based characterization of capacity for acyclic
networks presented in this chapter is due to Yan et al. [145]. Entropy-
based approaches and other techniques for bounding communication
rates in non-multicast problems have been given in various works in-
cluding [104, 38, 147, 86, 2, 53].
The conversion processThe study of network coding on the undirected
wired network model was initiated by Li and Li [89].
On the code construction side, the poison-antidote approach for multi-
ple unicast network coding was introduced by Ratnakar et al. [117, 118].
The information exchange scenario was introduced by Wu et al. [139].
The (cope) protocol was developed by Katti et al. [77, 78]. We discuss
work on subgraph selection for inter-session network coding in Chap-
ter 5.
4
Network Coding in Lossy Networks
In this chapter, we discuss the use of network coding, particularly ran-
dom linear network coding, in lossy networks with packet erasures. The
main result that we establish is that random linear network coding
achieves the capacity of a single connection (unicast or multicast) in
a given coding subgraph, i.e., by efficiently providing robustness, ran-
dom linear network coding allows a connection to be established at the
maximum throughput that is possible in a given coding subgraph.
Throughout this chapter, we assume that a coding subgraph is given;
we address the problem of subgraph selection in Chapter 5, where we
also discuss whether separating coding and subgraph selection is optimal.
The lossy coding subgraphs we consider here are applicable to various
types of network, including multi-hop wireless networks and peer-to-peer
networks. In the latter case, losses are not caused so much by unreliable
links, but rather by unreliable nodes that intermittently join and leave
the network.
We model coding subgraphs using the time-expanded subgraphs de-
scribed in Section 1.3. Recall that a time-expanded subgraph describes
the times and locations at which packet injections and receptions occur.
Since a coding subgraph specifies only to the times at which packet in-
jections occur, a time-expanded subgraph is in fact an element in the
random ensemble of a coding subgraph. A time-expanded subgraph is
shown in Figure 4.1.
For simplicity, we assume that links are delay-free in the sense that
the arrival time of a received packet corresponds to the time that it was
injected to the link. This assumption does not alter the results that
we derive in this chapter, and a link with delay can be converted into
delay-free links by introducing an artificial node implementing a trivial
coding function. On the left of Figure 4.2, we show a fragment of a
76
Network Coding in Lossy Networks 77
b3=f(b1, b2)
b1b2
time1 2 3
Fig. 4.1. Coding at a node in a time-expanded subgraph. Packet b1is received
at time 1, and packet b2is received at time 2. The thick, horizontal arcs have
infinite capacity, and represent data stored at a node. Thus, at time 3, packets
b1and b2can be used to form packet b3.
1
Node 1
Node 1′
Node 2
b1
2 time
b1
1
b1
2 time
Fig. 4.2. Conversion of a link with delay into delay-free links.
time-expanded subgraph that depicts a transmission with delay. The
packet b1is injected by node 1 at time 1, and this packet is not received
by node 2 until time 2. On the right, we depict the same transmission,
but we introduce an artificial node 1′. Node 1′does nothing but store b1
78 Network Coding in Lossy Networks
time1
Node 1
Node 2
Node 3
τ0
Fig. 4.3. A time-expanded subgraph where, between time 0 and time τ, a
packet is successfully transmitted from node 1 to node 2 and another is suc-
cessfully transmitted from node 1 to both nodes 2 and 3.
then transmit it out again at time 2. In this way, we can suppose that
there is a delay-free transmission from node 1 to node 1′at time 1 and
one from node 1′to node 2 at time 2.
Let AiJ be the counting process describing the arrival of packets that
are injected on hyperarc (i, J ), and let AiJ K be the counting process
describing the arrival of packets that are injected on hyperarc (i, J) and
received by exactly the set of nodes K⊂J; i.e., for τ≥0, AiJ (τ) is
the total number of packets that are injected on hyperarc (i, J ) between
time 0 and time τ, and AiJ K (τ) is the total number of packets that are
injected on hyperarc (i, J ) and received by all nodes in K(and no nodes
in N \ K) between time 0 and time τ. For example, suppose that three
packets are injected on hyperarc (1,{2,3}) between time 0 and time τ0
and that, of these three packets, one is received by node 2 only, one is
lost entirely, and one is received by both nodes 2 and 3; then we have
A1(23)(τ0) = 3, A1(23)∅(τ0) = 1, A1(23)2 τ0) = 1, A1(23)3 (τ0) = 0, and
A1(23)(23)(τ0) = 1. A possible time-expanded subgraph corresponding
to these events is shown in Figure 4.3.
We assume that AiJ has an average rate ziJ and that AiJ K has an
4.1 Random linear network coding 79
average rate ziJ K ; more precisely, we assume that
lim
τ→∞
AiJ (τ)
τ=ziJ
and that
lim
τ→∞
AiJK (τ)
τ=ziJK
almost surely. Hence we have ziJ =PK⊂JziJK and, if the link is
lossless, we have ziJK = 0 for all K(J. The vector z, consisting of
ziJ , (i, J)∈ A, is the coding subgraph that we are given.
In Section 4.1, we specify precisely what we mean by random linear
network coding in a lossy network, then, in Section 4.2, we establish the
main result of this chapter: we show that random linear network coding
achieves the capacity of a single connection in a given coding subgraph.
This result is concerned only with throughput and does not consider
delay: in Section 4.3, we strengthen the result in the special case of
Poisson traffic with i.i.d. losses by giving error exponents. These error
exponents allow us to quantify the rate of decay of the probability of
error with coding delay and to determine the parameters of importance
in this decay.
4.1 Random linear network coding
The specific coding scheme we consider is as follows. We suppose that,
at the source node, we have Kmessage packets w1, w2,...,wK, which
are vectors of length λover some finite field Fq. (If the packet length is
bbits, then we take λ=⌈b/ log2q⌉.) The message packets are initially
present in the memory of the source node.
The coding operation performed by each node is simple to describe
and is the same for every node: received packets are stored into the
node’s memory, and packets are formed for injection with random linear
combinations of its memory contents whenever a packet injection occurs
on an outgoing link. The coefficients of the combination are drawn
uniformly from Fq.
Since all coding is linear, we can write any packet uin the network
as a linear combination of w1, w2,...,wK, namely, u=PK
k=1 γkwk. We
call γthe global encoding vector of u, and we assume that it is sent
along with u, as side information in its header. The overhead this incurs
(namely, Klog2qbits) is negligible if packets are sufficiently large.
Nodes are assumed to have unlimited memory. The scheme can be
80 Network Coding in Lossy Networks
Initialization:
•The source node stores the message packets w1, w2,...,wKin its
memory.
Operation:
•When a packet is received by a node,
–the node stores the packet in its memory.
•When a packet injection occurs on an outgoing link of a node,
–the node forms the packet from a random linear combination
of the packets in its memory. Suppose the node has Lpackets
u1, u2,...,uLin its memory. Then the packet formed is
u0:=
L
X
l=1
αlul,
where αlis chosen according to a uniform distribution over the
elements of Fq. The packet’s global encoding vector γ, which
satisfies u0=PK
k=1 γkwk, is placed in its header.
Decoding:
•Each sink node performs Gaussian elimination on the set of global
encoding vectors from the packets in its memory. If it is able
to find an inverse, it applies the inverse to the packets to obtain
w1, w2,...,wK; otherwise, a decoding error occurs.
Fig. 4.4. Summary of the random linear network coding scheme used in this
chapter (cf. Section 2.5.1.1).
modified so that received packets are stored into memory only if their
global encoding vectors are linearly-independent of those already stored.
This modification keeps our results unchanged while ensuring that nodes
never need to store more than Kpackets.
A sink node collects packets and, if it has Kpackets with linearly-
independent global encoding vectors, it is able to recover the message
packets. Decoding can be done by Gaussian elimination. The scheme
can be run either for a predetermined duration or, in the case of rateless
operation, until successful decoding at the sink nodes. We summarize
the scheme in Figure 4.4.
The scheme is carried out for a single block of Kmessage packets at
the source. If the source has more packets to send, then the scheme is
repeated with all nodes flushed of their memory contents.
4.2 Coding theorems 81
21 3
Fig. 4.5. A network consisting of two point-to-point links in tandem.
Reprinted with permission from [93].
4.2 Coding theorems
In this section, we specify achievable rate intervals for random linear
network coding in various scenarios. The fact that the intervals we
specify are the largest possible (i.e., that random linear network coding
is capacity-achieving) can be seen by simply noting that the rate of a
connection must be limited by the rate at which distinct packets are
being received over any cut between the source and the sink. A formal
converse can be obtained using the cut-set bound for multi-terminal
networks (see [28, Section 14.10]).
4.2.1 Unicast connections
4.2.1.1 Two-link tandem network
We develop our general result for unicast connections by extending from
some special cases. We begin with the simplest non-trivial case: that of
two point-to-point links in tandem (see Figure 4.5).
Suppose we wish to establish a connection of rate arbitrarily close to
Rpackets per unit time from node 1 to node 3. Suppose further that
random linear network coding is run for a total time ∆, from time 0
until time ∆, and that, in this time, a total of Npackets is received by
node 2. We call these packets v1, v2,...,vN.
Any packet ureceived by a node is a linear combination of v1, v2,...,vN,
so we can write
u=
N
X
n=1
βnvn.
Now, since vnis formed by a random linear combination of the message
packets w1, w2,...,wK, we have
vn=
K
X
k=1
αnkwk
82 Network Coding in Lossy Networks
for n= 1,2,...,N. Hence
u=
K
X
k=1 N
X
n=1
βnαnk!wk,
and it follows that the kth component of the global encoding vector of
uis given by
γk=
N
X
n=1
βnαnk.
We call the vector βassociated with uthe auxiliary encoding vector of u,
and we see that any node that receives ⌊K(1 + ε)⌋or more packets with
linearly-independent auxiliary encoding vectors has ⌊K(1 + ε)⌋packets
whose global encoding vectors collectively form a random ⌊K(1+ε)⌋×K
matrix over Fq, with all entries chosen uniformly. If this matrix has rank
K, then node 3 is able to recover the message packets. The probability
that a random ⌊K(1+ε)⌋×Kmatrix has rank Kis, by a simple counting
argument, Q⌊K(1+ε)⌋
k=1+⌊K(1+ε)⌋−K(1 −1/qk), which can be made arbitrarily
close to 1 by taking Karbitrarily large. Therefore, to determine whether
node 3 can recover the message packets, we essentially need only to
determine whether it receives ⌊K(1 + ε)⌋or more packets with linearly-
independent auxiliary encoding vectors.
Our proof is based on tracking the propagation of what we call inno-
vative packets. Such packets are innovative in the sense that they carry
new, as yet unknown, information about v1, v2,...,vNto a node. It
turns out that the propagation of innovative packets through a network
follows the propagation of jobs through a queueing network, for which
fluid flow models give good approximations. We first give a heuristic
argument in terms of this fluid analogy before proceeding to a formal
argument.
Since the packets being received by node 2 are the packets v1, v2,...,vN
themselves, it is clear that every packet being received by node 2 is in-
novative. Thus, innovative packets arrive at node 2 at a rate of z122,
and this can be approximated by fluid flowing in at rate z122. These
innovative packets are stored in node 2’s memory, so the fluid that flows
in is stored in a reservoir.
Packets, now, are being received by node 3 at a rate of z233, but
whether these packets are innovative depends on the contents of node 2’s
memory. If node 2 has more information about v1, v2,...,vNthan node
3 does, then it is highly likely that new information will be described to
4.2 Coding theorems 83
2
3
z122
z233
Fig. 4.6. Fluid flow system corresponding to two-link tandem network.
Reprinted with permission from [93].
node 3 in the next packet that it receives. Otherwise, if node 2 and node
3 have the same degree of information about v1, v2,...,vN, then packets
received by node 3 cannot possibly be innovative. Thus, the situation
is as though fluid flows into node 3’s reservoir at a rate of z233, but the
level of node 3’s reservoir is restricted from ever exceeding that of node
2’s reservoir. The level of node 3’s reservoir, which is ultimately what
we are concerned with, can equivalently be determined by fluid flowing
out of node 2’s reservoir at rate z233.
We therefore see that the two-link tandem network in Figure 4.5 maps
to the fluid flow system shown in Figure 4.6. It is clear that, in this
system, fluid flows into node 3’s reservoir at rate min(z122 , z233). This
rate determines the rate at which packets with new information about
v1, v2,...,vN—and, therefore, linearly-independent auxiliary encoding
vectors—arrive at node 3. Hence the time required for node 3 to receive
⌊K(1 + ε)⌋packets with linearly-independent auxiliary encoding vectors
is, for large K, approximately K(1 + ε)/min(z122, z233), which implies
that a connection of rate arbitrarily close to Rpackets per unit time can
be established provided that
R≤min(z122, z233 ).(4.1)
The right-hand side of (4.1) is indeed the capacity of the two-link tandem
network, and we therefore have the desired result for this case.
We now proceed to establish the result formally. All packets received
by node 2, namely v1, v2, . . . , vN, are considered innovative. We asso-
ciate with node 2 the set of vectors U, which varies with time and is
initially empty, i.e., U(0) := ∅. If packet uis received by node 2 at time
τ, then its auxiliary encoding vector βis added to Uat time τ, i.e.,
U(τ+) := {β} ∪ U(τ).
We associate with node 3 the set of vectors W, which again varies with
time and is initially empty. Suppose packet u, with auxiliary encoding
84 Network Coding in Lossy Networks
vector β, is received by node 3 at time τ. Let µbe a positive integer,
which we call the innovation order. Then we say uis innovative if
β /∈span(W(τ)) and |U(τ)|>|W(τ)|+µ−1. If uis innovative, then β
is added to Wat time τ.†
The definition of innovative is designed to satisfy two properties: First,
we require that W(∆), the set of vectors in Wwhen the scheme termi-
nates, is linearly independent. Second, we require that, when a packet is
received by node 3 and |U(τ)|>|W(τ)|+µ−1, it is innovative with high
probability. The innovation order µis an arbitrary factor that ensures
that the latter property is satisfied.
Suppose |U(τ)|>|W(τ)|+µ−1. Since uis a random linear combi-
nation of vectors in U(τ), it follows that uis innovative with some non-
trivial probability. More precisely, because βis uniformly-distributed
over q|U(τ)|possibilities, of which at least q|U(τ)|−q|W(τ)|are not in
span(W(τ)), it follows that
Pr(β /∈span(W(τ))) ≥q|U(τ)|−q|W(τ)|
q|U(τ)|
= 1 −q|W(τ)|−|U(τ)|
≥1−q−µ.
Hence uis innovative with probability at least 1 −q−µ. Since we can
always discard innovative packets, we assume that the event occurs with
probability exactly 1 −q−µ. If instead |U(τ)| ≤ |W(τ)|+µ−1, then
we see that ucannot be innovative, and this remains true at least until
another arrival occurs at node 2. Therefore, for an innovation order of
µ, the propagation of innovative packets through node 2 is described by
the propagation of jobs through a single-server queueing station with
queue size (|U(τ)| − |W(τ)| − µ+ 1)+, where, for a real number x,
(x)+:= max(x, 0). We similarly define (x)−:= max(−x, 0).
The queueing station is serviced with probability 1−q−µwhenever the
queue is non-empty and a received packet arrives on arc (2,3). We can
equivalently consider “candidate” packets that arrive with probability
1−q−µwhenever a received packet arrives on arc (2,3) and say that
†This definition of innovative differs from merely being informative, which is the
sense in which innovative is used in [27]. Indeed, a packet can be informative,
in the sense that in gives a node some new, as yet unknown, information about
v1, v2, . . . , vN(or about w1, w2,...,wK), and not satisfy this definition of innova-
tive. We have defined innovative so that innovative packets are informative (with
respect to other innovative packets at the node), but not necessarily conversely.
This allows us to bound, or dominate, the behavior of random linear network
coding, though we cannot describe it exactly.
4.2 Coding theorems 85
the queueing station is serviced whenever the queue is non-empty and a
candidate packet arrives on arc (2,3). We consider all packets received
on arc (1,2) to be candidate packets.
The system we wish to analyze, therefore, is the following simple
queueing system: Jobs arrive at node 2 according to the arrival of re-
ceived packets on arc (1,2) and, with the exception of the first µ−1 jobs,
enter node 2’s queue. The jobs in node 2’s queue are serviced by the
arrival of candidate packets on arc (2,3) and exit after being serviced.
The number of jobs exiting is a lower bound on the number of packets
with linearly-independent auxiliary encoding vectors received by node
3.
We analyze the queueing system of interest using the fluid approxima-
tion for discrete-flow networks (see, e.g., [24, 25]). We do not explicitly
account for the fact that the first µ−1 jobs arriving at node 2 do not
enter its queue because this fact has no effect on job throughput. Let B1,
B, and Cbe the counting processes for the arrival of received packets on
arc (1,2), of innovative packets on arc (2,3), and of candidate packets
on arc (2,3), respectively. Let Q(τ) be the number of jobs queued for
service at node 2 at time τ. Hence Q=B1−B. Let X:= B1−Cand
Y:= C−B. Then
Q=X+Y. (4.2)
Moreover, we have
Q(τ)dY (τ) = 0,(4.3)
dY (τ)≥0,(4.4)
and
Q(τ)≥0 (4.5)
for all τ≥0, and
Y(0) = 0.(4.6)
We observe now that equations (4.2)–(4.6) give us the conditions for
a Skorohod problem (see, e.g., [25, Section 7.2]) and, by the oblique
reflection mapping theorem, there is a well-defined, Lipschitz-continuous
mapping Φ such that Q= Φ(X).
86 Network Coding in Lossy Networks
Let
¯
C(K)(τ) := C(Kτ )
K,
¯
X(K)(τ) := X(Kτ )
K,
and
¯
Q(K)(τ) := Q(Kτ )
K.
Recall that A233 is the counting process for the arrival of received
packets on arc (2,3). Therefore, C(τ) is the sum of A233(τ) Bernoulli-
distributed random variables with parameter 1 −q−µ. Hence
¯
C(τ) := lim
K→∞
¯
C(K)(τ)
= lim
K→∞(1 −q−µ)A233(Kτ )
Ka.s.
= (1 −q−µ)z233τa.s.,
where the last equality follows by the assumptions of the model. There-
fore
¯
X(τ) := lim
K→∞
¯
X(K)(τ) = (z122 −(1 −q−µ)z233)τa.s.
By the Lipschitz-continuity of Φ, then, it follows that ¯
Q:= limK→∞ ¯
Q(K)=
Φ( ¯
X), i.e., ¯
Qis, almost surely, the unique ¯
Qthat satisfies, for some ¯
Y,
¯
Q(τ) = (z122 −(1 −q−µ)z233)τ+¯
Y , (4.7)
¯
Q(τ)d¯
Y(τ) = 0,(4.8)
d¯
Y(τ)≥0,(4.9)
and
¯
Q(τ)≥0 (4.10)
for all τ≥0, and
¯
Y(0) = 0.(4.11)
A pair ( ¯
Q, ¯
Y) that satisfies (4.7)–(4.11) is
¯
Q(τ) = (z122 −(1 −q−µ)z233)+τ(4.12)
and
¯
Y(τ) = (z122 −(1 −q−µ)z233)−τ.
Hence ¯
Qis given by equation (4.12).
4.2 Coding theorems 87
L+ 121 ···
Fig. 4.7. A network consisting of Lpoint-to-point links in tandem. Reprinted
with permission from [93].
Recall that node 3 can recover the message packets with high proba-
bility if it receives ⌊K(1+ε)⌋packets with linearly-independent auxiliary
encoding vectors and that the number of jobs exiting the queueing sys-
tem is a lower bound on the number of packets with linearly-independent
auxiliary encoding vectors received by node 3. Therefore, node 3 can
recover the message packets with high probability if ⌊K(1 + ε)⌋or more
jobs exit the queueing system. Let νbe the number of jobs that have
exited the queueing system by time ∆. Then
ν=B1(∆) −Q(∆).
Take K=⌈(1 −q−µ)∆RcR/(1 + ε)⌉, where 0 < Rc<1. Then
lim
K→∞
ν
⌊K(1 + ε)⌋= lim
K→∞
B1(∆) −Q(∆)
K(1 + ε)
=z122 −(z122 −(1 −q−µ)z233)+
(1 −q−µ)RcR
=min(z122,(1 −q−µ)z233 )
(1 −q−µ)RcR
≥1
Rc
min(z122, z233 )
R>1
provided that
R≤min(z122, z233 ).(4.13)
Hence, for all Rsatisfying (4.13), ν≥ ⌊K(1 + ε)⌋with probability arbi-
trarily close to 1 for Ksufficiently large. The rate achieved is
K
∆≥(1 −q−µ)Rc
1 + εR,
which can be made arbitrarily close to Rby varying µ,Rc, and ε.
4.2.1.2 L-link tandem network
We extend our result to another special case before considering general
unicast connections: we consider the case of a tandem network consisting
of Lpoint-to-point links and L+ 1 nodes (see Figure 4.7).
88 Network Coding in Lossy Networks
L+ 1
2
zL(L+1)(L+1)
z233
z122
...
Fig. 4.8. Fluid flow system corresponding to L-link tandem network.
Reprinted with permission from [93].
This case is a straightforward extension of that of the two-link tandem
network. It maps to the fluid flow system shown in Figure 4.8. In this
system, it is clear that fluid flows into node (L+ 1)’s reservoir at rate
min1≤i≤L{zi(i+1)(i+1)}. Hence a connection of rate arbitrarily close to
Rpackets per unit time from node 1 to node L+ 1 can be established
provided that
R≤min
1≤i≤L{zi(i+1)(i+1)}.(4.14)
Since the right-hand side of (4.14) is indeed the capacity of the L-link
tandem network, we therefore have the desired result for this case.
The formal argument requires care. For i= 2,3,...,L+1, we associate
with node ithe set of vectors Vi, which varies with time and is initially
empty. We define U:= V2and W:= VL+1. As in the case of the two-link
tandem, all packets received by node 2 are considered innovative and,
if packet uis received by node 2 at time τ, then its auxiliary encoding
vector βis added to Uat time τ. For i= 3,4,...,L+ 1, if packet u,
with auxiliary encoding vector β, is received by node iat time τ, then
we say uis innovative if β /∈span(Vi(τ)) and |Vi−1(τ)|>|Vi(τ)|+µ−1.
If uis innovative, then βis added to Viat time τ.
This definition of innovative is a straightforward extension of that in
Section 4.2.1.1. The first property remains the same: we continue to
require that W(∆) is a set of linearly-independent vectors. We extend
the second property so that, when a packet is received by node ifor any
i= 3,4,...,L+ 1 and |Vi−1(τ)|>|Vi(τ)|+µ−1, it is innovative with
high probability.
Take some i∈ {3,4,...,L+1}. Suppose that packet u, with auxiliary
encoding vector β, is received by node iat time τand that |Vi−1(τ)|>
|Vi(τ)|+µ−1. Thus, the auxiliary encoding vector βis a random linear
combination of vectors in some set V0that contains Vi−1(τ). Hence,
4.2 Coding theorems 89
because βis uniformly-distributed over q|V0|possibilities, of which at
least q|V0|−q|Vi(τ)|are not in span(Vi(τ)), it follows that
Pr(β /∈span(Vi(τ))) ≥q|V0|−q|Vi(τ)|
q|V0|
= 1 −q|Vi(τ)|−|V0|
≥1−q|Vi(τ)|−|Vi−1(τ)|
≥1−q−µ.
Therefore uis innovative with probability at least 1 −q−µ. Following
the argument in Section 4.2.1.1, we see, for all i= 2,3,...,L, that the
propagation of innovative packets through node iis described by the
propagation of jobs through a single-server queueing station with queue
size (|Vi(τ)|−|Vi+1(τ)|−µ+1)+and that the queueing station is serviced
with probability 1−q−µwhenever the queue is non-empty and a received
packet arrives on arc (i, i +1). We again consider candidate packets that
arrive with probability 1 −q−µwhenever a received packet arrives on
arc (i, i + 1) and say that the queueing station is serviced whenever the
queue is non-empty and a candidate packet arrives on arc (i, i + 1).
The system we wish to analyze in this case is therefore the following
simple queueing network: Jobs arrive at node 2 according to the arrival
of received packets on arc (1,2) and, with the exception of the first µ−1
jobs, enter node 2’s queue. For i= 2,3,...,L−1, the jobs in node i’s
queue are serviced by the arrival of candidate packets on arc (i, i+1) and,
with the exception of the first µ−1 jobs, enter node (i+ 1)’s queue after
being serviced. The jobs in node L’s queue are serviced by the arrival of
candidate packets on arc (L, L + 1) and exit after being serviced. The
number of jobs exiting is a lower bound on the number of packets with
linearly-independent auxiliary encoding vectors received by node L+ 1.
We again analyze the queueing network of interest using the fluid
approximation for discrete-flow networks, and we again do not explicitly
account for the fact that the first µ−1 jobs arriving at a queueing
node do not enter its queue. Let B1be the counting process for the
arrival of received packets on arc (1,2). For i= 2,3,...,L, let Bi, and
Cibe the counting processes for the arrival of innovative packets and
candidate packets on arc (i, i+ 1), respectively. Let Qi(τ) be the number
of jobs queued for service at node iat time τ. Hence, for i= 2,3,...,L,
Qi=Bi−1−Bi. Let Xi:= Ci−1−Ciand Yi:= Ci−Bi, where C1:= B1.
Then, we obtain a Skorohod problem with the following conditions: For
90 Network Coding in Lossy Networks
all i= 2,3,...,L,
Qi=Xi−Yi−1+Yi.
For all τ≥0 and i= 2,3, . . . , L,
Qi(τ)dYi(τ) = 0,
dYi(τ)≥0,
and
Qi(τ)≥0.
For all i= 2,3,...,L,
Yi(0) = 0.
Let
¯
Q(K)
i(τ) := Qi(Kτ )
K
and ¯
Qi:= limK→∞ ¯
Q(K)
ifor i= 2,3,...,L. Then the vector ¯
Qis,
almost surely, the unique ¯
Qthat satisfies, for some ¯
Y,
¯
Qi(τ) =
(z122 −(1 −q−µ)z233)τ+¯
Y2(τ) if i= 2,
(1 −q−µ)(z(i−1)ii −zi(i+1)(i+1))τ
+¯
Yi(τ)−¯
Yi−1(τ)otherwise,(4.15)
¯
Qi(τ)d¯
Yi(τ) = 0,(4.16)
d¯
Yi(τ)≥0,(4.17)
and
¯
Qi(τ)≥0 (4.18)
for all τ≥0 and i= 2,3, . . . , L, and
¯
Yi(0) = 0 (4.19)
for all i= 2,3,...,L.
A pair ( ¯
Q, ¯
Y) that satisfies (4.15)–(4.19) is
¯
Qi(τ) = (min(z122,min
2≤j<i{(1 −q−µ)zj(j+1)(j+1) })
−(1 −q−µ)zi(i+1)(i+1))+τ(4.20)
and
¯
Yi(τ) = (min(z122,min
2≤j<i{(1 −q−µ)zj(j+1)(j+1) })
−(1 −q−µ)zi(i+1)(i+1))−τ.
4.2 Coding theorems 91
Hence ¯
Qis given by equation (4.20).
The number of jobs that have exited the queueing network by time ∆
is given by
ν=B1(∆) −
L
X
i=2
Qi(∆).
Take K=⌈(1 −q−µ)∆RcR/(1 + ε)⌉, where 0 < Rc<1. Then
lim
K→∞
ν
⌊K(1 + ε)⌋= lim
K→∞
B1(∆) −PL
i=2 Q(∆)
K(1 + ε)
=min(z122,min2≤i≤L{(1 −q−µ)zi(i+1)(i+1) })
(1 −q−µ)RcR
≥1
Rc
min1≤i≤L{zi(i+1)(i+1)}
R>1
(4.21)
provided that
R≤min
1≤i≤L{zi(i+1)(i+1)}.(4.22)
Hence, for all Rsatisfying (4.22), ν≥ ⌊K(1 + ε)⌋with probability arbi-
trarily close to 1 for Ksufficiently large. The rate can again be made
arbitrarily close to Rby varying µ,Rc, and ε.
4.2.1.3 General unicast connection
We now extend our result to general unicast connections. The strategy
here is simple: A general unicast connection can be formulated as a
flow, which can be decomposed into a finite number of paths. Each of
these paths is a tandem network, which is the case that we have just
considered.
Suppose that we wish to establish a connection of rate arbitrarily close
to Rpackets per unit time from source node sto sink node t. Suppose
further that
R≤min
Q∈Q(s,t)
X
(i,J)∈Γ+(Q)X
K6⊂Q
ziJK
,
where Q(s, t) is the set of all cuts between sand t, and Γ+(Q) denotes
the set of forward hyperarcs of the cut Q, i.e.,
Γ+(Q) := {(i, J )∈ A | i∈Q, J \Q6=∅}.
92 Network Coding in Lossy Networks
Therefore, by the max-flow/min-cut theorem (see, e.g., [5, Sections 6.5–
6.7], [11, Section 3.1]), there exists a flow vector xsatisfying
X
{J|(i,J)∈A} X
j∈J
xiJj −X
{j|(j,I)∈A,i∈I}
xjIi =
Rif i=s,
−Rif i=t,
0 otherwise,
for all i∈ N ,
X
j∈K
xiJj ≤X
{L⊂J|L∩K6=∅}
ziJL (4.23)
for all (i, J )∈ A and K⊂J, and xiJj ≥0 for all (i, J )∈ A and j∈J.
Using the conformal realization theorem (see, e.g., [11, Section 1.1]),
we decompose xinto a finite set of paths {p1, p2,...,pM}, each carrying
positive flow Rmfor m= 1,2,...,M, such that PM
m=1 Rm=R. We
treat each path pmas a tandem network and use it to deliver innova-
tive packets at rate arbitrarily close to Rm, resulting in an overall rate
for innovative packets arriving at node tthat is arbitrarily close to R.
Some care must be take in the interpretation of the flow and its path
decomposition because the same packet may be received by more than
one node.
Consider a single path pm. We write pm={i1, i2,...,iLm, iLm+1},
where i1=sand iLm+1 =t. For l= 2,3,...,Lm+ 1, we associate with
node ilthe set of vectors V(pm)
l, which varies with time and is initially
empty. We define U(pm):= V(pm)
2and W(pm):= V(pm)
Lm+1.
We note that the constraint (4.23) can also be written as
xiJj ≤X
{L⊂J|j∈L}
α(j)
iJL ziJL
for all (i, J )∈ A and j∈J, where Pj∈Lα(j)
iJL = 1 for all (i, J )∈ A and
L⊂J, and α(j)
iJL ≥0 for all (i, J )∈ A,L⊂J, and j∈L. Suppose
packet u, with auxiliary encoding vector β, is placed on hyperarc (i1, J)
and received by K⊂J, where K∋i2, at time τ. We associate with
uthe independent random variable Pu, which takes the value mwith
probability Rmα(i2)
i1JK /P{L⊂J|i2∈L}α(i2)
i1JL ziJL. If Pu=m, then we say
uis innovative on path pm, and βis added to U(pm)at time τ.
Take l= 2,3,...,Lm. Now suppose packet u, with auxiliary en-
coding vector β, is placed on hyperarc (il, J) and received by K⊂
J, where K∋il+1 , at time τ. We associate with uthe indepen-
dent random variable Pu, which takes the value mwith probability
4.2 Coding theorems 93
Rmα(il+1)
ilJK /P{L⊂J|il+1 ∈L}α(il+1)
ilJL ziJ L. We say uis innovative on path
pmif Pu=m,β /∈span(∪m−1
n=1 W(pn)(∆)∪V(pm)
l+1 (τ)∪∪M
n=m+1U(pn)(∆)),
and |V(pm)
l(τ)|>|V(pm)
l+1 (τ)|+µ−1.
This definition of innovative is somewhat more complicated than that
in Sections 4.2.1.1 and 4.2.1.2 because we now have Mpaths that we
wish to analyze separately. We have again designed the definition to
satisfy two properties: First, we require that ∪M
m=1W(pm)(∆) is linearly-
independent. This is easily verified: Vectors are added to W(p1)(τ) only
if they are linearly independent of existing ones; vectors are added to
W(p2)(τ) only if they are linearly independent of existing ones and ones
in W(p1)(∆); and so on. Second, we require that, when a packet is
received by node il,Pu=m, and |V(pm)
l−1(τ)|>|V(pm)
l(τ)|+µ−1, it is
innovative on path pmwith high probability.
Take l∈ {3,4,...,Lm+ 1}. Suppose that packet u, with auxiliary
encoding vector β, is received by node ilat time τ, that Pu=m, and
that |V(pm)
l−1(τ)|>|V(pm)
l(τ)|+µ−1. Thus, the auxiliary encoding vector
βis a random linear combination of vectors in some set V0that contains
V(pm)
l−1(τ). Hence βis uniformly-distributed over q|V0|possibilities, of
which at least q|V0|−qdare not in span(V(pm)
l(τ)∪˜
V\m), where d:=
dim(span(V0)∩span(V(pm)
l(τ)∪˜
V\m)). We have
d= dim(span(V0)) + dim(span(V(pm)
l(τ)∪˜
V\m))
−dim(span(V0∪V(pm)
l(τ)∪˜
V\m))
≤dim(span(V0\V(pm)
l−1(τ))) + dim(span(V(pm)
l−1(τ)))
+ dim(span(V(pm)
l(τ)∪˜
V\m))
−dim(span(V0∪V(pm)
l(τ)∪˜
V\m))
≤dim(span(V0\V(pm)
l−1(τ))) + dim(span(V(pm)
l−1(τ)))
+ dim(span(V(pm)
l(τ)∪˜
V\m))
−dim(span(V(pm)
l−1(τ)∪V(pm)
l(τ)∪˜
V\m)).
Since V(pm)
l−1(τ)∪˜
V\mand V(pm)
l(τ)∪˜
V\mboth form linearly-independent
sets,
dim(span(V(pm)
l−1(τ))) + dim(span(V(pm)
l(τ)∪˜
V\m))
= dim(span(V(pm)
l−1(τ))) + dim(span(V(pm)
l(τ))) + dim(span( ˜
V\m))
= dim(span(V(pm)
l(τ))) + dim(span(V(pm)
l−1(τ)∪˜
V\m)).
94 Network Coding in Lossy Networks
Hence it follows that
d≤dim(span(V0\V(pm)
l−1(τ))) + dim(span(V(pm)
l(τ)))
+ dim(span(V(pm)
l−1(τ)∪˜
V\m))
−dim(span(V(pm)
l−1(τ)∪V(pm)
l(τ)∪˜
V\m))
≤dim(span(V0\V(pm)
l−1(τ))) + dim(span(V(pm)
l(τ)))
≤ |V0\V(pm)
l−1(τ)|+|V(pm)
l(τ)|
=|V0| − |V(pm)
l−1(τ)|+|V(pm)
l(τ)|,
which yields
d− |V0| ≤ |V(pm)
l(τ)| − |V(pm)
l−1(τ)| ≤ −ρ.
Therefore, it follows that
Pr(β /∈span(V(pm)
l(τ)∪˜
V\m)) ≥q|V0|−qd
q|V0|= 1 −qd−|V0|≥1−q−µ.
We see then that, if we consider only those packets such that Pu=m,
the conditions that govern the propagation of innovative packets are
exactly those of an Lm-link tandem network, which we dealt with in
Section 4.2.1.2. By recalling the distribution of Pu, it follows that the
propagation of innovative packets along path pmbehaves like an Lm-link
tandem network with average arrival rate Rmon every link. Since we
have assumed nothing special about m, this statement applies for all
m= 1,2,...,M.
Take K=⌈(1 −q−µ)∆RcR/(1 + ε)⌉, where 0 < Rc<1. Then, by
equation (4.21),
lim
K→∞
|W(pm)(∆)|
⌊K(1 + ε)⌋>Rm
R.
Hence
lim
K→∞
| ∪M
m=1 W(pm)(∆)|
⌊K(1 + ε)⌋=
M
X
m=1
|W(pm)(∆)|
⌊K(1 + ε)⌋>
M
X
m=1
Rm
R= 1.
As before, the rate can be made arbitrarily close to Rby varying µ,Rc,
and ε.
4.2.2 Multicast connections
The result for multicast connections is, in fact, a straightforward exten-
sion of that for unicast connections. In this case, rather than a single
4.2 Coding theorems 95
sink t, we have a set of sinks T. As in the framework of static broad-
casting (see [127, 128]), we allow sink nodes to operate at different rates.
We suppose that sink t∈Twishes to achieve rate arbitrarily close to
Rt, i.e., to recover the Kmessage packets, sink twishes to wait for a
time ∆tthat is only marginally greater than K/Rt. We further suppose
that
Rt≤min
Q∈Q(s,t)
X
(i,J)∈Γ+(Q)X
K6⊂Q
ziJK
for all t∈T. Therefore, by the max-flow/min-cut theorem, there exists,
for each t∈T, a flow vector x(t)satisfying
X
{j|(i,J)∈A} X
j∈J
x(t)
iJj −X
{j|(j,I)∈A,i∈I}
x(t)
jIi =
Rif i=s,
−Rif i=t,
0 otherwise,
for all i∈ N ,
X
j∈K
x(t)
iJj ≤X
{L⊂J|L∩K6=∅}
ziJL
for all (i, J )∈ A and K⊂J, and x(t)
iJj ≥0 for all (i, J )∈ A and j∈J.
For each flow vector x(t), we go through the same argument as that
for a unicast connection, and we find that the probability of error at
every sink node can be made arbitrarily small by taking Ksufficiently
large.
We summarize our results with the following theorem statement.
Theorem 4.1 Consider the coding subgraph z. The random linear net-
work coding scheme described in Section 4.1 is capacity-achieving for
single connections in z, i.e., for Ksufficiently large, it can achieve, with
arbitrarily small error probability, a connection from source node sto
sink nodes in the set Tat rate arbitrarily close to Rtpackets per unit
time for each t∈Tif
Rt≤min
Q∈Q(s,t)
X
(i,J)∈Γ+(Q)X
K6⊂Q
ziJK
for all t∈T.
Remark. The capacity region is determined solely by the average rates
{ziJK }at which packets are received. Thus, the packet injection and
96 Network Coding in Lossy Networks
loss processes, which give rise to the packet reception processes, can in
fact take any distribution, exhibiting arbitrary correlations, as long as
these average rates exist.
4.3 Error exponents for Poisson traffic with i.i.d. losses
We now look at the rate of decay of the probability of error pein the
coding delay ∆. In contrast to traditional error exponents where coding
delay is measured in symbols, we measure coding delay in time units—
time τ= ∆ is the time at which the sink nodes attempt to decode the
message packets. The two methods of measuring delay are essentially
equivalent when packets arrive in regular, deterministic intervals.
We specialize to the case of Poisson traffic with i.i.d. losses. Thus, the
process AiJ K is a Poisson process with rate ziJK . Consider the unicast
case for now, and suppose we wish to establish a connection of rate R.
Let Cbe the supremum of all asymptotically-achievable rates.
We begin by deriving an upper bound on the probability of error. To
this end, we take a flow vector xfrom sto tof size Cand, following the
development in Section 4.2, develop a queueing network from it that de-
scribes the propagation of innovative packets for a given innovation order
µ. This queueing network now becomes a Jackson network. Moreover,
as a consequence of Burke’s theorem (see, e.g., [79, Section 2.1]) and
the fact that the queueing network is acyclic, the arrival and departure
processes at all stations are Poisson in steady-state.
Let Ψt(m) be the arrival time of the mth innovative packet at t, and
let C′:= (1 −q−µ)C. When the queueing network is in steady-state,
the arrival of innovative packets at tis described by a Poisson process
of rate C′. Hence we have
lim
m→∞
1
mlog E[exp(θΨt(m))] = log C′
C′−θ(4.24)
for θ < C′[14, 113]. If an error occurs, then fewer than ⌈R∆⌉innovative
packets are received by tby time τ= ∆, which is equivalent to saying
that Ψt(⌈R∆⌉)>∆. Therefore,
pe≤Pr(Ψt(⌈R∆⌉)>∆),
and, using the Chernoff bound, we obtain
pe≤min
0≤θ<C′exp (−θ∆ + log E[exp(θΨt(⌈R∆⌉))]) .
Let εbe a positive real number. Then using equation (4.24) we obtain,
4.3 Error exponents for Poisson traffic with i.i.d. losses 97
for ∆ sufficiently large,
pe≤min
0≤θ<C′exp −θ∆ + R∆log C′
C′−θ+ε
= exp(−∆(C′−R−Rlog(C′/R)) + R∆ε).
Hence, we conclude that
lim
∆→∞
−log pe
∆≥C′−R−Rlog(C′/R).(4.25)
For the lower bound, we examine a cut whose flow capacity is C. We
take one such cut and denote it by Q∗. It is clear that, if fewer than
⌈R∆⌉distinct packets are received across Q∗in time τ= ∆, then an
error occurs. The arrival of distinct packets across Q∗is described by a
Poisson process of rate C. Thus we have
pe≥exp(−C∆)
⌈R∆⌉−1
X
l=0
(C∆)l
l!
≥exp(−C∆)(C∆)⌈R∆⌉−1
Γ(⌈R∆⌉),
and, using Stirling’s formula, we obtain
lim
∆→∞
−log pe
∆≤C−R−Rlog(C/R).(4.26)
Since (4.25) holds for all positive integers µ, we conclude from (4.25)
and (4.26) that
lim
∆→∞
−log pe
∆=C−R−Rlog(C/R).(4.27)
Equation (4.27) defines the asymptotic rate of decay of the proba-
bility of error in the coding delay ∆. This asymptotic rate of decay
is determined entirely by Rand C. Thus, for a packet network with
Poisson traffic and i.i.d. losses employing random linear network coding
as described in Section 4.1, the flow capacity Cof the minimum cut of
the network is essentially the sole figure of merit of importance in deter-
mining the effectiveness of random linear network coding for large, but
finite, coding delay. Hence, in deciding how to inject packets to support
the desired connection, a sensible approach is to reduce our attention
to this figure of merit, which is indeed the approach that we take in
Chapter 5.
Extending the result from unicast connections to multicast connec-
tions is straightforward—we simply obtain (4.27) for each sink.
98 Network Coding in Lossy Networks
4.4 Notes and further reading
Network coding for lossy networks has been looked at in [51, 94, 80, 98,
32, 136]. In [51, 32], a capacity result is established; in [80], a capacity
result is established for the case where no side-information is placed in
packet headers, and a code construction based on maximum distance
separable (mds) codes is proposed; in [94, 98, 136], the use of random
linear network coding in lossy networks is examined. The exposition in
this chapter is derived from [93, 94, 98].
Random linear network coding originates from [62, 27, 66], which deal
with lossless networks. In [111, 99, 103], some variations to random lin-
ear network coding, as described in Section 4.1, are proposed. In [99],
a variation that reduces memory usage at intermediate nodes is exam-
ined, while, in [111, 103], variations that reduce encoding and decoding
complexity are examined. One of the schemes proposed in [103] achieves
linear encoding and decoding complexity.
5
Subgraph Selection
In the previous two chapters, we assumed that a coding subgraph spec-
ifying the times and locations of packet injections was given. We dealt
only with half the problem of establishing connections in coded packet
networks—the coding half. This chapter deals with the other half: sub-
graph selection.
Subgraph selection, which is the problem of determining the coding
subgraph to use, is the coded networks analog of the joint problems
of routing and scheduling in conventional, routed networks. Subgraph
selection and coding are very different problems, and the techniques
used in the this chapter differ significantly from those in the previous
two chapters. In particular, while the previous two chapters generally
used techniques from information theory and coding theory, this chapter
generally uses techniques from networking theory.
Subgraph selection is essentially a problem of network resource allo-
cation: We have a limited resource (packet injections) that we wish to
allocate to coded packets in such as way as to achieve certain commu-
nication objectives. We propose a number of solutions to the problem,
and we divide these solutions into two categories: flow-based approaches
(Section 5.1) and queue-length-based approaches (Section 5.2). In flow-
based approaches, we assume that the communication objective is to
establish a set of (unicast or multicast) connections at certain, given
flow rates while, in queue-length-based approaches, we suppose that the
flow rates, though existent, are not necessarily known, and we select
coding subgraphs using the state of packet queues.
As in the previous two chapters, we deal primarily with intra-session
coding—coding confined to a single session or connection. This allows
us to make use the various results that we have established for network
coding in a single session. Unfortunately, intra-session coding is sub-
99
100 Subgraph Selection
optimal. Recall the modified butterfly network (Figure 1.2) and mod-
ified wireless butterfly network (Figure 1.4). In both these examples,
a gain was achieved by using inter-session coding—coding across two
or more independent sessions. There are far fewer results concerning
inter-session coding than those concerning intra-session coding. One
thing that is known, though, is linear codes do not suffice in general
to achieve the inter-session coding capacity of a coding subgraph [38].
Since no non-linear network codes that seem practicable have yet been
found, we are therefore forced to find suboptimal approaches to linear
inter-session coding, or to simply use intra-session coding. We discuss
subgraph selection techniques for both intra-session coding and subop-
timal inter-session coding in this chapter.
We place particular emphasis on subgraph selection techniques that
can computed in a distributed manner, with each node making com-
putations based only on local knowledge and knowledge acquired from
information exchanges. Such distributed algorithms are inspired by ex-
isting, successful distributed algorithms in networking, such as the dis-
tributed Bellman-Ford algorithm (see, e.g., [13, Section 5.2]), which is
used to find routes in routed packet networks. In general, distributed
subgraph selection techniques currently exist only in cases where arcs
essentially behave independently and the capacities of separate arcs are
not coupled.
5.1 Flow-based approaches
We discuss flow-based approaches under the assumption that there is
a cost that we wish to minimize. This cost, which is a function of the
coding subgraph z, reflects some notion of network efficiency (we could
have, for example, an energy cost, a congestion cost, or even a mone-
tary cost), and it allows us to favor particular subgraphs in the class of
subgraphs that are capable of establishing the desired connections. A
cost-minimization objective is certainly not the only possible (through-
put maximization, for example, is another possibility), but it is very
general, and much of the following discussion applies to other objec-
tives also. Let fbe the cost function. We assume, for tractability and
simplicity, that fis convex.
We first discuss intra-session coding. For intra-session coding, we
formulate the problem and discuss methods for its solution then, in
Section 5.1.1.6, we consider applying these methods for communication
5.1 Flow-based approaches 101
over wireless networks, and we compare their performance to existing
methods. In Section 5.1.3, we discuss inter-session coding.
5.1.1 Intra-session coding
5.1.1.1 Problem formulation
We specify a multicast connection with a triplet (s, T , {Rt}t∈T), where
sis the source of the connection, Tis the set of sinks, and {Rt}t∈Tis
the set of rates to the sinks (see Section 4.2.2). Suppose we wish to
establish Cmulticast connections, (s1, T1,{Rt,1}),...,(sC, TC,{Rt,C }).
Using Theorem 4.1 and the max-flow/min-cut theorem, we see that sub-
graph selection in a lossy network with random linear network coding in
each session can be phrased as the following mathematical programming
problem:
minimize f(z)
subject to z∈Z,
C
X
c=1
y(c)
iJK ≤ziJ K ,∀(i, J )∈ A,K⊂J,
X
j∈K
x(t,c)
iJj ≤X
{L⊂J|L∩K6=∅}
y(c)
iJL ,
∀(i, J )∈ A,K⊂J,t∈Tc,c= 1,...,C,
x(t,c)∈F(t,c),∀t∈Tc,c= 1,...,C,
(5.1)
where x(t,c)is the vector consisting of x(t,c)
iJj , (i, J )∈ A,j∈J, and F(t,c)
is the bounded polyhedron of points x(t,c)satisfying the conservation of
flow constraints
X
{J|(i,J)∈A} X
j∈J
x(t,c)
iJj −X
{j|(j,I)∈A,i∈I}
x(t,c)
jIi =
Rt,c if i=sc,
−Rt,c if i=t,
0 otherwise,
∀i∈ N ,
and non-negativity constraints
x(t,c)
iJj ≥0,∀(i, J )∈ A,j∈J.
In this formulation, y(c)
iJK represents the average rate of packets that are
injected on hyperarc (i, J ) and received by exactly the set of nodes K
102 Subgraph Selection
(which occurs with average rate ziJK ) and that are allocated to connec-
tion c.
For simplicity, let us consider the case where C= 1. The extension
to C > 1 is conceptually straightforward and, moreover, the case where
C= 1 is interesting in its own right: whenever each multicast group
has a selfish cost objective, or when the network sets arc weights to
meet its objective or enforce certain policies and each multicast group
is subject to a minimum-weight objective, we wish to establish single
efficient multicast connections.
Let
biJK := P{L⊂J|L∩K6=∅} ziJ L
ziJ
,
which is the fraction of packets injected on hyperarc (i, J) that are re-
ceived by a set of nodes that intersects K. Problem (5.1) is now
minimize f(z)
subject to z∈Z,
X
j∈K
x(t)
iJj ≤ziJ biJ K ,∀(i, J )∈ A,K⊂J,t∈T,
x(t)∈F(t),∀t∈T .
(5.2)
In the lossless case, problem (5.2) simplifies to the following problem:
minimize f(z)
subject to z∈Z,
X
j∈J
x(t)
iJj ≤ziJ ,∀(i, J )∈ A,t∈T ,
x(t)∈F(t),∀t∈T .
(5.3)
As an example, consider the network depicted in Figure 5.1, which
consists only of point-to-point arcs. Suppose that the network is lossless,
that we wish to achieve multicast of unit rate from sto two sinks, t1and
t2, and that we have Z= [0,1]|A| and f(z) = P(i,j)∈A zij . An optimal
solution to problem (5.3) is shown in the figure. We have flows x(1) and
x(2) of unit size from sto t1and t2, respectively, and, for each arc (i, j),
zij = max(x(1)
ijj , x(2)
ijj ), as we expect from the optimization. For a simple
arc (i, j), it is unnecessary to write x(1)
ijj for the component of flow x(1)
on the arc; we can simply write x(1)
ij , and we shall do so as appropriate.
Under this abbreviated notation, we have zij = max(x(1)
ij , x(2)
ij ).
5.1 Flow-based approaches 103
t1
t2
s
(1/2,0,1/2)
(1/2,0,1/2)
(1/2,1/2,0)
(1/2,1/2,1/2)
(1/2,1/2,1/2)
(1/2,0,1/2)
(1/2,1/2,1/2)
(1/2,1/2,0)
(1/2,1/2,0)
Fig. 5.1. A network of lossless point-to-point arcs with multicast from sto
T={t1, t2}. Each arc is marked with the triple (zij, x(1)
ij , x(2)
ij ). Reprinted
with permission from [101].
s
t1
t2
(0,1/2)
(1/2,0)
1/2
1/2
(1/2,1/2)
(1/2,1/2)
1/2
(1/2,0)
(0,1/2)
1
(1/2,0)
(0,1/2)
Fig. 5.2. A network of lossless broadcast arcs with multicast from sto T=
{t1, t2}. Each hyperarc is marked with ziJ at its start and the pair (x(1)
iJj , x(2)
iJj )
at its ends.
The same multicast problem in a routed packet network would entail
minimizing the number of arcs used to form a tree that is rooted at
sand that reaches t1and t2—in other words, solving the Steiner tree
problem on directed graphs [115]. The Steiner tree problem on directed
graphs is well-known to be np-complete, but solving problem (5.3) is
not. In this case, problem (5.3) is in fact a linear optimization problem.
It is a linear optimization problem that can be thought of as a fractional
104 Subgraph Selection
relaxation of the Steiner tree problem [154]. This example illustrates
one of the attractive features of the coded approach: it allows us avoid
an np-complete problem and instead solve its fractional relaxation.
For an example with broadcast arcs, consider the network depicted in
Figure 5.2. Suppose again that the network is lossless, that we wish to
achieve multicast of unit rate from sto two sinks, t1and t2, and that
we have Z= [0,1]|A| and f(z) = P(i,J)∈A ziJ . An optimal solution to
problem (5.3) is shown in the figure. We still have flows x(1) and x(2)
of unit size from sto t1and t2, respectively, but now, for each hyperarc
(i, J ), we determine ziJ from the various flows passing through hyperarc
(i, J ), each destined toward a single node jin J, and the optimization
gives ziJ = max(Pj∈Jx(1)
iJj ,Pj∈Jx(2)
iJj ).
Neither problem (5.2) nor (5.3) as it stands is easy to solve. But the
problems are very general. Their complexities improve if we assume
that the cost function is separable and possibly even linear, i.e., if we
suppose f(z) = P(i,J )∈AfiJ (ziJ ), where fiJ is a convex or linear func-
tion, which is a very reasonable assumption in many practical situations.
For example, packet latency is usually assessed with a separable, convex
cost function and energy, monetary cost, and total weight are usually
assessed with separable, linear cost functions.
The complexities of problems (5.2) and (5.3) also improve if we make
some assumptions on the form of the constraint set Z, which is the case
in most practical situations.
A particular simplification applies if we assume that, when nodes
transmit in a lossless network, they reach all nodes in a certain re-
gion, with cost increasing as this region is expanded. This applies,
for example, if we are interested in minimizing energy consumption,
and the region in which a packet is reliably received expands as we ex-
pend more energy in its transmission. More precisely, suppose that we
have separable cost, so f(z) = P(i,J )∈A fiJ (ziJ ). Suppose further that
each node ihas Mioutgoing hyperarcs (i, J (i)
1),(i, J(i)
2),...,(i, J(i)
Mi) with
J(i)
1(J(i)
2(··· (J(i)
Mi. (We assume that there are no identical arcs,
as duplicate arcs can effectively be treated as a single arc.) Then, we
assume that fiJ(i)
1(ζ)< fiJ(i)
2(ζ)<···< fiJ(i)
Mi
(ζ) for all ζ≥0 and nodes
i.
Let us introduce, for (i, j )∈ A′:= {(i, j)|(i, J )∈A, J ∋j}, the
5.1 Flow-based approaches 105
variables
ˆx(t)
ij :=
Mi
X
m=m(i,j)
x(t)
iJ(i)
mj,
where m(i, j) is the unique msuch that j∈J(i)
m\J(i)
m−1(we define
J(i)
0:= ∅for all i∈ N for convenience). Now, problem (5.3) can be
reformulated as the following problem, which has substantially fewer
variables:
minimize X
(i,J)∈A
fiJ (ziJ )
subject to z∈Z
X
k∈J(i)
Mi\J(i)
m−1
ˆx(t)
ik ≤
Mi
X
n=m
ziJ(i)
n,∀i∈ N ,m= 1,...,Mi,t∈T ,
ˆx(t)∈ˆ
F(t),∀t∈T ,
(5.4)
where ˆ
F(t)is the bounded polyhedron of points ˆx(t)satisfying the con-
servation of flow constraints
X
{j|(i,j)∈A′}
ˆx(t)
ij −X
{j|(j,i)∈A′}
ˆx(t)
ji =
Rtif i=s,
−Rtif i=t,
0 otherwise,
∀i∈N,
and non-negativity constraints
0≤ˆx(t)
ij ,∀(i, j)∈ A′.
Proposition 5.1 Suppose that f(z) = P(i,J)∈A fiJ (ziJ )and that fiJ(i)
1(ζ)<
fiJ(i)
2(ζ)<··· < fiJ(i)
Mi
(ζ)for all ζ≥0and i∈ N . Then problem (5.3)
and problem (5.4) are equivalent in the sense that they have the same
optimal cost and zis part of an optimal solution for (5.3) if and only if
it is part of an optimal solution for (5.4).
Proof Suppose (x, z) is a feasible solution to problem (5.3). Then, for
106 Subgraph Selection
all (i, j)∈ A′and t∈T,
Mi
X
m=m(i,j)
ziJ(i)
m≥
Mi
X
m=m(i,j)X
k∈J(i)
m
x(t)
iJ(i)
mk
=X
k∈J(i)
Mi
Mi
X
m=max(m(i,j),m(i,k))
x(t)
iJ(i)
mk
≥X
k∈J(i)
Mi\J(i)
m(i,j)−1
Mi
X
m=max(m(i,j),m(i,k))
x(t)
iJ(i)
mk
=X
k∈J(i)
Mi\J(i)
m(i,j)−1
Mi
X
m=m(i,k)
x(t)
iJ(i)
mk
=X
k∈J(i)
Mi\J(i)
m(i,j)−1
ˆx(t)
ik .
Hence (ˆx, z ) is a feasible solution of problem (5.4) with the same cost.
Now suppose (ˆx, z) is an optimal solution of problem (5.4). Since fiJ (i)
1(ζ)<
fiJ(i)
2(ζ)<···< fiJ (i)
Mi
(ζ) for all ζ≥0 and i∈ N by assumption, it fol-
lows that, for all i∈ N , the sequence ziJ(i)
1, ziJ(i)
2,...,ziJ (i)
Mi
is given
recursively, starting from m=Mi, by
ziJ(i)
m= max
t∈T
X
k∈J(i)
Mi\J(i)
m−1
ˆx(t)
ik
−
Mi
X
m′=m+1
ziJ(i)
m′.
Hence ziJ(i)
m≥0 for all i∈ N and m= 1,2,...,Mi. We then set,
starting from m=Miand j∈J(i)
Mi,
x(t)
iJ(i)
mj:= min
ˆx(t)
ij −
Mi
X
l=m+1
xiJ(i)
lj, ziJ(i)
m−X
k∈J(i)
Mi\J(i)
m(i,j)
x(t)
iJ(i)
mk
.
It is now not difficult to see that (x, z) is a feasible solution of problem
(5.3) with the same cost.
Therefore, the optimal costs of problems (5.3) and (5.4) are the same
and, since the objective functions for the two problems are the same, z
is part of an optimal solution for problem (5.3) if and only if it is part
of an optimal solution for problem (5.4).
5.1 Flow-based approaches 107
2
1 3
Fig. 5.3. The slotted Aloha relay channel. Reprinted with permission from
[93].
5.1.1.2 Example: Slotted Aloha relay channel
This example, which we refer to as the slotted Aloha relay channel, re-
lates to multi-hop wireless networks. One of most important issues in
multi-hop wireless networks is medium access, i.e., determining how ra-
dio nodes share the wireless medium. A simple, yet popular, method
for medium access control is slotted Aloha (see, e.g., [13, Section 4.2]),
where nodes with packets to send follow simple random rules to deter-
mine when they transmit. In this example, we consider a multi-hop
wireless network using slotted Aloha for medium access control.
We suppose that the network has the simple topology shown in Fig-
ure 5.3 and that, in this network, we wish to establish a single unicast
connection of rate Rfrom node 1 to node 3. The random rule we take
for transmission is that the two transmitting nodes, node 1 and node
2, each transmit packets independently in a given time slot with some
fixed probability. In coded packet networks, nodes are never “unback-
logged” as they are in regular, routed slotted Aloha networks—nodes
can transmit coded packets whenever they are given the opportunity.
Hence z1(23), the rate of packet injection on hyperarc (1,{2,3}), is the
probability that node 1 transmits a packet in a given time slot, and
likewise z23, the rate of packet injection on hyperarc (2,3), is the prob-
ability that node 2 transmits a packet in a given time slot. Therefore,
Z= [0,1]2, i.e., 0 ≤z1(23) ≤1 and 0 ≤z23 ≤1.
If node 1 transmits a packet and node 2 does not, then the packet is
received at node 2 with probability p1(23)2 , at node 3 with probability
p1(23)3, and at both nodes 2 and 3 with probability p1(23)(23) (it is lost
entirely with probability 1 −p1(23)2 −p1(23)3 −p1(23)(23)). If node 2
transmits a packet and node 1 does not, then the packet is received at
node 3 with probability p233 (it is lost entirely with probability 1−p233 ).
108 Subgraph Selection
If both nodes 1 and 2 each transmit a packet, then the packets collide
and neither of the packets is received successfully anywhere.
It is possible that simultaneous transmission does not necessarily re-
sult in collision, with one or more packets being received. This phe-
nomenon is referred to as multipacket reception capability [48] and is
decided by lower-layer implementation details. In this example, however,
we simply assume that simultaneous transmission results in collision.
Hence, we have
z1(23)2 =z1(23)(1 −z23 )p1(23)2,(5.5)
z1(23)3 =z1(23)(1 −z23 )p1(23)3,(5.6)
z1(23)(23) =z1(23)(1 −z23)p1(23)(23),(5.7)
and
z233 = (1 −z1(23) )z23p233 .(5.8)
We suppose that our objective is to set up the desired connection while
minimizing the total number of packet transmissions for each message
packet, perhaps for the sake of energy conservation or conservation of
the wireless medium (to allow it to be used for other purposes, such as
other connections). Therefore
f(z1(23), z23) = z1(23) +z23.
The slotted Aloha relay channel is very similar to the relay channel
introduced by van der Meulen [134], and determining the capacity of the
latter is one of the famous, long-standing, open problems of information
theory. The slotted Aloha relay channel is related to the relay channel
(hence its name), but different. While the relay channel relates to the
physical layer, we are concerned with higher layers, and our problem is
soluble.
The relevant optimization problem to solve in this case is (5.2), and
it reduces to
minimize z1(23) +z23
subject to 0 ≤z1(23), z23 ≤1,
R≤z1(23)(1 −z23 )(p1(23)2 +p1(23)3 +p1(23)(23)),
R≤z1(23)(1 −z23 )(p1(23)3 +p1(23)(23)) + (1 −z1(23) )z23p233 .
Let us assume some values for the parameters of the problem and work
through it. Let R:= 1/8, p1(23)2 := 9/16, p1(23)3 := 1/16, p1(23)(23) :=
5.1 Flow-based approaches 109
1
4z1(23)(1 −z23 ) + 3
4(1 −z1(23))z23 =1
8
13
16 z1(23)(1 −z23) = 1
8
Z0
0.4
0.8
1
0 0.2 0.4 0.6 0.8
z23
z1(23)
1
0
0.2
0.6
Fig. 5.4. Feasible set of problem (5.9). Reprinted with permission from [93].
3/16, and p233 := 3/4. Then the optimization problem we have is
minimize z1(23) +z23
subject to 0 ≤z1(23), z23 ≤1,
1
8≤13
16z1(23)(1 −z23 ),
1
8≤1
4z1(23)(1 −z23 ) + 3
4(1 −z1(23))z23.
(5.9)
The feasible set of this problem is shown in Figure 5.4. It is the shaded
region labeled Z0. By inspection, the optimal solution of (5.9) is the
lesser of the two intersections between the curves defined by
13
16z1(23)(1 −z23) = 1
8
and
1
4z1(23)(1 −z23 ) + 3
4(1 −z1(23))z23 =1
8.
We obtain z∗
1(23) ≃0.179 and z∗
23 ≃0.141.
The problem we have just solved is by no means trivial. We have taken
110 Subgraph Selection
a wireless packet network subject to losses that are determined by a
complicated set of conditions—including medium contention—and found
a way of establishing a given unicast connection of fixed throughput
using the minimum number of transmissions per message packet. The
solution is that node 1 transmits a packet every time slot with probability
0.179, and node 2 transmits a packet every time slot independently with
probability 0.141. Whenever either node transmits a packet, they follow
the coding scheme of Section 4.1.
The network we dealt with was, unfortunately, only a small one, and
the solution method we used will not straightforwardly scale to larger
problems. But the solution method is conceptually simple, and there
are cases where the solution to large problems is computable—and com-
putable in a distributed manner. We deal with this topic next.
5.1.1.3 Distributed algorithms
In many cases, the optimization problems (5.2), (5.3), and (5.4) are con-
vex or linear problems and their solutions can, in theory, be computed.
For practical network applications, however, it is often important that
solutions can be computed in a distributed manner, with each node mak-
ing computations based only on local knowledge and knowledge acquired
from information exchanges. Thus, we seek distributed algorithms to
solve optimization problems (5.2), (5.3), and (5.4), which, when paired
with the random linear coding scheme of the previous chapter, yields a
distributed approach to efficient operation. The algorithms we propose
will generally take some time to converge to an optimal solution, but
it is not necessary to wait until the algorithms have converged before
transmission—we can apply the coding scheme to the coding subgraph
we have at any time, optimal or otherwise, and continue doing so while
it converges. Such an approach is robust to dynamics such as changes
in network topology that cause the optimal solution to change, because
the algorithms will simply converge toward the changing optimum.
To this end, we simplify the problem by assuming that the objective
function is of the form f(z) = P(i,J )∈A fiJ (ziJ ), where fiJ is a mono-
tonically increasing, convex function, and that, as ziJ is varied, ziJK /ziJ
is constant for all K⊂J. Therefore, biJ K is a constant for all (i, J )∈ A
and K⊂J. We also drop the constraint set Z, noting that separable
constraints, at least, can be handled by making fiJ approach infinity
as ziJ approaches its upper constraint. These assumptions apply if, at
least from the perspective of the connection we wish to establish, arcs
5.1 Flow-based approaches 111
essentially behave independently and the capacities of separate arcs are
not coupled.
With these assumptions, problem (5.2) becomes
minimize X
(i,J)∈A
fiJ (ziJ )
subject to X
j∈K
x(t)
iJj ≤ziJ biJ K ,∀(i, J )∈ A,K⊂J,t∈T ,
x(t)∈F(t),∀t∈T .
(5.10)
Since the fiJ are monotonically increasing, the constraint
X
j∈K
x(t)
iJj ≤ziJ biJ K ,∀(i, J )∈ A,K⊂J,t∈T(5.11)
gives
ziJ = max
K⊂J,t∈T(Pj∈Kx(t)
iJj
biJK ).(5.12)
Expression (5.12) is, unfortunately, not very useful for algorithm design
because the max function is difficult to deal with, largely as a result
of its not being differentiable everywhere. One way to overcome this
difficulty is to approximate ziJ by replacing the max in (5.12) with an
lm-norm (see [35]), i.e., to approximate ziJ with z′
iJ , where
z′
iJ :=
X
K⊂J,t∈T Pj∈Kx(t)
iJj
biJK !m
1/m
.
The approximation becomes exact as m→ ∞. Moreover, since z′
iJ ≥ziJ
for all m > 0, the coding subgraph z′admits the desired connection for
any feasible solution.
Now the relevant optimization problem is
minimize X
(i,J)∈A
fiJ (z′
iJ )
subject to x(t)∈F(t),∀t∈T ,
which is no more than a convex multicommodity flow problem. There
are many algorithms for convex multicommodity flow problems (see [109]
for a survey), some of which (e.g., the algorithms in [9, 12]) are well-
suited for distributed implementation. The primal-dual approach to
internet congestion control (see [129, Section 3.4]) can also be used to
112 Subgraph Selection
solve convex multicommodity flow problems in a distributed manner,
and we examine this method in Section 5.1.1.4.
There exist, therefore, numerous distributed algorithms for the sub-
graph selection problem—or, at least, for an approximation of the prob-
lem. What about distributed algorithms for the true problem? One clear
tactic for finding such algorithms is to eliminate constraint (5.11) using
Lagrange multipliers. Following this tactic, we obtain a distributed algo-
rithm that we call the subgradient method. We describe the subgradient
method in Section 5.1.1.5.
5.1.1.4 Primal-dual method
For the primal-dual method, we assume that the cost functions fiJ are
strictly convex and differentiable. Hence there is a unique optimal solu-
tion to problem (5.10). We present the algorithm for the lossless case,
with the understanding that it can be straightforwardly extended to the
lossy case. Thus, the optimization problem we address is
minimize X
(i,J)∈A
fiJ (z′
iJ )
subject to x(t)∈F(t),∀t∈T ,
(5.13)
where
z′
iJ :=
X
t∈T
X
j∈J
x(t)
iJj
m
1/m
.
Let (y)+
adenote the following function of y:
(y)+
a=(yif a > 0,
max{y, 0}if a≤0.
To solve problem (5.13) in a distributed fashion, we introduce ad-
ditional variables pand λand consider varying x,p, and λin time τ
according to the following time derivatives:
˙x(t)
iJj =−k(t)
iJj (x(t)
iJj ) ∂fiJ (z′
iJ )
∂x(t)
iJj
+q(t)
ij −λ(t)
iJj !,(5.14)
˙p(t)
i=h(t)
i(p(t)
i)(y(t)
i−σ(t)
i),(5.15)
˙
λ(t)
iJj =m(t)
iJj (λ(t)
iJj )−x(t)
iJj +
λ(t)
iJj
,(5.16)
5.1 Flow-based approaches 113
where
q(t)
ij := p(t)
i−p(t)
j,
y(t)
i:= X
{J|(i,J)∈A} X
j∈J
x(t)
iJj −X
{j|(j,I)∈A,i∈I}
x(t)
jIi,
and k(t)
iJj (x(t)
iJj )>0, h(t)
i(p(t)
i)>0, and m(t)
iJj (λ(t)
iJj )>0 are non-decreasing
continuous functions of x(t)
iJj ,p(t)
i, and λ(t)
iJj respectively.
Proposition 5.2 The algorithm specified by Equations (5.14)–(5.16) is
globally, asymptotically stable.
Proof We prove the stability of the primal-dual algorithm by using the
theory of Lyapunov stability (see, e.g., [129, Section 3.10]). This proof
is based on the proof of Theorem 3.7 of [129].
The Lagrangian for problem (5.13) is as follows:
L(x, p, λ) = X
(i,J)∈A
fiJ (z′
iJ )
+X
t∈T
X
i∈N
p(t)
i
X
{J|(i,J)∈A} X
j∈J
x(t)
iJj −X
{j|(j,I)∈A,i∈I}
x(t)
jIi −σ(t)
i
−X
(i,J)∈A X
j∈J
λ(t)
iJj x(t)
iJj
,(5.17)
where
σ(t)
i=
Rtif i=s,
−Rtif i=t,
0 otherwise.
Since the objective function of problem (5.13) is strictly convex, it has a
unique minimizing solution, say ˆx, and Lagrange multipliers, say ˆpand
ˆ
λ, which satisfy the following Karush-Kuhn-Tucker conditions:
∂L(ˆx, ˆp, ˆ
λ)
∂x(t)
iJj
= ∂fiJ (z′
iJ )
∂x(t)
iJj
+ˆp(t)
i−ˆp(t)
j−ˆ
λ(t)
iJj != 0,
∀(i, J )∈ A,j∈J,t∈T , (5.18)
114 Subgraph Selection
X
{J|(i,J)∈A} X
j∈J
ˆx(t)
iJj −X
{j|(j,I)∈A,i∈I}
ˆx(t)
jIi =σ(t)
i,∀i∈ N ,t∈T ,
(5.19)
ˆx(t)
iJj ≥0∀(i, J )∈ A,j∈J,t∈T , (5.20)
ˆ
λ(t)
iJj ≥0∀(i, J )∈ A,j∈J,t∈T , (5.21)
ˆ
λ(t)
iJj ˆx(t)
iJj = 0 ∀(i, J )∈ A,j∈J,t∈T . (5.22)
Using equation (5.17), we see that (ˆx, ˆp, ˆ
λ) is an equilibrium point of
the primal-dual algorithm. We now prove that this point is globally,
asymptotically stable.
Consider the following function as a candidate for the Lyapunov func-
tion:
V(x, p, λ) = X
t∈T
X
(i,J)∈A X
j∈J Zx(t)
iJj
ˆx(t)
iJj
1
k(t)
iJj (σ)(σ−ˆx(t)
iJj )dσ
+Zλ(t)
iJj
ˆ
λ(t)
iJj
1
m(t)
iJj (γ)(γ−ˆ
λ(t)
iJj )dγ!+X
i∈N Zp(t)
i
ˆp(t)
i
1
h(t)
i(β)(β−ˆp(t)
i)dβ).
Note that V(ˆx, ˆp, ˆ
λ) = 0. Since, k(t)
iJj (σ)>0, if x(t)
iJj 6= ˆx(t)
iJj , we have
Zx(t)
iJj
ˆ
x(t)
iJj
1
k(t)
iJj (σ)(σ−ˆx(t)
iJj )dσ > 0.
This argument can be extended to the other terms as well. Thus, when-
ever (x, p, λ)6= (ˆx, ˆp, ˆ
λ), we have V(x, p, λ)>0.
Now,
˙
V=X
t∈T
X
(i,J)∈A X
j∈J−x(t)
iJj +
λ(t)
iJj
(λ(t)
iJj −ˆ
λ(t)
iJj )
− ∂fiJ (z′
iJ )
∂x(t)
iJj
+q(t)
iJj −λ(t)
iJj !·(x(t)
iJj −ˆx(t)
iJj )#
+X
i∈N
(y(t)
i−σ(t)
i)(p(t)
i−ˆp(t)
i)).
Note that
−x(t)
iJj +
λ(t)
iJj
(λ(t)
iJj −ˆ
λ(t)
iJj )≤ −x(t)
iJj (λ(t)
iJj −ˆ
λ(t)
iJj ),
5.1 Flow-based approaches 115
since the inequality is an equality if either x(t)
iJj ≤0 or λ(t)
iJj ≥0; and,
in the case when x(t)
iJj >0 and λ(t)
iJj <0, we have (−x(t)
iJj )+
λ(t)
iJj
= 0 and,
since ˆ
λ(t)
iJj ≥0, −x(t)
iJj (λ(t)
iJj −ˆ
λ(t)
iJj )≥0. Therefore,
˙
V≤X
t∈T
X
(i,J)∈A X
j∈Jh−x(t)
iJj (λ(t)
iJj −ˆ
λ(t)
iJj )
− ∂fiJ (z′
iJ )
∂x(t)
iJj
+q(t)
iJj −λ(t)
iJj !·(x(t)
iJj −ˆx(t)
iJj )#
+X
i∈N
(y(t)
i−σ(t)
i)(p(t)
i−ˆp(t)
i))
= (ˆq−q)′(x−ˆx) + ( ˆp−p)′(y−ˆy)
+X
t∈T
X
(i,J)∈A X
j∈Jh−ˆx(t)
iJj (λ(t)
iJj −ˆ
λ(t)
iJj )
− ∂fiJ (z′
iJ )
∂x(t)
iJj
+ ˆq(t)
iJj −ˆ
λ(t)
iJj !·(x(t)
iJj −ˆx(t)
iJj )#
+X
i∈N
(ˆy(t)
i−σ(t)
i)(p(t)
i−ˆp(t)
i))
=X
t∈TX
(i,J)∈A X
j∈J ∂fiJ (ˆz′
iJ )
∂ˆx(t)
iJj
−∂fiJ (z′
iJ )
∂x(t)
iJj !(x(t)
iJj −ˆx(t)
iJj )−λ′ˆx,
where the last line follows from Karush-Kuhn-Tucker conditions (5.18)–
(5.22) and the fact that
p′y=X
t∈TX
i∈N
p(t)
i
X
{J|(i,J)∈A} X
j∈J
x(t)
iJj −X
{j|(j,I)∈A,i∈I}
x(t)
jIi
=X
t∈TX
(i,J)∈A X
j∈J
x(t)
iJj (p(t)
i−p(t)
j) = q′x.
Thus, owing to the strict convexity of the functions {fiJ }, we have ˙
V≤
−λ′ˆx, with equality if and only if x= ˆx. So it follows that ˙
V≤0 for all
λ≥0, since ˆx≥0.
If the initial choice of λis such that λ(0) ≥0, we see from the primal-
dual algorithm that λ(τ)≥0. This is true since ˙
λ≥0 whenever λ≤0.
116 Subgraph Selection
Thus, it follows by the theory of Lyapunov stability that the algorithm
is indeed globally, asymptotically stable.
The global, asymptotic stability of the algorithm implies that no mat-
ter what the initial choice of (x, p) is, the primal-dual algorithm will
converge to the unique solution of problem (5.13). We have to choose
λ, however, with non-negative entries as the initial choice. Further,
there is no guarantee that x(τ) yields a feasible solution for any given
τ. Therefore, a start-up time may be required before a feasible solution
is obtained.
The algorithm that we currently have is a continuous time algorithm
and, in practice, an algorithm operating in discrete message exchanges is
required. To discretize the algorithm, we consider time steps n= 0,1,...
and replace the derivatives by differences:
x(t)
iJj [n+ 1] = x(t)
iJj [n]−α(t)
iJj [n] ∂fiJ (z′
iJ [n])
∂x(t)
iJj [n]+q(t)
ij [n]−λ(t)
iJj [n]!,
(5.23)
p(t)
i[n+ 1] = p(t)
i[n] + β(t)
i[n](y(t)
i[n]−σ(t)
i),(5.24)
λ(t)
iJj [n+ 1] = λ(t)
iJj [n] + γ(t)
iJj [n]−x(t)
iJj [n]+
λ(t)
iJj [n],(5.25)
where
q(t)
ij [n] := p(t)
i[n]−p(t)
j[n],
y(t)
i[n] := X
{J|(i,J)∈A} X
j∈J
x(t)
iJj [n]−X
{j|(j,I)∈A,i∈I}
x(t)
jIi[n],
and α(t)
iJj [n]>0, β(t)
i[n]>0, and γ(t)
iJj [n]>0 are step sizes. This dis-
cretized algorithm operates in synchronous rounds, with nodes exchang-
ing information in each round. It is expected that this synchronicity can
be relaxed in practice.
We associate a processor with each node. We assume that the pro-
cessor for node ikeeps track of the variables pi,{xiJj }{J,j|(i,J)∈A,j ∈J},
and {λiJj }{J,j|(i,J )∈A,j∈J}. With such an assignment of variables to pro-
cessors, the algorithm is distributed in the sense that a node exchanges
information only with its neighbors at every iteration of the primal-dual
algorithm. We summarize the primal-dual method in Figure 5.5.
5.1 Flow-based approaches 117
(i) Each node iinitializes pi[0], {xiJj [0]}{J,j|(i,J )∈A,j∈J}, and
{λiJj [0]}{J,j|(i,J )∈A,j∈J}such that λiJ j [0] ≥0 for all (J, j)
such that (i, J)∈ A and j∈J. Each node isends pi[0],
{xiJj [0]}j∈J, and {λiJ j [0]}j∈Jover each outgoing hyperarc
(i, J).
(ii) At the nth iteration, each node icomputes pi[n+ 1],
{xiJj [n+ 1]}{J,j|(i,J )∈A,j∈J}, and {λiJ j [n+ 1]}{J,j|(i,J )∈A,j∈J}
using equations (5.23)–(5.25). Each node isends pi[n+ 1],
{xiJj [n+ 1]}j∈J, and {λiJ j [n+ 1]}j∈Jover each outgoing hy-
perarc (i, J ).
(iii) The current coding subgraph z′[n] is computed. For each node
i, we set
z′
iJ [n] := X
t∈T X
j∈J
x(t)
iJj [n]!m!1/m
for all outgoing hyperarcs (i, J ).
(iv) Steps (ii) and (iii) are repeated until the sequence of coding
subgraphs {z′[n]}converges.
Fig. 5.5. Summary of the primal-dual method.
5.1.1.5 Subgradient method
We present the subgradient method for linear cost functions; with some
modifications, it may be made to apply also to convex ones. Thus, we
assume that the objective function fis of the form
f(z) := X
(i,J)∈A
aiJ ziJ ,
where aiJ >0.
Consider the Lagrangian dual of problem (5.10):
maximize X
t∈T
q(t)(p(t))
subject to X
t∈TX
K⊂J
p(t)
iJK =aiJ ∀(i, J )∈ A,
p(t)
iJK ≥0,∀(i, J )∈ A,K⊂J,t∈T ,
(5.26)
where
q(t)(p(t)) := min
x(t)∈F(t)X
(i,J)∈A X
j∈J
X
{K⊂J|K∋j}
p(t)
iJK
biJK
xiJj .(5.27)
In the lossless case, the dual problem defined by equations (5.26) and
(5.27) simplifies somewhat, and we require only a single dual variable
118 Subgraph Selection
p(t)
iJJ for each hyperarc (i, J ). In the case that relates to optimization
problem (5.4), the dual problem simplifies more still, as there are fewer
primal variables associated with it. Specifically, we obtain, for the La-
grangian dual,
maximize X
t∈T
ˆq(t)(p(t))
subject to X
t∈T
p(t)
iJ(i)
m
=siJ(i)
m,∀i∈ N ,m= 1,...,Mi,
p(t)
iJ ≥0,∀(i, J)∈ A,t∈T ,
(5.28)
where
siJ(i)
m:= aiJ(i)
m−aiJ(i)
m−1
,
and
ˆq(t)(p(t)) := min
ˆx(t)∈ˆ
F(t)X
(i,j)∈A′
m(i,j)
X
m=1
p(t)
iJ(i)
m
ˆx(t)
ij .(5.29)
Note that, by the assumptions of the problem, siJ >0 for all (i, J)∈ A.
In all three cases, the dual problems are very similar, and essentially
the same algorithm can be used to solve them. We present the subgra-
dient method for the case that relates to optimization problem (5.4)—
namely, the primal problem
minimize X
(i,J)∈A
aiJ ziJ
subject to X
k∈J(i)
Mi\J(i)
m−1
ˆx(t)
ik ≤
Mi
X
n=m
ziJ(i)
n,
∀i∈ N ,m= 1,...,Mi,t∈T ,
ˆx(t)∈ˆ
F(t),∀t∈T
(5.30)
with dual (5.28)—with the understanding that straightforward modifi-
cations can be made for the other cases.
We first note that problem (5.29) is, in fact, a shortest path problem,
which admits a simple, asynchronous distributed solution known as the
distributed asynchronous Bellman-Ford algorithm (see, e.g., [13, Section
5.2]).
Now, to solve the dual problem (5.28), we employ subgradient opti-
mization (see, e.g., [10, Section 6.3.1] or [108, Section I.2.4]). We start
with an iterate p[0] in the feasible set of (5.28) and, given an iterate p[n]
5.1 Flow-based approaches 119
for some non-negative integer n, we solve problem (5.29) for each tin T
to obtain x[n]. Let
g(t)
iJ(i)
m
[n] := X
k∈J(i)
Mi\J(i)
m−1
ˆx(t)
ik [n].
We then assign
piJ [n+ 1] := arg min
v∈PiJ X
t∈T
(v(t)−(p(t)
iJ [n] + θ[n]g(t)
iJ [n]))2(5.31)
for each (i, J )∈ A, where PiJ is the |T|-dimensional simplex
PiJ =(vX
t∈T
v(t)=siJ , v ≥0)
and θ[n]>0 is an appropriate step size. In other words, piJ [n+ 1] is set
to be the Euclidean pro jection of piJ [n] + θ[n]giJ [n] onto PiJ .
To perform the projection, we use the following proposition.
Proposition 5.3 Let u:= piJ [n] + θ[n]giJ [n]. Suppose we index the
elements of Tsuch that u(t1)≥u(t2)≥... ≥u(t|T|). Take ˆ
kto be the
smallest ksuch that
1
k siJ −
tk
X
r=1
u(r)!≤ −u(tknn+1)
or set ˆ
k=|T|if no such kexists. Then the projection (5.31) is achieved
by
p(t)
iJ [n+ 1] =
u(t)+siJ −Ptˆ
k
r=1 u(r)
ˆ
kn if t∈ {t1,...,tˆ
k},
0otherwise.
Proof We wish to solve the following problem.
minimize X
t∈T
(v(t)−u(t))2
subject to v∈PiJ .
First, since the objective function and the constraint set PiJ are both
convex, it is straightforward to establish that a necessary and sufficient
condition for global optimality of ˆv(t)in PiJ is
ˆv(t)>0⇒(u(t)−ˆv(t))≥(u(r)−ˆv(r)),∀r∈T(5.32)
120 Subgraph Selection
(see, e.g., [10, Section 2.1]). Suppose we index the elements of Tsuch
that u(t1)≥u(t2)≥... ≥u(t|T|). We then note that there must be an
index kin the set {1,...,|T|} such that v(tl)>0 for l= 1,...,k and
v(tl)= 0 for l > kn + 1, for, if not, then a feasible solution with lower
cost can be obtained by swapping around components of the vector.
Therefore, condition (5.32) implies that there must exist some dsuch
that ˆv(t)=u(t)+dfor all t∈ {t1,...,tkn}and that d≤ −u(t)for all
t∈ {tkn+1,...,t|T|}, which is equivalent to d≤ −u(tk+1 ). Since ˆv(t)is
in the simplex PiJ , it follows that
kd +
tk
X
t=1
u(t)=siJ ,
which gives
d=1
k siJ −
tk
X
t=1
u(t)!.
By taking k=ˆ
k, where ˆ
kis the smallest ksuch that
1
k siJ −
tk
X
r=1
u(r)!≤ −u(tk+1),
(or, if no such kexists, then ˆ
k=|T|), we see that we have
1
ˆ
k−1 siJ −
tk−1
X
t=1
u(t)!>−u(tk),
which can be rearranged to give
d=1
ˆ
k siJ −
tk
X
t=1
u(t)!>−u(tk).
Hence, if v(t)is given by
v(t)=
u(t)+siJ −Ptˆ
k
r=1 u(r)
ˆ
kif t∈ {t1,...,tˆ
k},
0 otherwise,(5.33)
then v(t)is feasible and we see that the optimality condition (5.32) is
satisfied. Note that, since d≤ −u(tk+1), equation (5.33) can also be
written as
v(t)= max
0, u(t)+1
ˆ
k
siJ −
tˆ
k
X
r=1
u(r)
.(5.34)
5.1 Flow-based approaches 121
The disadvantage of subgradient optimization is that, whilst it yields
good approximations of the optimal value of the Lagrangian dual prob-
lem (5.28) after sufficient iteration, it does not necessarily yield a primal
optimal solution. There are, however, methods for recovering primal so-
lutions in subgradient optimization. We employ the following method,
which is due to Sherali and Choi [125].
Let {µl[n]}l=1,...,n be a sequence of convex combination weights for
each non-negative integer n, i.e., Pn
l=1 µl[n] = 1 and µl[n]≥0 for all
l= 1,...,n. Further, let us define
γln := µl[n]
θ[n], l = 1,...,n,n= 0,1,...,
and
∆γmax
n:= max
l=2,...,n{γln −γ(l−1)n}.
Proposition 5.4 If the step sizes {θ[n]}and convex combination weights
{µl[n]}are chosen such that
(i) γln ≥γ(l−1)nfor all l= 2,...,n and n= 0,1,...,
(ii) ∆γmax
n→0as n→ ∞, and
(iii) γ1n→0as n→ ∞ and γnn ≤δfor all n= 0,1,... for some
δ > 0,
then we obtain an optimal solution to the primal problem from any ac-
cumulation point of the sequence of primal iterates {˜x[n]}given by
˜x[n] :=
n
X
l=1
µl[n]ˆx[l], n = 0,1,.... (5.35)
Proof Suppose that the dual feasible solution that the subgradient
method converges to is ¯p. Then, using (5.31), there exists some msuch
that for n≥m
p(t)
iJ [n+ 1] = p(t)
iJ [n] + θ[n]g(t)
iJ [n] + ciJ [n]
for all (i, J )∈ A and t∈Tsuch that ¯p(t)
iJ >0.
Let ˜g[n] := Pn
l=1 µl[n]g[l]. Consider some (i, J )∈ A and t∈T. If
122 Subgraph Selection
¯p(t)
iJ >0, then for n > m we have
˜g(t)
iJ [n] =
m
X
l=1
µl[n]g(t)
iJ [l] +
n
X
l=m+1
µl[n]g(t)
iJ [l]
=
m
X
l=1
µl[n]g(t)
iJ [l] +
n
X
l=m+1
µl[n]
θ[n](p(t)
iJ [n+ 1] −p(t)
iJ [n]−ciJ [n])
=
m
X
l=1
µl[n]g(t)
iJ [l] +
n
X
l=m+1
γln(p(t)
iJ [n+ 1] −p(t)
iJ [n])
−
n
X
l=m+1
γlnciJ [n].
(5.36)
Otherwise, if ¯p(t)
iJ = 0, then from equation (5.34), we have
p(t)
iJ [n+ 1] ≥p(t)
iJ [n] + θ[n]g(t)
iJ [n] + ciJ [n],
so
˜g(t)
iJ [n]≤
m
X
l=1
µl[n]g(t)
iJ [l]+
n
X
l=m+1
γln(p(t)
iJ [n+1]−p(t)
iJ [n])−
n
X
l=m+1
γlnciJ [n].
(5.37)
It is straightforward to see that the sequence of iterates {˜x[n]}is primal
feasible, and that we obtain a primal feasible sequence {z[n]}by setting
ziJ(i)
m[n] := max
t∈T
X
k∈J(i)
Mi\J(i)
m−1
˜x(t)
ik [n]
−
Mi
X
m′=m+1
ziJ(i)
m′[n]
= max
t∈T˜giJ(i)
m−
Mi
X
m′=m+1
ziJ(i)
m′[n]
recursively, starting from m=Miand proceeding through to m= 1.
Sherali and Choi [125] showed that, if the required conditions on the
step sizes {θ[n]}and convex combination weights {µl[n]}are satisfied,
then
m
X
l=1
µl[n]g(t)
iJ [l] +
n
X
l=m+1
γln(p(t)
iJ [n+ 1] −p(t)
iJ [n]) →0
as k→ ∞; hence we see from equations (5.36) and (5.37) that, for k
5.1 Flow-based approaches 123
sufficiently large,
Mi
X
m′=m
ziJ(i)
m′[n] = −
n
X
l=m+1
γlnciJ (i)
m[n].
Recalling the primal problem (5.30), we see that complementary slack-
ness with ¯pholds in the limit of any convergent subsequence of {˜x[n]}.
The required conditions on the step sizes and convex combination
weights are satisfied by the following choices ([125, Corollaries 2–4]):
(i) step sizes {θ[n]}such that θ[n]>0, limn→0θ[n] = 0, P∞
n=1 θn=
∞, and convex combination weights {µl[n]}given by µl[n] =
θ[l]/Pn
k=1 θ[k] for all l= 1,...,n,n= 0,1,...;
(ii) step sizes {θ[n]}given by θ[n] = a/(b+cn) for all n= 0,1,...,
where a > 0, b≥0 and c > 0, and convex combination weights
{µl[n]}given by µl[n] = 1/n for all l= 1,...,n,n= 0,1,...; and
(iii) step sizes {θ[n]}given by θ[n] = n−αfor all n= 0,1,..., where
0< α < 1, and convex combination weights {µl[n]}given by
µl[n] = 1/n for all l= 1,...,n,n= 0,1,....
Moreover, for all three choices, we have µl[n+ 1]/µl[n] independent of l
for all n, so primal iterates can be computed iteratively using
˜x[n] =
n
X
l=1
µl[n]ˆx[l]
=
n−1
X
l=1
µl[n]ˆx[l] + µn[n]ˆx[n]
=φ[n−1]˜x[n−1] + µn[n] ˆx[n],
where φ[n] := µl[n+ 1]/µl[n].
This gives us our distributed algorithm. We summarize the subgradi-
ent method in Figure 5.6. We see that, although the method is indeed a
distributed algorithm, it again operates in synchronous rounds. Again,
it is expected that this synchronicity can be relaxed in practice.
5.1.1.6 Application: Minimum-transmission wireless unicast
We have now discussed sufficient material to allow us to establish coded
connections. But is it worthwhile to do so? Surely, coding should not
be used for all network communications; in some situations, the gain
from coding is not sufficient to justify the additional work, and packets
124 Subgraph Selection
(i) Each node icomputes siJ for its outgoing hyperarcs and ini-
tializes piJ [0] to a point in the feasible set of (5.28). For
example, we take p(t)
iJ [0] := siJ /|T|. Each node isends siJ
and piJ [0] over each outgoing hyperarc (i, J).
(ii) At the nth iteration, use p(t)[n] as the hyperarc costs and
run a distributed shortest path algorithm, such as distributed
Bellman-Ford, to determine ˆx(t)[n] for all t∈T.
(iii) Each node icomputes piJ [n+ 1] for its outgoing hyperarcs
using Proposition 5.3. Each node isends piJ [n+ 1] over each
outgoing hyperarc (i, J ).
(iv) Nodes compute the primal iterate ˜x[n] by setting
˜x[n] :=
n
X
l=1
µl[n]ˆx[l].
(v) The current coding subgraph z[n] is computed using the pri-
mal iterate ˜x[n]. For each node i, we set
ziJ(i)
m[n] := max
t∈T
X
k∈J(i)
Mi\J(i)
m−1
˜x(t)
ik [n]
−
Mi
X
m′=m+1
ziJ(i)
m′[n]
recursively, starting from m=Miand proceeding through to
m= 1.
(vi) Steps (ii)–(v) are repeated until the sequence of primal iterates
{˜x[n]}converges.
Fig. 5.6. Summary of the subgradient method.
should simply be routed. In this section, we describe an application
where coding is worthwhile.
We consider the problem of minimum-transmission wireless unicast—
the problem of establishing a unicast connection in a lossy wireless net-
work using the minimum number of transmissions per packet. This
efficiency criterion is the same as that in Section 5.1.1.2; it is a generic
efficiency criterion reflects the fact that sending packets unnecessarily
wastes both energy and bandwidth.
There are numerous approaches to wireless unicast; we consider five,
three of which (approaches (i)–(iii)) are routed approaches and two of
which (approaches (iv) and (v)) are coded approaches:
(i) End-to-end retransmission: A path is chosen from source to
sink, and packets are acknowledged by the sink, or destination
node. If the acknowledgment for a packet is not received by the
source, the packet is retransmitted. This represents the situation
5.1 Flow-based approaches 125
where reliability is provided by a retransmission scheme above
the arc layer, e.g., by the transmission control protocol (tcp) at
the transport layer, and no mechanism for reliability is present
at the arc layer.
(ii) End-to-end coding: A path is chosen from source to sink, and
an end-to-end forward error correction (fec) code, such as a
Reed-Solomon code, an lt code [92], or a Raptor code [102, 126],
is used to correct for packets lost between source and sink. This
is the Digital Fountain approach to reliability [18].
(iii) Arc-by-arc retransmission: A path is chosen from source to
sink, and automatic repeat request (arq) is used at the arc layer
to request the retransmission of packets lost on every arc in the
path. Thus, on every arc, packets are acknowledged by the in-
tended receiver and, if the acknowledgment for a packet is not
received by the sender, the packet is retransmitted.
(iv) Path coding: A path is chosen from source to sink, and every
node on the path employs coding to correct for lost packets. The
most straightforward way of doing this is for each node to use
an fec code, decoding and re-encoding packets it receives. The
drawback of such an approach is delay. Every node on the path
codes and decodes packets in a block. A way of overcoming this
drawback is to use codes that operate in more of a “convolutional”
manner, sending out coded packets formed from packets received
thus far, without decoding. The random linear network coding
scheme of Section 4.1 is such a code. A variation, with lower
complexity, is described in [111].
(v) Full coding: In this case, paths are eschewed altogether, and we
use our solution to the efficient operation problem. Problem (5.2)
is solved to find a subgraph, and the random linear coding scheme
of Section 4.1 is used. This represents the limit of achievability
provided that we are restricted from modifying the design of the
physical layer and that we do not exploit the timing of packets
to convey information.
We consider the following experiment: Nodes are placed randomly
according to a uniform distribution over a square region whose size is
set to achieve unit node density. In the network, transmissions are sub-
ject to distance attenuation and Rayleigh fading, but not interference
(owing to scheduling). So, when node itransmits, the signal-to-noise
ratio (snr) of the signal received at node jis γd(i, j)−α, where γis an
126 Subgraph Selection
2 4 6 8 10 12
0
1
2
3
4
5
6
7
8
9
10
Network size (Number of nodes)
Average number of transmissions per packet
End−to−end retransmission
End−to−end coding
Link−by−link retransmission
Path coding
Full coding
Fig. 5.7. Average number of transmissions per packet as a function of network
size for various wireless unicast approaches. Reprinted with permission from
[101].
exponentially-distributed random variable with unit mean, d(i, j ) is the
distance between node iand node j, and αis an attenuation parameter
that was taken to be 2. A packet transmitted by node iis successfully
received by node jif the received snr exceeds β, i.e., γd(i, j)−α≥β,
where βis a threshold that was taken to be 1/4. If a packet is not
successfully received, then it is completely lost. If acknowledgments are
sent, acknowledgments are subject to loss in the same way that packets
are and follow the reverse path.
The average number of transmissions required per packet using the
various approaches in random networks of varying size is shown in Fig-
ure 5.7. Paths or subgraphs were chosen in each random instance to min-
imize the total number of transmissions required, except in the cases of
end-to-end retransmission and end-to-end coding, where they were cho-
sen to minimize the number of transmissions required by the source node
(the optimization to minimize the total number of transmissions in these
cases cannot be done straightforwardly by a shortest path algorithm).
We see that, while end-to-end coding and arc-by-arc retransmission al-
5.1 Flow-based approaches 127
ready represent significant improvements on end-to-end retransmission,
the coded approaches represent more significant improvements still. By
a network size of nine nodes, full coding already improves on arc-by-arc
retransmission by a factor of two. Moreover, as the network size grows,
the performance of the various schemes diverges.
Here, we discuss performance simply in terms of the number of trans-
missions required per packet; in some cases, e.g., congestion, the per-
formance measure increases super-linearly in this quantity, and the per-
formance improvement is even greater than that depicted in Figure 5.7.
We see, at any rate, that coding yields significant gains, particularly for
large networks.
5.1.2 Computation-constrained coding
In the previous section, we assumed that all nodes in the network are ca-
pable of coding, and we focused on the problem of minimizing resources
that can be expressed as a function of the coding subgraph. But what if
the computation required for coding is itself a scarce resource? This is
a concern, for example, in currently-deployed networks that only have
routing capability—each node that needs to be upgraded to add coding
capability will incur some cost. In the computation-constrained case, we
wish to restrict coding to only a subset of the network nodes, trading
off resources for transmission with resources for computation.
The computation-constrained problem is, in general, a hard one. Sim-
ply determining a minimal set of nodes where coding is required in a
given subgraph is np-hard [81], suggesting that heurstics are necessary
for multicast connections involving more than a few sinks. When there
are only a small number of sinks, however, an optimal solution can be
found using a flow-based approach. This approach, due to Bhattad et al.
[17], partitions flows not only into sinks, but into sets of sinks. Thus, for
a multicast with sinks in the set T, we not only have a flow x(t)for each
t∈T, but we have a flow x(T′)for each T′⊂T. The flow formulation
by Bhattad et al. involves a number of variables and constraints that
grows exponentially with the number of sinks, so it is feasible only when
|T|is small, but, under this restriction, it allows optimal solutions to be
found.
When dealing with a larger number of sinks, suboptimal heuristics
provide a solution. Kim et al. [81] have proposed an evolutionary ap-
proach based on a genetic algorithm that shows good empircal perfor-
mance.
128 Subgraph Selection
s2
t2
t1
s1
1 2
b1⊕b2
b1
b1⊕b2
b1⊕b2
b1
b2
b2
Fig. 5.8. The modified butterfly network. In this network, every arc represents
a directed arc that is capable of carrying a single packet reliably.
5.1.3 Inter-session coding
Optimal inter-session coding, as we have mentioned, is very difficult.
In fact, even good inter-session coding is very difficult. One of the few
methods for finding non-trivial inter-session coding solutions was pro-
posed by Koetter and M´edard [85]; their method searched only within
a particular, limited class of linear codes and, even then, its complexity
scaled exponentially in the size of the network. Since we must maintain
an eye on practicability, we must not be too ambitious in our search for
inter-session codes.
As discussed in Section 3.5.1, a modest approach that we can take is
the following: Since our most familiar examples of inter-session coding
are the modified butterfly network (Figure 1.2) and modified wireless
butterfly network (Figure 1.4), we seek direct extensions of the inter-
session coding opportunities exemplified by these two cases.
For starters, let us consider the lossless wireline case. We show the
modified butterfly network again in Figure 5.8. Without intersession
coding, we would require two unit-sized flows x(1) and x(2), originating
and ending at s1and t1and s2and t2, respectively. There is only one
possible solution for each of the two flows:
x(1) = (x(1)
s11, x(1)
s21, x(1)
s1t2, x(1)
12 , x(1)
s2t1, x(1)
2t2, x(1)
2t1) = (1,0,0,1,0,0,1),
x(2) = (x(2)
s11, x(2)
s21, x(2)
s1t2, x(2)
12 , x(2)
s2t1, x(1)
2t2, x(2)
2t1) = (0,1,0,1,0,1,0).
This solution, as we know, is not feasible because it violates the capacity
constraint on arc (1,2). Without inter-session coding, the total rate of
5.1 Flow-based approaches 129
packet injections on arc (1,2) would be two, which violates its capacity
of one. We also know, however, that a simple inter-session code where
packets from each of the two sessions are xored at node 1 resolves this
situation by reducing the rate of packet injections on arc (1,2) from two
to one and increasing the rate of packet injections on arc (s1, t2) and
(s2, t1) from zero to one. If we can formulate the effect of this code
as flow equations, then we can hope to develop a flow-based approach
for systematically finding inter-session coding opportunities of the type
exemplified by the modified butterfly network.
Such a flow formulation was developed by Traskov et al. [132]. In the
formulation, three variables are introduced for each coding opportunity:
p, the poison variable, q, the antidote request variable, and r, the anti-
dote variable. The poison prepresents the effect of coding two sessions
together with an xor. The poison on arc (i, j), pij , is strictly negative
if the arc carries poisoned flow, i.e., it carries packets xored from two
separate sessions; otherwise, it is zero. Such a flow is “poisoned” be-
cause the xored packets, by themselves, are not useful to their intended
destinations. The antidote rrepresents the extra “remedy” packets that
must be set so that the effect of the poison can be reversed, i.e., so that
the xored packets can be decoded to recover the packets that are ac-
tually useful. The antidote on arc (i, j ), rij , is strictly positive if the
arc carries remedy packets; otherwise, it is zero. The antidote request
qis essentially an imaginary variable in that it need not correspond to
any real physical entity, It could, however, correspond to actual protocol
messages requesting remedy packets to be sent. The antidote request
connects the coding node to the nodes from which remedy packets are
sent, thus making a cycle from p,q, and rand facilitating a flow formu-
lation. The antidote request on arc (i, j ), qij, is strictly negative if the
arc carries antidote requests; otherwise, it is zero.
The Traskov et al. flow formulation is best understood using an ex-
ample. We take, as our example, the modified butterfly network and, in
Figure 5.9, we show the poison, antidote request, and antidote variables
for this network. We have two of each variable: one, 1 →2, relates to
the impact of coding on flow two, while the other, 2 →1, relates to
the impact of coding on flow one. Note that p(1 →2), q(1 →2), and
r(1 →2) form a cycle, as do p(2 →1), q(2 →1), and r(2 →1).
This formulation, once extended to a general lossless wireline network
that allows coding at all nodes, yields the following formulation of the
130 Subgraph Selection
s1
2
s2t1
1
t2
q(2 →1) = −1
p(1 →2) = −1
r(2 →1) = 1
r(1 →2) = 1
p(2 →1) = −1
q(1 →2) = −1p(1 →2) = −1
p(2 →1) = −1
Fig. 5.9. The modified butterfly network with poison, antidote request, and
antidote variables shown. Reprinted with permission from [132].
subgraph selection problem:
minimize f(z)
subject to z∈Z,
X
{j|(i,j)∈A}
x(c)
ij −X
{j|(j,i)∈A}
x(c)
ji =
Rcif i=sc,
−Rcif j=tc,
0 otherwise,
∀i∈ N ,c= 1,...,C,
x≥0
x∈ T (z),
(5.38)
where T(z) for a given zis the set of xsatisfying, for some {pij(c→
d, k)},{qij (c→d, k)}, and {rij (c→d, k)}, the following equalities and
5.1 Flow-based approaches 131
inequalities:
X
{j|(j,i)∈A}
(pji (c→d, k) + qji (c→d, k) + rji (c→d, k))
=X
{j|(i,j)∈A}
(pij (c→d, k) + qij (c→d, k) + rij (c→d, k)) ,
∀i, k ∈ N ,c, d = 1,...,C,
X
{j|(j,i)∈A}
pji (c→d, k)−X
{j|(i,j)∈A}
pij (c→d, k)(≥0 if i=k,
≤0 otherwise,
∀i, k ∈ N ,c, d = 1,...,C,
X
{j|(j,i)∈A}
qji (c→d, k)−X
{j|(i,j)∈A}
qij (c→d, k)(≤0 if i=k,
≥0 otherwise,
∀i, k ∈ N ,c, d = 1,...,C,
pij (d→c, i) = pij (c→d, i),
∀j∈ {j|(i, j)∈ A},c, d = 1,...,C,
C
X
c=1
xij (c) + X
kX
d>c
pmax
ij (c, d, k) + X
kX
d6=c
rij (c→d, k)
≤zij ,
∀(i, j)∈ A,
xij (d) + X
kX
c
(pij (c→d, k) + qij (d→c, k)) ≥0,
∀(i, j)∈ A,d= 1,...,C,
p≤0, r ≥0, s ≤0,
where
pmax
ij (c, d, k),max(pij (c→d, k), pij (d→c, k)).
In this formulation, multicast is not explicitly considered—only inter-
session coding for multiple unicast sessions is considered. It allows for
packets from two separate sessions to be xored at any node and for
remedy packets to be sent and decoding to be performed at any nodes
at which these operations are valid. It does not allow, however, for
poisoned flows to be poisoned again. That this formulation is indeed
correct, given these restrictions, is shown in [132].
This formulation can be straightforwardly extended to the case of
lossless wireless networks, thus allowing us to extend the modified wire-
132 Subgraph Selection
less butterfly network in the same way. The precise equations that are
obtained are given in [42].
5.2 Queue-length-based approaches
Queue-length-based, or back-pressure, algorithms were first introduced
in [131, 6] for multicommodity flow problems, i.e. multiple unicast net-
work problems without coding. The basic idea can be summed up as
follows. Each node ikeeps track of the number U(c)
iof packets of each
unicast session c. It is convenient to think of U(c)
ias the length of a queue
of packets Q(c)
imaintained by node ifor each session c. Each queue has a
potential that is an increasing function of its length. At each step, packet
transmissions are prioritized according to the potential difference across
their corresponding start and end queues, so as to maximize the total
potential decrease in the network, subject to network constraints. In the
simplest version, the potential is equal to the queue length, and trans-
missions are prioritized according to the queue length difference, i.e., for
a given arc (i, j), packets of session arg maxc(U(c)
i−U(c)
j) have priority
for transmission. This policy gives rise to queue length gradients from
each session’s source to its sink. We can draw an analogy with pressure
gradients and think of packets as “flowing down” these gradients.
Different versions and extensions of the basic back-pressure algorithm
have been proposed for finding asymptotically optimal solutions for var-
ious types of multicommodity flow problems with different objectives.
For instance, back-pressure algorithms with potential given by queue
lengths are proposed for dynamic control of routing and scheduling
in time-varying networks, in [131] and other subsequent works. Such
approaches are extended to the problem of minimum-energy routing
in [107]. In [7] a back-pressure algorithm with an exponential potential
function is proposed as a low-complexity approximation algorithm for
constructing a solution to a feasible multicommodity flow problem.
The back-pressure approach can be generalized to optimize over differ-
ent classes of network codes. The underlying idea is the introduction of
an appropriately defined system of virtual queues and/or virtual trans-
missions. The back-pressure principle of maximizing total potential de-
crease at each step, subject to network constraints, can then be applied
to obtain control policies for network coding, routing and scheduling
based on the virtual queue lengths. Extensions to various types of net-
5.2 Queue-length-based approaches 133
work problems with different objectives can be obtained analogously to
the multicommodity routing case.
5.2.1 Intra-session network coding for multiple multicast
sessions
In this section, we consider a dynamically varying network problem with
a set Cof multicast sessions. Each session c∈ C has a set Scof source
nodes whose data is demanded by a set Tcof sink nodes. We describe
an extension of the basic queue-length-based approach to the case of
multiple multicast sessions with intra-session network coding, i.e. only
packets from the same session are coded together.
5.2.1.1 Network model
We consider a lossless network comprised of a set Nof N=|N | nodes,
with a set Aof communication arcs between them that are fixed or time-
varying according to some specified processes. There is a set of multicast
sessions Csharing the network. Each session c∈ C is associated with
a set Sc⊂ N of source nodes, and an arrival process, at each source
node, of exogenous session cpackets to be transmitted to each of a set
Tc⊂ N \Scof sink nodes. We denote by τmax the maximum number of
sinks of a session.
Time is assumed to be slotted, with the time unit normalized so that
time slots correspond to integral times τ= 0,1,2,.... For simplicity, we
assume fixed length packets and arc transmission rates that are restricted
to integer numbers of packets per slot. We assume that the channel
states, represented by a vector S(τ) taking values in a finite set, are fixed
over the duration of each slot τ, and known at the beginning of the slot.
For simplicity of exposition we assume that the exogenous packet arrival
and channel processes are independent and identically distributed (i.i.d.)
across slots. The queue-length-based policy described below applies also
in the case of stationary ergodic processes. The analysis below can be
generalized to the latter case using an approach similar to that in [106]†.
We consider both wired and wireless networks. In our model, for
the wired case the network connectivity and the arc rates are explic-
itly specified. For wireless networks, the network connectivity and arc
transmission rates depend on the signal and interference powers and the
†Groups of Mtimeslots are considered as a “super timeslot”, where Mis sufficiently
large that the time averages of the channel and arrival processes differ from their
steady-state values by no more than a given small amount.
134 Subgraph Selection
channel states. We assume that the vector of transmit powers P(τ) at
each time τtakes values in a given set Π and is constant over each slot.
We also assume we are given a rate function µ(P , S ) specifying the vec-
tor of instantaneous arc rates µ(τ) = (µiJ (τ)) as a function of the vector
of transmit powers P(τ) and channel states S(τ).
In this section, all arc, source and flow rates are in packets per unit
time. We assume upper bounds µout
max and µin
max on the total flow rate
into and out of a node respectively.
5.2.1.2 Network coding and virtual queues
We use the approach of distributed random linear network coding with
coefficient vectors, described in Section 2.5.1.1. For simplicity, we do not
explicitly consider batch restrictions in the descriptions and analysis of
the policies in this section. Thus, the results represent an upper bound
on performance that is approached only asymptotically (in the batch size
and packet length). If the policies are operated across multiple batches
(the only change from the policy described below being an additional
restriction not to code across batches), there is some capacity loss which
decreases with increasing batch size and depends on the detailed source
and channel statistics.
Recall from Section 2.3 that for network coding within a multicast
session, a solution is given by a union of individual flow solutions for
each sink. Here, we define virtual queues to keep track of the individual
sinks’ flow solutions as follows.
Each node iconceptually maintains, for each sink tof each session
c, a virtual queue Q(t,c)
iwhose length U(t,c)
iis the number of session c
packets queued at node ithat are intended for sink t. A single physical
session cpacket corresponds to a packet in the virtual queue Q(t,c)
iof
each sink tfor which it is intended. For instance, in the butterfly network
of Figure 5.10, each physical session cpacket at source node sis intended
to be multicast to the two sink nodes t1, t2, and so corresponds to one
packet in each virtual queue Q(t1,c)
sand Q(t2,c)
s. Each packet in a virtual
queue corresponds to a distinct physical packet; thus there is a one-to-
many correspondence between physical packets and packets in virtual
queues.
A packet in a virtual queue Q(t,c)
ican be transferred over an arc (i, j)
to the corresponding virtual queue Q(t,c)
jat the arc’s end node j; this is
called a virtual transmission. With network coding, for any subset T′⊂
Tcof a session’s sinks, a single physical transmission of a packet on an
arc (i, j) can simultaneously accomplish, for each sink t∈ T ′, one virtual
5.2 Queue-length-based approaches 135
t2
s
t1
1
3 4
2
b1
b2
b1
b1⊕b2
b2
b1
b2
b1⊕b2
b1⊕b2
Fig. 5.10. The butterfly network with a single multicast session c, one source
node sand two sink nodes t1, t2.
transmission from Q(t,c)
ito Q(t,c)
j. The physically transmitted packet is a
random linear coded combination of the physical packets corresponding
to the virtually transmitted packets. In the case of a wireless broadcast
transmission from a node ito a set of nodes J, although the nodes in
Jall receive the transmitted packet, they update their virtual queues
selectively, according to control information included in the packet, such
that each constituent virtual transmission is point-to-point, i.e. from one
queue Q(t,c)
ito one queue Q(t,c)
jat some end node j∈J, which may
differ for different sinks t. Thus, there is conservation of virtual packets
(virtual flows); we can draw an analogy with the no-coding case where
physical packets (physical flows) are conserved. An illustration is given
in Figure 5.11 of a physical broadcast transmission which accomplishes
two virtual transmissions, for a multicast session with two sinks.
Let w(c)
ibe the average arrival rate of exogenous session cpackets
at each node i;w(c)
i= 0 for i6∈ Sc. Each source node i∈ Scforms
coded source packets at an average rate r(c)
i=w(c)
i+ǫfor some ǫ > 0,
slightly higher than its exogenous packet arrival rate w(c)
i. Each coded
source packet is formed as an independent random linear combination
of previously arrived exogenous packets, and is “added” to each queue
Q(t,c)
i, t ∈ Tc. In order for each sink to be able to decode all source
packets, w(c)
i
r(c)
i
should be at most the ratio of the total number of packets
reaching each sink to the total number of coded source packets. As we
136 Subgraph Selection
Sink 1
Sink 2
Sink 1
Sink 2
Sink 1
Sink 2
Fig. 5.11. Illustration of a physical broadcast transmission comprising two
virtual transmissions. Each oval corresponds to a node. The left node broad-
casts a physical packet received by the two right nodes, one of which adds the
packet to the virtual queue for sink 1, and the other, to the virtual queue for
sink 2.
will see in the next section, for sufficiently large times t, this condition
is satisfied and decoding is successful with high probability.†
Let A(c)
i(τ) be the number of coded session csource packets formed
at node iin timeslot τ. Thus we have
r(c)
i=E{A(c)
i(τ)}.(5.39)
We assume that the second moment of the total number of source packets
formed at each node in each timeslot is bounded by a finite maximum
value A2
max, i.e.
E
X
c
A(c)
i(τ)!2
≤A2
max,(5.40)
which implies
E(X
c
A(c)
i(τ))≤Amax.(5.41)
†If we employ batch coding where each batch contains a fixed number of exogenous
packets, feedback from the sinks can be used to signal when the sources should
stop forming coded packets of each batch. This determines the effective value of ǫ
for each batch.
5.2 Queue-length-based approaches 137
5.2.1.3 Problem and notation
We consider a multiple multicast network problem where the average
network capacity is slightly higher than the minimum needed to support
the source rates. We would like a control policy for dynamic subgraph
selection (i.e. scheduling of packets/transmitters, and routing) which,
coupled with random linear network coding, stably supports the given
source rates. The problem is formulated more precisely as follows.
Let Uc
i(τ) be the number of physical session cpackets queued at node
iat time τ. Stability is defined in terms of an “overflow” function
γ(c)
i(M) = lim sup
τ→∞
1
τ
τ
X
τ′=0
Pr{Uc
i(τ′)> M}(5.42)
The session cqueue at node iis considered stable if γc
i(M)→0 as
M→ ∞. A network of queues is considered stable iff each individual
queue is stable. Since
Uc
i(τ)≤X
t
U(t,c)
i(τ)≤τmaxUc
i(τ)∀i, c, τ
the network is stable iff all virtual queues are stable.
Let ziJ denote the average value of the time-varying rate µiJ(τ) of
hyperarc (i, J ). We use x(t,c)
iJj to denote average virtual flow rate, over
arc (i, J )∈ A, from Q(t,c)
ito Q(t,c)
j,j∈J. We use y(c)
iJ to denote average
session cphysical flow rate on (i, J )∈ A.
For brevity of notation, we use the convention that any term with
subscript iJj equals zero unless (i, J )∈ A, j ∈J, and any term with
superscript (t, c) equals zero unless c∈ C, t ∈ Tc.
Let πSdenote the probability in each slot that the channel states take
the value S. Let Zbe the set consisting of all rate vectors z= (ziJ ) that
can represented as z=PSπSzSfor some set of rate vectors zS, each of
which is in the convex hull of the set of rate vectors {µiJ (P , S)|P∈Π}.
Zrepresents the set of all long-term average transmission rates (ziJ )
supportable by the network [106, 59].
Let Λ be the set of all rate vectors (r(c)
i) such that there exist values
138 Subgraph Selection
for (ziJ )∈Zand {x(t,c)
iJj , y (t,c)
iJ }satisfying:
x(t,c)
tJi = 0 ∀c, t, J, i (5.43)
x(t,c)
iJj ≥0∀i, j, c, t, J (5.44)
r(c)
i≤X
J,j
x(t,c)
iJj −X
j,I
x(t,c)
jIi ∀i, c, t ∈ Tc, t 6=i(5.45)
X
j∈J
x(t,c)
iJj ≤y(c)
iJ ∀i, J, c, t (5.46)
X
c
y(c)
iJ ≤ziJ ∀i, J (5.47)
Equations (5.43)–(5.47) correspond to the feasible set of problem (5.1)
in the lossless case, where, rather than a single source node scfor each
session c, there may be multiple source nodes, described by the set Sc.
The variables {x(t,c)
ab }for a (session, sink) pair (c, t ∈ Tc) define a flow
carrying rate at least r(c)
ifrom each source node ito t(Inequalities
(5.44)–(5.45)), in which virtual flow that is intended for tis not retrans-
mitted away from t(Equation (5.46)).
We describe below a queue-length-based policy that is stable and
asymptotically achieves the given source rates for any network prob-
lem where (r(c)
i+ǫ′)∈Λ.†This condition implies the existence of a
solution, but we assume that the queue-length-based policy operates
without knowledge of the solution variables.
5.2.1.4 Control policies
We consider policies that make control decisions at the start of each time
slot τand operate as follows.
•Power allocation: A vector of transmit powers P(τ) = (PiJ (τ)) is
chosen from the set Π of feasible power allocations. This, together
with the channel state S(τ), determines the arc rates µ(τ) = (µiJ (τ)),
assumed constant over the time slot.
•Session scheduling, rate allocation and network coding: For each arc
(i, J ), each sink tof each session cis allocated a transmission rate
†This particular problem formulation, which provisions sufficient network capacity
to support, for each session c, an additional source rate of ǫ′at each node, affords
a simple solution and analysis. We could alternatively use a slightly more com-
plicated formulation which includes for each session can additional source rate of
ǫ′only at the actual source nodes s∈ Sc, similarly to that in [67] for the case of
correlated sources.
5.2 Queue-length-based approaches 139
µ(t,c)
iJj (τ) for each destination node j∈J. These allocated rates must
satisfy the overall arc rate constraint
µiJ (τ)≥X
c∈C
max
t∈TcX
j∈J
µ(t,c)
iJj (τ).
µ(t,c)
iJj (τ) gives the maximum rate of virtual transmissions from Q(t,c)
i
to Q(t,c)
j. Besides this limit on virtual transmissions for pairs of queues
over each link, the total number of virtual transmissions out of Q(t,c)
i
over all links with start node iis also limited by the queue length
U(t,c)
i(τ) at the start of the time slot. Each session cpacket physically
transmitted on arc (i, J ) is a random linear combination, in Fq, of
packets corresponding to a set of virtual transmissions on (i, J ), each
associated with a different sink in Tc. Thus, the rate allocated to
session con (i, J ) is the maximum, over sinks t∈ Tc, of each sink t’s
total allocated rate Pj∈Jµ(t,c)
iJj (τ).
The following dynamic policy relies on queue length information to
make control decisions, without requiring knowledge of the input or
channel statistics. The intuition behind the policy is that it seeks to
maximize the total weight of virtual transmissions for each time slot,
subject to the above constraints.
Back-pressure policy
For each time slot τ, the transmit powers (PiJ (τ)) and allocated rates
(µ(t,c)
iJj (τ)) are chosen based on the queue lengths (U(t,c)
i(τ)) at the start
of the slot, as follows.
•Session scheduling: For each arc (i, J),
–for each session cand sink t∈ Tc, one end node
j(t,c)∗
iJ = arg max
j∈JU(t,c)
i−U(t,c)
j
= arg min
j∈J
U(t,c)
j
is chosen. Let U(t,c)∗
iJ denote U(t,c)
j(t,c)∗
iJ
for brevity.
–one session
c∗
iJ = arg max
c(X
t∈Tc
max U(t,c)
i−U(t,c)∗
iJ ,0)
140 Subgraph Selection
is chosen. Let
w∗
iJ =X
t∈Tc∗
iJ
max U(t,c)
i−U(t,c)∗
iJ ,0(5.48)
be the weight of the chosen session.
•Power control: The state S(τ) is observed, and a power allocation
P(τ) = arg max
P∈ΠX
i,J
µiJ (P , S(τ))w∗
iJ (5.49)
is chosen.
•Rate allocation: For each arc (i, J),
µ(t,c)
iJj (τ) = (µiJ (τ) if c=c∗
iJ , t ∈ Tc, j =j(t,c)∗
iJ and U(t,c)
i−U(t,c)
j>0
0 otherwise .
(5.50)
In a network where simultaneous transmissions interfere, optimizing
(5.49) requires a centralized solution. If there are enough channels for
independent transmissions, the optimization can be done independently
for each transmitter.
The stability of the back-pressure policy is shown by comparison with
a randomized policy that assumes knowledge of a solution based on the
long-term input and channel statistics. We will show that the random-
ized policy is stable, and that stability of the randomized policy implies
stability of the back-pressure policy.
Randomized policy
Assume given values of (ziJ)∈Zand {x(t,c)
iJj , y (t,c)
iJ }satisfying
x(t,c)
tJi = 0 ∀c, t, J, i (5.51)
x(t,c)
iJj ≥0∀i, j, c, t, J (5.52)
r(c)
i+ǫ′≤X
J,j
x(t,c)
iJj −X
j,I
x(t,c)
jIi ∀i, c, t ∈ Tc, t 6=i(5.53)
X
j∈J
x(t,c)
iJj ≤y(c)
iJ ∀i, J, c, t (5.54)
X
c
y(c)
iJ ≤ziJ ∀i, J (5.55)
are given.
The following lemma shows that for any rate vector (ziJ )∈Z, power
can be allocated according to the time-varying channel state S(τ) such
that the time average link rates converge to (ziJ ).
5.2 Queue-length-based approaches 141
Lemma 5.1 Consider a rate vector (ziJ)∈Z. There exists a stationary
randomized power allocation policy which gives link rates µiJ (τ)satisfy-
ing
lim
τ→∞
1
τ
τ
X
0
µiJ (τ′) = ziJ
with probability 1 for all (i, J )∈ A, where, for each time slot τin which
channel state S(τ)takes value S, the power allocation is chosen ran-
domly from a finite set {PS,1,...,PS,m}according to stationary proba-
bilities {qS,1,...,qS ,m}.
Proof (Outline) From the definition of Zin Section 5.2.1.3 and by
Carath´eodory’s Theorem (see, e.g. [8]), (ziJ) = PSπSzSfor some set
of rate vectors zS, each of which is a convex combination of vectors
in {µiJ (P , S)|P∈Π}. The probabilities of the stationary randomized
power allocation policy are chosen according to the weights of the convex
combinations for each state S.
The randomized policy is designed such that
E{µ(t,c)
iJj (τ)}=x(t,c)
iJj .(5.56)
For each time slot τ, transmit powers (PiJ (τ)) and allocated rates (µ(t,c)
iJj (τ))
are chosen based on the given values of (ziJ) and {x(t,c)
iJj , y (t,c)
iJ }as well
as the channel state S(τ), as follows.
•Power allocation: The channel state S(τ) is observed, and power is
allocated according to the algorithm of Lemma 5.1, giving instanta-
neous arc rates µiJ (τ) and long-term average rates ziJ .
•Session scheduling and rate allocation: For each arc (i, J), one session
c=ciJ is chosen randomly with probability y(c)
iJ
Pcy(c)
iJ
. Each of its sinks
t∈ Tcis chosen independently with probability Pjx(t,c)
iJj
y(c)
iJ
. Let TiJ ⊂ Tc
denote the set of chosen sinks. For each t∈ TiJ , one destination node
j=j(t,c)
iJ in Jis chosen with probability x(t,c)
iJj
Pjx(t,c)
iJj
. The corresponding
allocated rates are
µ(t,c)
iJj (τ) = (Pcy(c)
iJ
ziJ µiJ (τ) if c=ciJ , t ∈ TiJ and j=j(t,c)
iJ
0 otherwise .
(5.57)
142 Subgraph Selection
Theorem 5.1 If input rates (r(c)
i)are such that (r(c)
i+ǫ′)∈Λ,ǫ′>
0, both the randomized policy and the back pressure policy stabilize the
system with average total virtual queue length bounded as
X
i,c,t
U(t,c)
i= lim sup
τ→∞
1
τ
τ−1
X
τ′=0 X
i,c,t
E{U(t,c)
i(τ′)} ≤ BN
ǫ′(5.58)
where
B=τmax
2(Amax +µin
max)2+ (µout
max)2.(5.59)
The proof of this theorem uses the following result:
Theorem 5.2 Let U(τ) = (U1(τ),...,Un(τ)) be a vector of queue
lengths. Define the Lyapunov function L(U(τ)) = Pn
j=1[Uj(τ)]2. If
for all τ
E{L(U(τ+ 1)) −L(U(τ))|U(τ)} ≤ C1−C2
n
X
j=1
Uj(τ) (5.60)
for some positive constants C1, C2, and if E{L(U(0))}<∞, then
n
X
j=1
Uj= lim sup
τ→∞
1
τ
τ−1
X
τ′=0
n
X
j=1
E{Uj(τ′)} ≤ C1
C2
(5.61)
and each queue is stable.
Proof Summing over τ= 0,1,...,T −1 the expectation of (5.60) over
the distribution of U(τ), we have
E{L(U(T)) −L(U(0))} ≤ T C1−C2
T−1
X
τ=0
n
X
j=1
E{Uj(τ)}.
Since L(U(T)) >0,
1
T
T−1
X
τ=0
n
X
j=1
E{Uj(τ)} ≤ C1
C2
+1
T C2
E{L(U(0))}.
5.2 Queue-length-based approaches 143
Taking the lim sup as T→ ∞ gives (5.61). Each queue is stable since
γj(M) = lim sup
τ→∞
1
τ
τ
X
τ′=0
Pr{Uj(τ′)> M}
≤lim sup
τ→∞
1
τ
τ
X
τ′=0
E{Uj(τ′)}/M (5.62)
≤C1
C2M→0 as M→ ∞,(5.63)
where (5.62) holds since Uj(τ′) is nonnegative.
Proof of Theorem 5.1: The queue lengths evolve according to:
U(t,c)
i(τ+ 1) ≤max
U(t,c)
i(τ)−X
J,j
µ(t,c)
iJj (τ),0
+X
j,I
µ(t,c)
jIi (τ) + A(c)
i(τ) (5.64)
which reflects the policy that the total number of virtual transmissions
out of Q(t,c)
iis limited by the queue length U(t,c)
i(τ).
Define the Lyapunov function L(U) = Pi,c,t(U(t,c)
i)2. Squaring (5.64)
and dropping some negative terms from the right hand side, we obtain
[U(t,c)
i(τ+ 1)]2≤[U(t,c)
i(τ)]2+
A(c)
i+X
j,I
µ(t,c)
jIi
2
+
X
J,j
µ(t,c)
iJj
2
−2U(t,c)
i(τ)
X
J,j
µ(t,c)
iJj −X
j,I
µ(t,c)
jIi −A(c)
i
(5.65)
where the time dependencies of µ(t,c)
iJj and A(c)
iare not shown for brevity,
since these remain constant over the considered time slot.
Taking expectations of the sum of (5.65) over all i, c, t, noting that
X
i,c,t
X
j,Z
µ(t,c)
iJj
2
≤X
i,c
τmax
max
t∈TcX
j,Z
µ(t,c)
iJj
2
≤X
i
τmax
X
c
max
t∈TcX
j,Z
µ(t,c)
iJj
2
(5.66)
≤Nτmax µout
max2,
144 Subgraph Selection
and
X
i,c,t
A(c)
i+X
j,I
µ(t,c)
jIi
2
≤X
i,c
τmax
A(c)
i+ max
t∈TcX
j,I
µ(t,c)
jIi
2
≤X
i
τmax
X
c
A(c)
i+ max
t∈TcX
j,I
µ(t,c)
jIi
2
(5.67)
≤Nτmax Amax +µin
max2
(where the Cauchy-Schwarz inequality is used in steps (5.66) and (5.67)),
and using (5.39), (5.40), we obtain the drift expression
E{L(U(τ+ 1)) −L(U(τ))|U(τ)} ≤ 2BN −
2X
i,c,t
U(t,c)
i(τ)
E
X
J,j
µct
iJj −X
j,I
µ(t,c)
jIi
U(τ)
−r(c)
i
.(5.68)
Substituting (5.53) and (5.56) into (5.68) gives
E{L(U(τ+ 1)) −L(U(τ))|U(τ)} ≤ 2BN −2ǫ′X
i,c,t
U(t,c)
i(τ) (5.69)
where Bis defined in (5.59).
Applying Theorem 5.2 gives
X
i,c,t
U(t,c)
i≤BN
ǫ′.(5.70)
Thus the randomized policy satisfies the queue occupancy bound (5.58).
For the back pressure policy, E{µ(t,c)
iJj (τ)|U(τ)}is dependent on U(τ).
The drift expression (5.71) can be expressed as
E{L(U(τ+ 1)) −L(U(τ))|U(τ)} ≤ 2BN −2
D−X
i,c,t
U(t,c)
i(τ)r(c)
i
where
D=X
i,c,t
U(t,c)
i(τ)
E
X
J,j
µ(ct)
iJj −X
j,I
µ(t,c)
jIi
U(τ)
,(5.71)
which is the portion of the drift expression that depends on the policy,
can be rewritten as
D=X
i,J,j X
c,t
Enµ(t,c)
iJj |U(τ)oU(t,c)
i(τ)−U(t,c)
j(τ).
5.2 Queue-length-based approaches 145
We compare the values of (5.72) for the two policies, giving
Drand =X
i,J,j X
c,t
x(t,c)
iJj U(t,c)
i−U(t,c)
j
≤X
j,I X
c
yc
iJ X
t
max
j∈ZU(t,c)
i−U(t,c)
j
≤X
j,I X
c
yc
iJ w∗
iJ
≤X
j,I
ziJ w∗
iJ
=X
j,I
X
S
πSzS
iJ
w∗
iJ
≤X
S
πSmax
P∈ΠX
j,I
µiJ (P , S )w∗
iJ
=Dbackpressure
where the last step follows from (5.49)-(5.50). Since the Lyapunov drift
for the back-pressure policy is more negative than the drift for the ran-
domized policy, the bound (5.70) also applies for the back-pressure pol-
icy. This completes the proof.
The queue-length-based policy can be simplified in the wired network
case where each arc (i, j ) has a destination node set of size 1 and a
capacity µij that does not depend on Por S.
Back-pressure policy for wired networks
For each time slot τand each arc (i, j),
•Session scheduling: one session
c∗
ij = arg max
c(X
t∈Tc
max U(t,c)
i−U(t,c)
j,0)
is chosen.
•Rate allocation: the maximum rate of virtual transmissions from Q(t,c)
i
to Q(t,c)
jis set as
µ(t,c)
ij (τ) = (µij if c=c∗
ij , t ∈ Tc,and U(t,c)
i−U(t,c)
j>0
0 otherwise.(5.72)
•Network coding: each session cpacket physically transmitted on arc
146 Subgraph Selection
(i, j) is a random linear combination, in Fq, of packets correspond-
ing to a set of virtual transmissions on (i, j), each associated with a
different sink in Tc.
Theorem 5.1 implies that each sink can receive packets at a rate
asymptotically close to the source rate. To retrieve the actual infor-
mation, each sink must also be able to decode the coded packets. The
following theorem shows that the probability that not all sinks are able
to decode the information tends to zero exponentially in the coding block
length.
Theorem 5.3 For exogenous arrival rates w(c)
i=r(c)
i−ǫ, if (r(c)
i)is
strictly interior to Λ, then for sufficiently large time τ, the probabil-
ity that not every sink is able to decode its session’s exogenous packets
decreases exponentially in the length of the code.
Proof As described in Section 2.5, we can draw a parallel between a
given sequence Sof packet transmissions and a corresponding static
network Gwith the same node set Nand with links corresponding to
transmissions in S. The following analysis is an extension, based on
this correspondence, of the analysis in Section 2.4.2 of random network
coding for static networks.
Consider any session c. Let the randomly chosen network coding coef-
ficients associated with the session cpackets be represented by a vector
ξ= (ξ1,...,ξν). Consider any sink t∈ Tc. It follows from Theorem 5.1
that over some sufficiently large time τ, with high probability there is a
virtual flow of rc
iτ−PjU(t,c)
j(τ)≥(r(c)
i−ǫ)τpackets from each session
csource node ito t, corresponding to coded combinations of (r(c)
i−ǫ)τ
exogenous packets. Consider any (r(c)
i−ǫ)τof the packets received by
tfrom each session csource node i. We denote by d(t,c)(ξ) the determi-
nant, as a polynomial in ξ, of the matrix whose rows equal the coefficient
vectors of these packets. Consider the physical packet transmissions cor-
responding to this virtual flow, which are transmissions involving queues
Q(t,c)
j. These physical transmissions would constitute an uncoded phys-
ical flow if their originating transmissions from the source nodes were
uncoded independent packets and there were no other sinks/virtual flows
in the network. We denote by ˜
ξthe value of ξcorresponding to this case,
noting that d(t,c)(˜
ξ) = ±1.†Thus, d(t,c)(ξ) is not identically zero.
†For this uncoded flow case, the coefficient vectors of the (r(c)
i−ǫ)τsession cpackets
received by tform the rows of the identity matrix.
5.2 Queue-length-based approaches 147
Since the product Qc,t∈Tcd(t,c)(ξ) as a function of the network code
coefficients ξis not identically zero, by the Schwartz-Zippel theorem,
choosing the code coefficients uniformly at random from a finite field of
size qyields a zero value with probability inversely proportional to q.
The result follows since qis exponential in the length of the code.
5.2.2 Inter-session coding
Queue-length-based approaches can also be extended to subgraph selec-
tion for simple inter-session network codes such as those described in
Section 3.5. In different classes of network coding strategies, different
aspects of the coding/routing history of packets restrict whether and
how they can be coded/decoded/removed at particular nodes. Using
these aspects to define different commodities or queues allows the effect
of such restrictions to be propagated to influence control decisions at
other nodes. The class of strategies over which we optimize, as well
as the complexity and convergence rate, is determined by the choice of
commodities and algorithm details.
Queue-length-based algorithms for optimizing over the class of pair-
wise poison-antidote codes (ref Section 3.5.1) are given in [42, 61]. A
common feature of both algorithms is that they make coding decisions
by treating an xor coding operation as a type of virtual transmission
that, unlike the virtual transmissions of the previous section, does not
occur over a physical network arc. A coding operation is analogous to
an actual arc transmission in that it removes one packet from each of a
set of start queues and adds one packet to each of a set of end queues.
For pairwise poison-antidote coding, there are two start queues corre-
sponding to the packets being coded together, and, depending on the
variant of the algorithm, the end queues correspond to the resulting
poison packet and/or antidote packets†. At each step, besides prioritiz-
ing among physical transmissions over arcs analogously to the previous
section, the algorithms also choose, among the coding possibilities at
each node, the one with the largest positive potential difference across
start and end queues. In [61] there are also virtual decoding transmis-
sions that determine, based on local queue lengths, where each poison
packet is decoded. Interested readers are referred to [42, 61] for details.
Simpler algorithms in this vein can be used to optimize over classes of
strategies involving the canonical wireless one-hop xor coding scenarios
†Separate control messages must be sent to the nodes from which the antidote
packets are to originate.
148 Subgraph Selection
of Section 3.5.2.1: for controlling routing/MAC to optimally create and
exploit opportunities for coding. Such approaches generalize the cope
protocol which, as discussed in Sections 3.5.2.2-3.5.2.3, assumes given
protocols for routing and MAC that do not take coding into account.
cope has been shown experimentally to yield significant performance
improvements for udp sessions over 802.11 wireless networks [77, 78],
but it has not been rigorously studied under a theoretical model. Queue-
length-based approaches offer one possible approach for distributed op-
timization over various classes of wireless one-hop xor codes.
5.3 Notes and further reading
The first papers to broach the sub ject of subgraph selection in coded
packet networks are due to Cui et al. [31], Lun et al. [95, 96, 97, 100,
101], and Wu et al. [140, 141]. These papers all describe flow-based
approaches for intra-session coding. Subsequent extensions of this work
are plentiful and include [15, 17, 81, 91, 121, 122, 130, 137, 142, 143].
The distributed algorithms that we describe, the primal-dual method
and the subgradient method, first appear in [100] and [95], respectively.
The flow-based approach for inter-session coding that we discuss is due
to Traskov et al. [132]. Another, that deals with cope-like coding in
wireless networks, has recently been put forth by Sengupta et al. [124].
Queue-length-based approaches to subgraph selection in coded packet
networks first appear in [67] for the case of intra-session multicast coding.
The approach and analysis presented in this chapter, from [59], is based
on and generalizes that in [106] for the case of multi-commodity routing.
Queue-length-based approaches for inter-session coding are still in their
infancy, though they show promise. Some recent work on this topic is
described in [42, 61].
An approach to subgraph selection for wireless xor coding in trian-
gular grid networks is given in [39].
6
Security Against Adversarial Errors
6.1 Introduction
Multicast in decentralized settings, such as wireless ad hoc and peer to
peer networks, is seen as a potential application area that can benefit
from distributed network coding and its robustness to arc failures and
packet losses. In such settings, packets are coded and forwarded by end
hosts to other end hosts. It is thus important to consider security against
compromised nodes.
Network coding presents new capabilities as well as challenges for
network security. One advantage of multicast network coding is that it
facilitates the use of a subgraph containing multiple paths to each sink
node. Coding across multiple paths offers useful possibilities for infor-
mation theoretic security against adversaries that observe or control a
limited subset of arcs/transmissions in the network. By adding appro-
priately designed redundancy, error detection or error correction capa-
bilities can be added to a distributed multicast scheme based on random
linear network coding, as described in the following. On the other hand,
coding at intermediate nodes poses a problem for traditional security
techniques. For instance, coded combinations involving an erroneous
packet result in more erroneous packets, so traditional error correction
codes that deal with a limited proportion of erroneous packets are less
effective. Also, traditional signature schemes do not allow for coding
at non-trusted intermediate nodes. A homomorphic signature scheme
by [22], which is based on elliptic curves, allows nodes to sign linear
combinations of packets; under the assumption of the hardness of the
computational co-Diffie-Hellman problem on elliptic curves, it prevents
forging of signatures and detects corruption of packets. In this chapter
149
150 Security Against Adversarial Errors
we focus on the problem of detection and correction of adversarial errors
in multicast network coding, taking an information theoretic approach.
6.1.1 Notational conventions
We denote matrices with bold uppercase letters and vectors with bold
lowercase letters. All vectors are row vectors unless indicated otherwise
with a subscript T. We denote by [x,y] the concatenation of two row vec-
tors xand y. For any vector (or matrix) whose entries (rows/columns)
are indexed by the arcs of a network, we assume a consistent ordering of
the vector entries (matrix rows/columns) corresponding to a topological
ordering of the arcs.
6.2 Error correction
We consider network coded multicast on an acyclic graph G= (N,A),
with a single source node s∈ N and a set T ⊂ N of sink nodes. The
problem is to correct errors introduced on an unknown subset Z ⊂ A of
arcs (or packets†), so as to allow reliable communication. The maximum
rate at which reliable communication is possible depends on |Z| and the
minimum source-sink cut capacity m= mint∈T R(s, t), where R(s, t) is
the minimum cut capacity between sand t. We discuss below some
theoretical bounds as well as constructions of network error-correcting
codes.
6.2.1 Error correction bounds for centralized network coding
6.2.1.1 Model and problem formulation
The case where the network code is centrally designed and known to all
parties (source, sinks and adversary) is the most direct generalization
from traditional algebraic coding theory to network coding.
The problem formulation here is similar to that of Section 3.2 in that
all arcs are assumed to have the same capacity (there can be multiple
arcs connecting a pair of nodes), and we are interested in how large the
source rate can be relative to the arc capacity. We use the term network
to refer to a tuple (G, s, T), or equivalently, (N,A, s, T).
Without loss of generality, we assume that the source node has in-
degree 0. Since the network is acyclic, we can adopt a delay-free network
†See Section 2.5 for a discussion of the correspondence between the static arc-based
and dynamic packet-based network models.
6.2 Error correction 151
coding model, i.e. the nth symbol of an arc lis transmitted only after
o(l) has received the nth symbol of of its input processes. We restrict
consideration to scalar network coding, i.e. the nth symbol transmitted
on an arc lis a function of the nth symbol of each input process of node
o(l), and this function is the same for all n. The transmitted symbols
are from an arc alphabet Yassumed to be equal to Fqfor some prime
power q. In these respects the coding model resembles the delay-free
scalar linear coding model of Section 2.2. However, we allow arbitrary
(possibly nonlinear) coding operations, and allow the source alphabet
Xto be different from the arc alphabet Y; instead of a given number of
fixed-rate source processes, we have a single source process whose rate
log |X | we seek to bound relative to the arc capacity log |Y |.
As in Section 2.2, we can focus on a single symbol for the source and
each arc. The coding operation for an arc lis a function φl:X → Y
if o(l) = s, or φl:Qk:d(k)=o(l)Y → Y if o(l)6=s. The set of coding
operations for all network arcs defines a network code φ={φl:l∈ A}.
Let Xdenote the random source symbol, and Ylthe random symbol
received by the end node d(l) of arc l.†Let
YI(l):= X o(l) = s
{Yk:d(k) = o(l)}o(l)6=s
denote the set of input symbols of an arc l.
We index and consider the arcs l∈ A in topological order, i.e. lower-
indexed arcs are upstream of higher-indexed arcs. For brevity we will
refer to the arc and its index interchangeably. If no arc error occurs on
l,Yl=φl(YI(l)). An arc error is said to occur on lif Yl6=φl(YI(l)).
We say that a z-error occurs (in the network) if errors occur on exactly
zof the arcs. A network code is said to correct az-error if, upon
occurrence of the error, each sink in Tis still able to reproduce the
source symbol. A network code is z-error-correcting if it can correct all
z′-errors for all z′≤z. For a set Zof arcs, if an error occurs on each
arc l∈ Z and no errors occur on other arcs, an Z-error is said to occur;
the set Zis called the error pattern of the error. A Z-error-correcting
code corrects all Z-errors.
The proofs of the network error correction bounds below use a number
of additional definitions. For each arc l∈ A, we define the (error-free)
†The transmitted symbol may differ from the received symbol, e.g. if the error is
caused by interference on the arc. An error can also be caused by an adversarial
or faulty node transmitting a value different from that specified by the network
code. The following analysis, which focuses on the received symbol, applies to
both cases.
152 Security Against Adversarial Errors
global coding function ˜
φl:X → Y , where ˜
φl(X) = Ylwhen all arcs are
error-free. Γ+(Q) := {(i, j ) : i∈Q, j /∈Q}denotes the set of forward
arcs of a cut Q;|Γ+(Q)|is called the size of the cut.
6.2.1.2 Upper bounds
In this section we present upper bounds on the size of the source alpha-
bet, which are analogs of the classical Hamming and Singleton bounds
for point-to-point error-correcting codes. Here we consider arbitrary
(possibly nonlinear) coding functions φl, and define the error value el
associated with arc las
el:= Yl−φl(YI(l))mod q. (6.1)
Note that elis defined relative to the values of the arc inputs YI(l). This
allows us to think of elsimply as an input of node d(l); for a given code
and arc error values, we can inductively determine the arc values Ylin
topological order using
Yl=φl(YI(l)) + elmod q. (6.2)
The same result is obtained if we first find the arc values when the
network is error-free, then “apply” the arc errors in topological order,
i.e. for each arc lfor which el6= 0, we add elto Ylmod qand change
the values of higher-indexed arcs accordingly. Note that an error on arc
ldoes not affect lower-indexed arcs. A (network) error is defined by
the vector e:= (el:l∈ A)∈ Y|A| ; we will refer to an error and its
corresponding vector interchangeably.
Theorem 6.1 (Generalized Hamming Bound) Let (G, s, T)be an
acyclic network, and let m= mint∈T R(s, t). If there exists a z-error-
correcting code on (G, s, T)for an information source with alphabet X,
where z≤m, then
|X | ≤ qm
Pz
i=0 m
i(q−1)i,
where qis the size of arc alphabet Y.
Proof For a given network code φand a set Lof arcs, we denote by
out(φ, z, L, x) the set of all possible values of the vector (Yl:l∈ L) when
the source value is x∈ X and at most zerrors occur in the network.
Suppose φis a z-error-correcting code. Consider any cut Qseparating
the source node sand a sink node t, and any pair of distinct source
6.2 Error correction 153
values x, x′∈ X . To ensure tcan distinguish between xand x′for up to
zerrors, we must have
out(φ, z, Γ+(Q), x)∩out(φ, z, Γ+(Q), x′) = ∅.(6.3)
Consider the set Econsisting of Z-errors where Z ⊂ Γ+(Q),|Z | ≤ z.
Let e,e′be two distinct errors in E, and let k0be the smallest arc
index ksuch that the kth entry of eand e′differ. For a fixed source
value x∈ X , the values of Yl, l < k0are the same under both errors
e,e′while the value of Yk0differs for the two errors. Thus, the value
of (Yl:l∈Γ+(Q)) differs for any pair of distinct errors in E. Let
|Γ+(Q)|=j. Since |E| =Pz
i=0 j
i(q−1)i, we have
|out(φ, z, Γ+(Q), x)| ≥
z
X
i=0 j
i(q−1)i.(6.4)
From (6.13) and (6.4), since there are only qjpossible values for (Yl:
l∈Γ+(Q)), the number of source values is bounded by
|X | ≤ qj
Pz
i=0 j
i(q−1)i.
The theorem follows by noting that the bound holds for any source-sink
cut.
Theorem 6.2 (Generalized Singleton Bound) Let (G, s, T)be an
acyclic network, and let m= mint∈T R(s, t). If there exists a z-error-
correcting code on (G, s, T)for an information source with alphabet X,
where m > 2z, then
log |X | ≤ (m−2z) log q.
Proof Suppose {φl:l∈ A} is a z-error-correcting code for an informa-
tion source with alphabet X, where m > 2zand
|X | > qm−2z.(6.5)
We will show that this leads to a contradiction.
Consider a sink t∈ T for which there exists a cut Qof size mbetween the
source and t. Let k1,...,kmbe the arcs of Γ+(Q), ordered topologically,
i.e. k1< k2<···< km. By (6.5), there exist two distinct source symbols
x, x′∈ X such that ˜
φki(x) = ˜
φki(x′)∀i= 1,2,...,m−2z, so we can
write
˜
φk1(x),..., ˜
φkm(x)= (y1,...,ym−2z, u1,...,uz, w1,...,wz) (6.6)
154 Security Against Adversarial Errors
˜
φk1(x′),..., ˜
φkm(x′)= (y1,...,ym−2z, u′
1,...,u′
z, w′
1,...,w′
z) (6.7)
Suppose the source symbol is x. Let Zbe the set of arcs
{km−2z+1,...,km−z}.
We can construct a Z-error that changes the value of (Yk1,...,Ykm)
from its error-free value in (6.6) to the value
(y1,...,ym−2z, u′
1,...,u′
z, w′′
1,...,w′′
z),(6.8)
as follows. We start with the error-free value of (Yk1,...,Ykm), and
apply errors on the arcs of Zin topological order. First, we apply an
error of value (u′
1−Ykm−2z+1 )mod q= (u′
1−u1)mod qon arc km−2z+1 ,
which causes Ykm−2z+1 to change value from u1to u′
1. Note that this
may change the values of Yj, j > km−2z+1 but not the values of Yj, j <
km−2z+1. We proceed similarly for arcs km−2z+i, i = 2,...,z, in turn:
we apply an error of value (u′
i−Ykm−2z+i)mod qand update the values
of Yj, j > km−2z+iaccordingly. The value of (Yk1,...,Ykm) at the end
of this procedure is given by (6.8).
For the source symbol x′, we can follow a similar procedure to construct,
for the set of arcs Z′={km−z+1,...,km}, a Z′-error that changes the
value of (Yk1,...,Ykm) from its error-free value in (6.7) to the value in
(6.8).
Thus, sink tcannot reliably distinguish between the source symbols x
and x′, which gives a contradiction.
6.2.1.3 Generic linear network codes
Before developing lower bounds on the source alphabet size in the next
section, we introduce the notion of a generic linear network code, which
will be used in constructing network error-correcting codes that prove
the bounds. Intuitively speaking, a generic linear code is a scalar linear
code satisfying the following maximal independence property: for every
subset of arcs, if their (global) coding vectors can be linearly independent
in some network code, then they are linearly independent in a generic
linear code. A generic linear code is formally defined as follows.
Definition 6.1 For an acyclic network (N,A, s, T), let a linear network
code with n-dimensional coding vectors {cl:l∈ A}, whose entries are
elements of a field Fq, be given. Let each node i∈ N be associated with
6.2 Error correction 155
a linear subspace Wiof Fn
q, where
Wi=Fn
qi=s
span({cl:d(l) = i})i∈ N \s.
The network code is generic if, for any subset of arcs S ⊂ A,
Wo(l)6⊂ span({ck:k∈ S\l})∀l∈ S (6.9)
⇒Coding vectors cl, l ∈ S are linearly independent. (6.10)
Note that (6.9) is a necessary condition for (6.10); the definition of a
generic code requires the converse to hold.
Generic linear codes satisfy a stronger linear independence require-
ment compared to multicast linear codes (ref Section 2.2) which require
only that each sink node has a full rank set of inputs. Thus, a generic
linear code is also a multicast linear code, but a multicast linear code is
not in general generic.
Generic linear codes can be constructed using techniques analogous to
those we have seen for constructing multicast linear codes. The random
linear coding technique introduced for constructing multicast codes in
Section 2.4.2 can also construct generic linear network codes with proba-
bility asymptotically approaching 1 in the field size, though significantly
larger field sizes may be required to achieve similar success probabilities
in constructing generic codes, as compared to multicast codes, on a given
network. We can also take a centralized deterministic approach similar
to that in Section 2.4.1. For any positive integer nand an acyclic network
(N,A, s, T), the following algorithm constructs a generic linear network
code over a finite field Fwith more than |A|+n−1
n−1elements. The algo-
rithm is similar to Algorithm 1 for constructing a multicast linear code,
in that it sets the coding vectors†of the network arcs in topological or-
der, starting with nvirtual arcs connecting a virtual source node s′to
the actual source node s.
Note that at step A, there are at most |A|+n−1
n−1sets S, so it is always
possible to find a vector wsatisfying the condition. It can be shown by
induction that the network code constructed by Algorithm 2 is always
generic. The proof is given in [149].
6.2.1.4 Lower bounds
Next we derive lower bounds on the size of the source alphabet, which
generalize the classical Gilbert and Varshamov bounds for point-to-point
†The coding coefficients for an arc can be obtained from the coding vector of the
arc.
156 Security Against Adversarial Errors
Algorithm 2: Centralized algorithm for generic linear network code
construction
Input:N,A, s, T, n
N′:= N ∪ {s′}
A′:= A ∪ {l1,...,lm}where o(li) = s′, d(li) = sfor i= 1,...,m
foreach i= 1,...,m do cli:= [0i−1,1,0n−i]
foreach l∈ A do Initialize cl:= 0;
foreach i∈ N in topological order do
foreach l∈ O(i)do
Achoose cl:= w∈span({ck:d(k) = i}) where
w6∈ span({ck:k∈ S}) for any set Sof n−1 arcs in A\l
such that span({ck:d(k) = i})6⊂ span({ck:k∈ S})
error correcting codes. These bounds give sufficient conditions for the
existence of network error-correcting codes with parameters satisfying
the bounds.
The proofs are by construction. To construct the network error-
correcting codes in this section, we use a generalization of scalar lin-
ear network coding where the arc alphabet Yis a finite field Fqand the
source alphabet Xis a subset of an n-dimensional linear space Fn
q, where
we set nequal to the minimum source-sink cut size m= mint∈T R(s, t).
(For basic scalar linear network coding (ref Section 2.2), the source al-
phabet is the entire n-dimensional linear space Fn
q, where nis equal to
the number of source processes.) For a linear network error-correcting
code, Xis a k-dimensional subspace of Fm
qfor some k≤m. We will con-
sider the source values x∈ X as length-mrow vectors with entries in Fq.
To distinguish between the set of arc coding operations φ={φl:l∈ A}
(defined in Section 6.2.1.1) and the complete network code, which in-
cludes the choice of X ⊂ Fm
qas well, we refer to the former as the
underlying (scalar linear network) code. In this section, we use a generic
linear code as the underlying code φ, which can be constructed for a
given network as described in the previous section. Our remaining task
is to choose Xsuch that its values can be distinguished under a given
set of error events.
The error value elassociated with an arc lis defined as the difference,
6.2 Error correction 157
in Fq, between Yland φl(YI(l)). In place of (6.1)-(6.2) we have
el:= Yl−φl(YI(l)) (6.11)
Yl=φl(YI(l)) + el(6.12)
where all operations are in Fq. An error is defined by the vector e:=
(el:l∈ A)∈F|A|
q. Since Ylis given recursively by (6.12) which is a
linear relation, the value of Ylcan be expressed as the sum
˜
φl(x) + θl(e)
where ˜
φland θlare linear functions, determined by φ, whose arguments
x,eare the source and error values respectively. ˜
φl(x), the error-free
global coding function for arc l, is given by xcT
lwhere clis the global
coding vector of arc lin the underlying code φ.
Consider a set Υ of error patterns, and let Υ∗be the set of all errors
whose error pattern is in Υ. A pair of distinct source values x,x′is said
to be Υ-separable at a sink node tif tcan distinguish between xand x′
for any errors in Υ∗. In other words, ∀e,e′∈Υ∗,
(˜
φl(x) + θl(e) : l∈ I (t)) 6= ( ˜
φl(x′) + θl(e′) : l∈ I (t)).(6.13)
A pair x,x′is said to be Υ-separable if it is Υ-separable at every sink
node.
We wish to translate this condition on separability into a restriction
on the set Xof source values. We assume without loss of generality
that the number of input arcs at each sink is exactly m, the source-sink
minimum cut size. Let Ctdenote the matrix whose columns correspond
to the coding vectors cl(x) of t’s input arcs l∈ I(t). Let pt(e) denote
the row vector (θl(e) : l∈ I(t)). Then (6.13) can be written as
xCt+pt(e)6=x′Ct+pt(e′).
Right-multiplying both sides by C−1
t†, we obtain the following equivalent
condition for x,x′to be Υ-separable at t:∀e,e′∈Υ∗,
x+pt(e)C−1
t6=x′+pt(e′)C−1
t.
Defining the sets
Ξ(φ, Υ, t) := {pt(e)C−1
t:e∈Υ∗}(6.14)
∆(φ, Υ) := [
t∈T
{w=u′−u:u,u′∈Ξ(φ, Υ, t)}(6.15)
†The inverse exists since the underlying code is generic.
158 Security Against Adversarial Errors
and denoting by x+ ∆(φ, Υ) the set {x+w:w∈∆(φ, Υ)}, we have
the following lemma.
Lemma 6.1 (a) A pair of source values x,x′∈Fm
qis Υ-separable if
and only if
x′6∈ x+ ∆(φ, Υ).(6.16)
(b) The network code obtained from the underlying generic code φby
restricting the source alphabet to a set X ⊂ Fm
qis a Υ-error-correcting
code for the network if and only if the vectors in Xare pairwise Υ-
separable.
Let K=|A| be the number of arcs in the network, and let
Υj:= {Z :|Z| =j, Z ∈ Υ}(6.17)
be the subset of error patterns in Υ with exactly jarcs.
Theorem 6.3 (Generalized Gilbert-Varshamov Bound) For any
given error pattern set Υand any positive integer Asatisfying
(A−1)|T |
K
X
j=0
|Υj|(q−1)j
2
< qm,(6.18)
one can construct an Υ-error-correcting code with source alphabet size
|X | =A. For any positive integer ksatisfying
|T |
K
X
j=0
|Υj|(q−1)j
2
< qm−k,
one can construct a k-dimensional linear Υ-error-correcting code, i.e. |X | =
qk.
Proof We first consider a given underlying code φ. By Lemma 6.1, if we
can find a set X ⊂ Fm
qsuch that (6.16) holds for any pair x,x′∈ X , then
the network code obtained from φby restricting the source alphabet to
Xis an Υ-error-correcting code.
For the first part of the theorem, which generalizes the Gilbert bound,
we use a greedy approach similar to that in Gilbert [49]. We show that
for any positive integer Asatisfying
(A−1)|∆(φ, Υ)|< qm,(6.19)
6.2 Error correction 159
one can construct an Υ-error-correcting code with source alphabet size
|X | =A. First, a candidate set Wof source values is initialized as Fm
q.
For i= 1,...,A −1, at the ith step, an arbitrary vector xi∈ W is
chosen, xiis added to X, and all vectors in the set (x+∆(φ, Υ)) ∩ W are
removed from W. This is possible since the number of vectors removed
at each step is at most |xi+ ∆(φ, Υ)|=|∆(φ, Υ)|, so that at each step
i≤A−1, Whas size at least
|Fm
q| − i|∆(φ, Υ)| ≥ qm−(A−1)|∆(φ, Υ)|
>0
by condition (6.19). For the set Xconstructed by this procedure, any
pair x,x′∈ X satisfies (6.16).
For the second part of the theorem, which generalizes the Varshamov
bound, we use an approach similar to that used in Varshamov [135]. We
show that for any positive integer ksatisfying
|∆(φ, Υ)|< qm−k,(6.20)
one can construct a k-dimensional linear Υ-error-correcting code, i.e. |X | =
qk. For a linear code, the source alphabet Xis a linear subspace of Fm
q,
and the condition that (6.16) holds for any pair x,x′∈ X is equivalent
to the condition that
∆(φ, Υ) ∩ X ={0}.(6.21)
We construct Xby constructing its parity check matrix H, which is an
(m−k)×mmatrix with full row rank such that
X={x:x∈Fm
q,HxT=0T}.(6.22)
Defining
∆∗(φ, Υ) := ∆(φ, Υ)\{0},(6.23)
we have that (6.21) is equivalent to the condition that
HwT6=0T∀w∈∆∗(φ, Υ).(6.24)
To construct H, we first partition ∆∗(φ, Υ) into subsets ∆1(φ, Υ),...,∆m(φ, Υ)
such that ∆i(φ, Υ) contains all vectors w∈∆∗(φ, Υ) whose last nonzero
entry is the ith entry, i.e.
∆i(φ, Υ) := {w∈∆∗(φ, Υ) : w= (w1, w2,...,wi,0,...,0), wi6= 0}.
(6.25)
Let hT
ibe the column vector corresponding to the ith column of H,
160 Security Against Adversarial Errors
i.e. H= [hT
1...hT
m]. We set hT
1equal to any nonzero vector in Fm−k
q.
For i= 2,3,...,m, we recursively set hT
iequal to any vector in Fm−k
q\Ki(hT
1,...,hT
i−1),
where
Ki(hT
1,...,hT
i−1) :=
kT∈Fm−k
q:wikT+
i−1
X
j=1
wjhT
j=0Tfor some w∈∆i(φ, Υ)
;
this is possible since the number of possible choices for hT
iis at least
|Fm−k
q| − |Ki(hT
1, . . . , hT
i−1)| ≥ qm−k− |∆i(φ, Υ)|
≥qm−k− |∆(φ, Υ)|
>0.
By construction, (6.24) is satisfied.
To obtain bounds that are independent of the choice of underlying code
φ, note that
|∆(φ, Υ)| ≤ X
t∈T
|Ξ(φ, Υ, t)|2(6.26)
≤ |T | |Υ∗|2(6.27)
where (6.26) follows from the definition of ∆(φ, Υ) in (6.15), and (6.27)
follows from the definition of Ξ(φ, Υ, t) in (6.14). Recall that
Υj:= {Z :|Z| =j, Z ∈ Υ}(6.28)
is the subset of error patterns in Υ with exactly jarcs. Then the number
of errors in Υ∗is given by
|Υ∗|=
K
X
j=0
|Υj|(q−1)j.(6.29)
The theorem follows from combining (6.19)-(6.20) with (6.26), (6.27)
and (6.29).
We next consider the case where Υ is the collection of subsets of zor
fewer arcs, where z≤m. We can use a different proof approach, along
with a bound on |∆(φ, Υ)|which tightens the more general bound (6.27)
for this case, to obtain the following bound which is tighter than that
obtained by simply specializing Theorem 6.3 to this case.
Theorem 6.4 (Strengthened Generalized Varshamov Bound)
For any fixed acyclic network with minimum cut m= mint∈T R(s, t),
6.2 Error correction 161
k=m−2z > 0, and qsufficiently large, there exists a k-dimensional
linear z-error-correcting code for the network.
Proof As in the proof of Theorem 6.3, we construct Xby constructing
its (2z)×mparity check matrix H(ref (6.22)). We need matrix Hto
satisfy (6.24) for the case where Υ is the collection of subsets of zor
fewer arcs. For this case, the sets
∆∗(φ, Υ),∆i(φ, Υ),1≤i≤m,
defined in (6.23) and (6.25) respectively, are denoted by
∆∗(φ, z),∆i(φ, z),1≤i≤m.
Each set ∆i(φ, z),1≤i≤m, is partitioned into |∆i(φ, z)|/(q−1) equiv-
alence classes each of size q−1, such that the vectors in each equivalence
class are nonzero scalar multiples of each other. For any particular one
of these equivalence classes Q ⊂ ∆i(φ, z), a matrix Hsatisfies
HwT=0T(6.30)
for all vectors w∈ Q if and only if it satisfies (6.30) for the vector
(w1,...,wi−1,1,0,...,0) ∈ Q, or equivalently,
hi=−
i−1
X
j=1
wjhj.
Thus, there are exactly q2z(m−1) values for H(corresponding to arbitrary
values for h1,...,hi−1,hi+1,...,hm, the first i−1 of which determine
hi) such that there exists w∈ Q satisfying (6.30). The number of values
for Hsuch that there exists w∈∆∗(φ, z) satisfying (6.30) is then at
most
m
X
i=1
q2z(m−1)|∆i(φ, z)|/(q−1) = q2z(m−1)|∆∗(φ, z )|/(q−1)(6.31)
≤q2z(m−1)|∆(φ, z)|/(q−1).(6.32)
We obtain a bound on |∆(φ, z)|that is tighter than the more general
bound (6.27) as follows. From (6.14) and (6.15), we have
∆(φ, z) = [
t∈T
{pt(e)C−1
t−pt(e′)C−1
t:wH(e)≤z, wH(e′)≤z}
where wHdenotes Hamming weight. Since pt(e) is a linear function in
162 Security Against Adversarial Errors
e, we have
pt(e)C−1
t−pt(e′)C−1
t=pt(e−e′)C−1
t,
and
{e−e′:wH(e)≤z, wH(e′)≤z}={d:wH(d)≤2z}.
Thus,
∆(φ, z) = [
t∈T
{pt(d)C−1
t:wH(d)≤2z}(6.33)
which gives
|∆(φ, z)| ≤ |T |
2z
X
i=0 K
i(q−1)i
<|T |(q−1)2z2m
≤ |T |(q−1)2z2K(6.34)
Using (6.34) we can upper bound (6.32) by
q2z(m−1)|T |2K(q−1)2z−1<|T |2Kq2zm−1(6.35)
If q≥2K|T |, then (6.35) is upper bounded by q2zm. Since the number of
2z×mmatrices over Fqis q2zm, for q≥2K|T |, there exists some value
for Hsuch that HwT6=0T∀w∈∆∗(φ, z), which gives the desired
network code.
6.2.2 Distributed random network coding and
polynomial-complexity error correction
In this section, we consider network error correction in a distributed
packet network setting. The model and approach differ from that of the
previous section in two ways. First, instead of a centrally-designed net-
work code known in advance by all parties, we use distributed random
linear network coding. Second, we allow a fixed amount of overhead
in each packet that can be amortized over large packets. We describe,
for this setting, constructions of asymptotically optimal network error-
correcting codes with polynomial-complexity coding and decoding algo-
rithms.
6.2 Error correction 163
6.2.2.1 Coding vector approach
We consider multicasting of a batch of rpackets from source node sto
the set of sink nodes T, using the distributed random coding approach
with coding vectors described in Section 2.5.1.1. A non-adversarial
packet is formed as a random linear combination†, in Fq, of its input
packets, i.e. packets received by its origin node prior to its formation.
An adversary can arbitrarily corrupt the coding vector as well as the
data symbols of a packet. A packet that is not a linear combination of
its input packets is called adversarial.
We describe below the construction of a network error-correcting code
whose parameters depend on the maximum number zoof adversarial
packets as well as m, the minimum source-sink cut capacity (maximum
error-free multicast rate) in units of packets over the batch. The number
of source packets in the batch is set as
r=m−zo.(6.36)
The proportion of redundant symbols in each packet, denoted ρ, is set
as
ρ= (zo+ǫ)/r (6.37)
for some ǫ > 0.
For i= 1,...,r, the ith source packet is represented as a length-n
row vector xiwith entries in a finite field Fq. The first n−ρn −r
entries of the vector are independent exogenous data symbols, the next
ρn are redundant symbols, and the last rsymbols form the packet’s
coding vector (the unit vector with a single nonzero entry in the ith
position). The corresponding information rate of the code is m−2zo−
ǫ−r2/n, where the r2/n term, due to the overhead of including the
coding vector, decreases with the length nof the packet. We will show
that the probability of error decreases exponentially with ǫ.
We denote by Xthe r×nmatrix whose ith row is xi; it can be written
in the block form U R I where Udenotes the r×(n−ρn −r)
matrix of exogenous data symbols, Rdenotes the r×ρn matrix of
redundant symbols and Iis the r×ridentity matrix.
The rρn redundant symbols are obtained as follows. For any matrix
M, let vT
Mdenote the column vector obtained by stacking the columns of
Mone above the other, and vMits transpose, a row vector. Matrix X,
represented in column vector form, is given by vT
X= [vU,vR,vI]T. Let
†All nsymbols of a packet undergo the same random linear coding operations.
164 Security Against Adversarial Errors
Dbe an rρn ×rn matrix obtained by choosing each entry independently
and uniformly at random from Fq. The redundant symbols constituting
vR(or R) are obtained by solving the matrix equation
D[vU,vR,vI]T=0(6.38)
for vR. The value of Dis known to all parties.
Let yudenote the vector representing a non-adversarial packet u. If
there are no errors in the network, then for all packets u,yu=tuX
where tuis the packet’s coding vector. An adversarial packet can be
viewed as an additional source packet. The vector representing the ith
adversarial packet is denoted zi. Let Zdenote the matrix whose ith row
is zi.
For the rest of this section, we focus on any one of the sink nodes t∈ T .
Let wbe the number of linearly independent packets received by t; let
Y∈Fw×n
qdenote the matrix whose ith row is the vector representing
the ith of these packets. Since all coding operations in the network are
scalar linear operations in Fq,Ycan be be expressed as
Y=GX +KZ (6.39)
where matrices G∈Fw×r
qand K∈Fw×z
qrepresent the linear mappings
from the source and adversarial packets respectively to the sink’s set of
linearly independent input packets.
Since the matrix formed by the last rcolumns of Xis the identity
matrix, the matrix G′formed by the last rcolumns of Yis given by
G′=G+KL,(6.40)
where Lis the matrix formed by the last rcolumns of Z. In the error-
free setting, G′=G; in the presence of errors, the sink knows G′but
not G. Thus, we rewrite (6.39) as
Y=G′X+K(Z−LX) (6.41)
=G′X+E.(6.42)
Matrix Ecan be intuitively interpreted as the effective error seen by
the sink. It gives the difference between the data values in the received
packets and the data values corresponding to their coding vectors; its
last rcolumns are all zero.
Lemma 6.2 With probability at least (1 −1/q)|A| >1− |A|/q where
|A| is the number of arcs in the network, the matrix G′has full column
6.2 Error correction 165
rank, and the column spaces of G′and Kare disjoint except in the zero
vector.
Proof If the adversarial packets were replaced by additional source pack-
ets, the total number of source packets would be at most r+zo=m, by
(6.36). By Theorems 2.3 and 2.6, with probability at least (1 −1/q)|A| ,
random linear network coding in Fqallows tto decode the original source
packets†. This corresponds to Ghaving full column rank and the column
spaces of Gand Kbeing disjoint except in the zero vector. The result
follows by noting from (6.40) that any linear combination of columns of
G′corresponds to a linear combination of one or more columns of G
and zero or more columns of K, which is nonzero and not in the column
space of K.
The decoding process at sink tis as follows. First, the sink determines
z, the minimum cut from the adversarial packets to the sink. This is
with high probability equal to w−r, the difference between the number
of linearly independent packets received by the sink and the number of
source packets. Next, it chooses zcolumns of Ythat, together with
the columns of G′, form a basis for the column space of Y. We assume
without loss of generality that the first zcolumns are chosen, and we
denote the corresponding submatrix G′′. Matrix Y, rewritten in the
basis corresponding to the matrix [G′′ G′], takes the form
Y= [G′′ G′]IzYZ0
0 YXIr
=G′′ IzYZ0+G′0 YXIr(6.43)
where YZ,YXare z×(n−z−r) and r×(n−z−r) matrices respectively.
Let X1,X2,X3be the submatrices of Xconsisting of its first zcolumns,
the next n−z−rcolumns of X, and the last rcolumns respectively.
Lemma 6.3
G′X2=G′(YX+X1YZ) (6.44)
Proof Equating (6.41) and (6.43), we have
G′X+K(Z−LX) = G′′ IzYZ0+G′0 YXIr.(6.45)
†though some of the additional adversarial source packets might not be decodable
if the minimum cut between them and tis less than zo
166 Security Against Adversarial Errors
The column space of G′′ is spanned by the column space of [G′K], so
we can rewrite the equation in the form
G′X+K(Z−LX) = (G′M1+KM2)IzYZ0+G′0 YXIr.
(6.46)
From Lemma 6.2, the column spaces of G′and Kare disjoint except in
the zero vector, so we can equate the terms involving G′:
G′X1X2X3=G′M1IzYZ0+G′0 YXIr.
(6.47)
The leftmost zcolumns of the matrix equation give X1=M1. Substi-
tuting this into the next n−z−rcolumns, we obtain (6.44).
Lemma 6.4 With probability approaching 1 in q, the system of equations
(6.38) and (6.44) can be solved simultaneously to recover X.
Proof We rewrite the system of equations with the matrices X1,X2,YX,I
expressed in column vector form†vT
X1,vT
X2,vT
YX,vT
I.
Let D= [D1D2D3], where D1comprises the first rz columns of D,
D2the next r(n−r−z) columns and D3the remaining r2columns of
D. Let
α=n−r−z(6.48)
and let yi,j denote the (i, j)th entry of matrix YZ. We can write the
system of equations (6.38) and (6.44) in block matrix form as follows:
AvT
X1
vT
X2=G′vT
YX
−D3vT
I(6.49)
where Ais given by
−y1,1G′−y2,1G′... −yz,1G′G′0... ... 0
−y1,2G′−y2,2G′... −yz,2G′0G′0... 0
−y1,3G′−y2,3G′... −yz,3G′.
.
. 0 G′0 0
.
.
..
.
..
.
..
.
..
.
..
.
. 0 ...0
−y1,αG′−y2,α G′... −yz,αG′0 0 0 0 G′
D1D2
.
†Recall that for any matrix M,vT
Mdenotes the column vector obtained by stacking
the columns of Mone above the other.
6.2 Error correction 167
For j= 1,...,α, the jth row of matrices in Acorresponds to the jth
column of (6.44), equivalently written as
−G′X1YZ+G′X2=G′YX.
The bottom submatrix [D1D2] of Acorresponds to (6.38). We will
show that with probability approaching 1 in q,Ahas full column rank
(i.e. the columns of Aare linearly independent) which allows (6.49) to
be solved.
By Lemma 6.2, with probability approaching 1 in q, the columns of
matrix G′, and thus the rightmost αr columns of A, are linearly inde-
pendent. The upper left submatrix of Acan be zeroed out by column
operations involving the right submatrix (rightmost αr columns) of A.
The original matrix Ahas full column rank iff the matrix resulting from
these column operations (or, equivalently, its lower left rρn ×rz sub-
matrix, denoted B) has full column rank. Let dT
k, k = 1, . . . , rz, denote
the kth column of D1. Consider any fixed value of YZ(whose entries
are the yi,j variables), and denote by bT
k, k = 1, . . . , rz , the kth column
of B.bT
kis equal to the sum of dT
kand a linear combination, deter-
mined by the values of the yi,j variables, of the columns of D2. Since
the entries of D1are independently and uniformly distributed in Fq, so
are the entries of B. The probability that Bdoes not have full column
rank is 1 −Qrz
l=1 1−1/qrρn−l+1 , which is upper bounded by qrz−rρn
for sufficiently large q. Using a union bound over the qαz possible values
of YZ, the probability that Bdoes not have full column rank for one or
more values of YZis upper bounded by
qrz−rρn+αz =qrz−n(zo+ǫ)+(n−r−z)z(6.50)
< q−nǫ
where (6.50) follows from (6.37) and (6.48).
The decoding algorithm’s most complex step is solving the system of
equations (6.38) and (6.44), or equivalently the matrix equation (6.49)
which has dimension O(nm). Thus, the decoding algorithm has com-
plexity O(n3m3).
If instead of an omniscient adversary we assume that the adversary
observes only a limited number of packets, or that the source and sinks
share a secret channel, then it is possible to achieve a higher communica-
tion rate of m−zo. Such non-omniscient adversary models are analyzed
in [71, 69].
168 Security Against Adversarial Errors
6.2.2.2 Vector space approach
The vector space approach of Section 2.5.1.2 can be applied directly to
the problem of network error and erasure correction.
Distributed random linear network coding on an unknown network is
modeled as an operator channel, defined as follows.
Definition 6.2 An operator channel associated with ambient space W
is a channel with input and output alphabets equal to the set P(W)of all
subspaces of W. The input Vand output Uof the channel are related
by
U=gk(V)⊕E
where gkis an erasure operator that projects Vonto a random k-dimensional
subspace of V, and E∈ P(W)is the error vector space.
This casts the problem as a point-to-point channel coding problem.
This channel coding formulation admits a Reed-Solomon like code
construction, where the basis for the transmitted vector space is obtained
by evaluating a linearized message polynomial. We consider F=F2mas
a vector space of dimension mover Fq. Let u= (u0, u1,...,uj−1)∈Fj
be the source symbols, and
f(x) :=
j−1
X
i=0
uixqi
the corresponding linearized polynomial. Let A={α1,...,αl}be a set
of l≥jlinearly independent elements of Fspanning an l-dimensional
vector space hAiover Fq. The ambient space Wis given by {(α, β) :
α∈ hAi, β ∈F}which is regarded as a (l+m)-dimensional vector space
over Fq. The vector space Vtransmitted by the source is then the span
of the set
{(α1, f (α1)),...,(αl, f (αl))}.
In [83] is shown that these Reed-Solomon like codes are nearly Singleton
Bound-achieving, and admit an efficient decoding algorithm.
6.3 Detection of adversarial errors
In this section we consider information theoretic detection of errors in-
troduced by an adversary who knows the entire message and coding
6.3 Detection of adversarial errors 169
strategy except for some of the random coding coefficients. Such a sit-
uation may arise, for instance, if an adversary compromises a sink node
that was originally an intended recipient of the source message.
Suppose the adversary sends zerroneous adversarial packets. Each
sink receives packets that are random linear combinations of these r+z
packets. In the error correction case, the minimum overhead (i.e. propor-
tion of redundant information) depends on the number of arcs/transmissions
controlled by the adversary as a proportion of the source-sink minimum
cut. In the error detection case, there is no minimum overhead; it can
be traded off flexibly against the detection probability and coding field
size. An error detection scheme can be used for low overhead monitoring
during normal conditions when no adversary is known to be present, in
conjunction with a higher overhead error correction scheme activated
upon detection of an adversarial error.
Error detection capability is added to random linear coding by in-
cluding a flexible number of hash symbols in each packet. With this
approach, a sink node can detect adversarial modifications with high
probability. The only condition needed, which we will make precise be-
low, is the adversary’s incomplete knowledge of the random network
code when designing its packets. The adversary can have the same
(or greater) transmission capacity compared to the source, even to the
extent where every packet received by a sink is corrupted with an inde-
pendent adversarial packet.
6.3.1 Model and problem formulation
Each packet pin the network is represented by a row vector wpof
d+c+rsymbols from a finite field Fq, where the first dentries are
data symbols, the next care redundant hash symbols and the last r
form the packet’s (global) coefficient vector tp. The hash symbols in
each exogenous packet are given by a function ψd:Fd
q→Fc
qof the
data symbols. The coding vector of the ith exogenous packet is the unit
vector with a single nonzero entry in the ith position.
Let row vector mi∈F(c+d)
qrepresent the concatenation of the data
and hash symbols for the ith exogenous packet, and let Mbe the ma-
trix whose ith row is mi. A packet pis genuine if its data/hash symbols
are consistent with its coding vector, i.e. wp= [tpM,tp]. The exoge-
nous packets are genuine, and any packet formed as a linear combina-
tion of genuine packets is also genuine. Adversarial packets, i.e. packets
transmitted by the adversary, may contain arbitrary coding vector and
170 Security Against Adversarial Errors
data/hash values. An adversarial packet pcan be represented in general
by [tpM+vp,tp], where vpis an arbitrary vector Fc+d
q. If vpis nonzero,
p(and linear combinations of pwith genuine packets) are non-genuine.
A set Sof packets can be represented as a block matrix [TSM+VS|TS]
whose ith row is wpiwhere piis the ith packet of the set. A sink node
tattempts to decode when it has collected a decoding set consisting of
rlinearly independent packets (i.e. packets whose coding vectors are
linearly independent). For a decoding set D, the decoding process is
equivalent to pre-multiplying the matrix [TDM+VD|TD] with T−1
D.
This gives M+T−1
DVD|I, i.e. the receiver decodes to M+˜
M, where
˜
M=T−1
DVD(6.51)
gives the disparity between the decoded packets and the original packets.
If at least one packet in a decoding set is non-genuine, VD6=0, and the
decoded packets will differ from the original packets. A decoded packet
is inconsistent if its data and hash values do not match, i.e. applying
the function ψdto its data values does not yield its hash values. If one
or more decoded packets are inconsistent, the sink declares an error.
The coding vector of a packet transmitted by the source is uniformly
distributed over Fr
q; if a packet whose coding vector has this uniform dis-
tribution is linearly combined with other packets, the resulting packet’s
coding vector has the same uniform distribution. We are concerned with
the distribution of decoding outcomes conditioned on the adversary’s in-
formation, i.e. the adversary’s observed and transmitted packets, and its
information on independencies/dependencies among packets. Note that
in this setup, scaling a packet by some scalar element of Fqdoes not
change the distribution of decoding outcomes.
For given M, the value of a packet pis specified by the row vector
up= [tp,vp]. We call a packet psecret if, conditioned on the value of
vpand the adversary’s information, its coding vector tpis uniformly
distributed over Fr
q\Wfor some (possibly empty) subspace or affine
space W⊂Fr
q. Intuitively, secret packets include genuine packets whose
coding vectors are unknown (in the above sense) to the adversary, as
well as packets formed as linear combinations involving at least one se-
cret packet. A set Sof secret packets is secrecy-independent if each
of the packets remains secret when the adversary is allowed to observe
the other packets in the set; otherwise it is secrecy-dependent. Secrecy-
dependencies arise from the network transmission topology, for instance,
if a packet pis formed as a linear combination of a set Sof secret pack-
ets (possibly with other non-secret packets), then S ∪ {p}is secrecy-
6.3 Detection of adversarial errors 171
dependent. To illustrate these definitions, suppose that the adversary
knows that a sink’s decoding set contains an adversarial packet p1and
a packet p4formed as a linear combination of a non-genuine adversarial
packet p2with a genuine packet p3, and suppose that the adversary does
not observe any packets dependent on p3. Since a decoding set consists
of packets with linearly independent coding vectors, the distribution of
tp4, conditioned on the adversary’s information and any potential value
k2vp2for vp4, is uniform over Fr
q\{ktp1−k2tp2:k∈Fq}. Also, packets
p3and p4are secrecy-dependent.
Consider a decoding set Dcontaining one or more secret packets.
Choosing an appropriate packet ordering, we can express [TD|VD] in
the form
[TD|VD] =
A+B1V1
NA +B2V2
B3V3
(6.52)
where for any given values of Bi∈Fsi×r
q,Vi∈Fsi×(d+c)
q,i= 1,2,3, and
N∈Fs2×s1
q, the matrix A∈Fs1×r
qhas a conditional distribution that
is uniform over all values for which TDis nonsingular. The first s1+s2
rows correspond to secret packets, and the first s1rows correspond to
a set of secrecy-independent packets. s2= 0 if there are no secrecy-
dependencies among the secret packets in D.
6.3.2 Detection probability
In the following we consider decoding from a set of packets that contains
some non-genuine packet, which causes the decoded packets to differ
from the original exogenous packets. The first part of the theorem gives
a lower bound on the number of equally likely potential values of the
decoded packets– the adversary cannot narrow down the set of possible
outcomes beyond this regardless of how it designs its adversarial packets.
The second part provides, for a simple polynomial hash function, an
upper bound on the proportion of potential decoding outcomes that can
have consistent data and hash values, in terms of k=d
c, the ceiling of
the ratio of the number of data symbols to hash symbols. Larger values
for kcorrespond to lower overheads but lower probability of detecting
an adversarial modification. This tradeoff is a design parameter for the
network.
172 Security Against Adversarial Errors
Theorem 6.5 Consider a decoding set Dcontaining a secrecy-independent
subset of s1secret (possibly non-genuine) packets, and suppose the de-
coding set contains at least one non-genuine packet.
a) The adversary cannot determine which of a set of at least (q−1)s1
equally likely values of the decoded packets will be obtained at the sink.
In particular, there will be at least s1packets such that, for each of these,
the adversary cannot determine which of a set of at least q−1equally
likely values will be obtained.
b) Let ψ:Fk
q→Fqbe the function mapping (x1,...,xk),xi∈Fq, to
ψ(x1,...,xk) = x2
1+···+xk+1
k(6.53)
where k=d
c. Suppose the function ψdmapping the data symbols
x1,...,xdto the hash symbols y1,...,ycin an exogenous packet is de-
fined by
yi=ψ(x(i−1)k+1,...,xik)∀i= 1,...,c−1
yc=ψ(x(c−1)k+1,...,xd)
Then the probability of not detecting an error is at most k+1
qs1.
Proof See Appendix 6.A.
Corollary 6.1 Let the hash function ψdbe defined as in Theorem 6.5b.
Suppose a sink obtains more than rpackets, including a secrecy-independent
set of ssecret packets, and at least one non-genuine packet. If the sink
decodes using two or more decoding sets whose union includes all its re-
ceived packets, then the probability of not detecting an error is at most
k+1
qs.
Example: With 2% overhead (k= 50), code length=7, s= 5, the
detection probability is at least 98.9%; with 1% overhead (k= 100),
code length=8, s= 5, the detection probability is at least 99.0%.
While this approach works under relatively mild assumptions as de-
scribed above, it fails in the case where the adversary knows that the
genuine packets received at a sink have coding vectors that lie in some
w-dimensional subspace W⊂Fr
q, the following strategy allows it to con-
trol the decoding outcome and so ensure that the decoded packets have
consistent data and hash values.
The adversary ensures that the sink receives wgenuine packets with
linearly independent coefficient vectors in W, by supplying additional
6.4 Notes and further reading 173
such packets if necessary. The adversary also supplies the sink with
r−wnon-genuine packets whose coding vectors t1,...,tr−ware not in
W. Let tr−w+1,...,trbe a set of basis vectors for W, and let Tbe the
matrix whose ith row is ti. Then the coding vectors of the rpackets can
be represented by the rows of the matrix
I 0
0 K T
where Kis a nonsingular matrix in Fw×w
q. From (6.54), we have
I 0
0 K T˜
M="˜
V
0#
˜
M=T−1I 0
0 K−1"˜
V
0#
=T−1"˜
V
0#
Since the adversary knows Tand controls ˜
V, it can determine ˜
M.
6.4 Notes and further reading
Yeung and Cai were the first to study error correction in network coding.
They developed the bounds on centralized network error correction pre-
sented in this chapter [148, 20]. Low-complexity methods for detection
and correction of adversarial errors in distributed random network cod-
ing were given in Ho et al. [63] and Jaggi et al. [70, 71] respectively. The
vector space approach for correction of errors and erasures in distributed
random network coding was developed by Koetter and Kschischang [83].
In other related work on network coding security not covered in this
section, Charles et al. [22] and Zhao et al. [153] have developed signature
schemes for multicast network coding. The problem of ensuring secrecy
for multicast network coding in the presence of a wire tap adversary has
been considered in [19, 44, 16, 130].
6.A Appendix: Proof of results for adversarial error detection
We first establish two results that are used in the proof of Theorem 6.5.
Consider the hash function defined in (6.53). We call a vector (x1,...,xk+1 )∈
Fk+1
qconsistent if xk+1 =ψ(x1,...,xk).
174 Security Against Adversarial Errors
Lemma 6.5 At most k+ 1 out of the qvectors in a set
{u+γv:γ∈Fq},
where u= (u1,...,uk+1 )is a fixed vector in Fk+1
qand v= (v1,...,vk+1 )
is a fixed nonzero vector in Fk+1
q, can be consistent.
Proof Suppose some vector u+γvis consistent, i.e.
uk+1 +γvk+1 = (u1+γv1)2+···+ (uk+γvk)k+1
Note that for any fixed value of uand any fixed nonzero value of v,
(6.54) is a polynomial equation in γof degree equal to 1 + ˜
k, where
˜
k∈[1, k] is the highest index for which the corresponding vk′is nonzero,
i.e. v˜
k6= 0, vk′= 0 ∀k′>˜
k. By the fundamental theorem of algebra,
this equation can have at most 1 + ˜
k≤1 + kroots. Thus, the property
can be satisfied for at most 1 + kvalues of γ.
Corollary 6.2 Let ube a fixed row vector in Fn
qand Ya fixed nonzero
matrix in Fn×(k+1)
q. If row vector gis distributed uniformly over Fn
q,
then the vector u+gY is consistent with probability at most k+1
q.
Proof Suppose the ith row of Y, denoted yi, is nonzero. We can parti-
tion the set of possible values for gsuch that each partition consists of
all vectors that differ only in the ith entry gi. For each partition, the cor-
responding set of values of u+gY is of the form {u′+giyi:gi∈Fq}.
The result follows from Lemma 6.5 and the fact that giis uniformly
distributed over Fq.
Proof of Theorem 6.5: Writing A′=A+B1,TDbecomes
A′
N(A′−B1) + B2
B3
From (6.51), we have
A′
N(A′−B1) + B2
B3
˜
M=
V1
V2
V3
A′
−NB1+B2
B3
˜
M=
V1
V2−NV1
V3
6.A Appendix: Proof of results for adversarial error detection 175
which we can simplify to
A′
B′˜
M=V1
V′
2(6.54)
by writing
B′=−NB1+B2
B3,V′
2=V2−NV1
V3
Since
A′
B′is nonsingular ⇔TDis nonsingular,
for given values of B′,V1and V′
2, matrix A′∈Fs1×r
qhas a conditional
distribution that is uniform over the set Aof values for which A′
B′
is nonsingular.
The condition that the decoding set contains at least one non-genuine
packet corresponds to the condition VD6=0. We consider two cases. In
each case we show that we can partition the set Asuch that at most a
fraction k+1
qs1of values in each partition give decoding outcomes M+
˜
Mwith consistent data and hash values. The result then follows since
the conditional distribution of values within each partition is uniform.
Case 1: V′
26=0. Let vibe some nonzero row of V′
2, and bithe
corresponding row of B′. Then bi˜
M=vi.
We first partition Ainto cosets
An={An+rTbi:r∈Fs1
q}, n = 1,2,...,χ
where
χ=|A|
qs1
This can be done by the following procedure. Any element of Acan
be chosen as A1. Matrices A2,A3,...,Aχare chosen sequentially; for
each j= 2,...,χ,Ajis chosen to be any element of Anot in the cosets
An, n < j . Note that this forms a partition of A, since the presence of
some element cin two sets Anand Aj,n < j, implies that Ajis also
in An, which is a contradiction. It is also clear that each coset has size
{r:r∈Fs1
q}=qs1.
176 Security Against Adversarial Errors
For each such coset An, the corresponding values of ˜
Msatisfy, from
(6.54),
An+rTbi
B′˜
M=V1
V′
2
An
B′˜
M=V1−rTvi
V′
2
˜
M=An
B′−1V1−rTvi
V′
2
Let Ube the submatrix consisting of the first s1columns of An
B′−1
.
Since Uis nonsingular, we can find a set J ⊂ {1,...,r}of s1indexes
that correspond to independent rows of U. Consider sequentially the
corresponding rows of M+˜
M. The set of potential values for each of
these s1rows, for any given value of the previously considered rows, is
or can be partitioned into sets of the form {u+γvi:γ∈Fq}. Applying
Lemma 6.5 yields the result for this case.
Case 2: V′
2=0, i.e. V2−NV1=V3=0. Then V16=0, since other-
wise V1=V2=0and VD=0which would contradict the assumption
that there is at least one non-genuine packet.
We partition Asuch that each partition consists of all matrices in A
that have the same row space:
An=RAn:R∈Fs1×s1
q,det(R)6= 0, n = 1,2,...,χ
where
|An|=
s1−1
Y
i=0 qs1−qi, χ =|A|
|An|
This can be done by choosing any element of Aas A1, and choosing
An, n = 2,...,χ sequentially such that Anis any element of Anot in
Aj, j < n.
6.A Appendix: Proof of results for adversarial error detection 177
For each An, n = 1,...,χ, the corresponding values of ˜
Msatisfy, from
(6.54),
RAn
B′˜
M=V1
0
An
B′˜
M=R−1V1
0
˜
M=An
B′−1R−1V1
0
Let Ube the submatrix consisting of the first s1columns of An
B′−1
.
We can find an ordered set J={i1,...,is1:i1<··· < is1} ⊂
{1,...,r}of s1indexes that correspond to linearly independent rows
of U. Let UJand MJbe the submatrices of Uand Mrespectively
consisting of the s1rows corresponding to J. Then UJis nonsingular,
and the value of the matrix representation of the corresponding decoded
packets is uniformly distributed over the set
MJ+R′V1:R′∈Fs1×s1
q,det(R′)6= 0(6.55)
Let νbe the rank of V1. Consider a set of νindependent rows of V1.
Denote by Ithe corresponding set of row indexes, and denote by VI
the submatrix of V1consisting of those rows. We can write
V1=LVI
where L∈Fs1×ν
qhas full rank ν. We define RI=R′L, noting that
RIVI=R′LVI=R′V1
and that RIis uniformly distributed over all matrices in Fs1×ν
qthat
have full rank ν. Thus, (6.55) becomes
MJ+RIVI:RI∈Fs1×ν
q,rank(RI) = ν(6.56)
Denote by r1,...,rs1the rows of RI, and by Rnthe submatrix of RI
consisting of its first nrows. We consider the rows sequentially, starting
with the first row r1. For n= 1,...,s1, we will show that conditioned on
any given value of Rn−1, the probability that the inth decoded packet
Min+rnVIis consistent is at most k+1
q. Note that the conditional
distribution of rnis the same for any values of Rn−1that have the same
rank.
178 Security Against Adversarial Errors
Case A: Rn−1has zero rank. This is the case if n= 1, or if n > 1 and
Rn−1=0.
Suppose we remove the restriction rank(RI) = ν, so that rnis uni-
formly distributed over Fν
q. By Corollary 6.2, min+rnVIwould have
consistent data and hash values with probability at most k+1
q. With the
restriction rank(RI) = ν, the probability of rnbeing equal to 0is low-
ered. Since the corresponding decoded packet min+rnVIis consistent
for rn=0, the probability that it is consistent is less than k+1
q.
Case B: n > 1 and Rn−1has nonzero rank.
Conditioned on rnbeing in the row space of Rn−1,rn=gRn−1
where gis uniformly distributed over Fn−1
q. Since Rn−1VI6=0, by
Corollary 6.2, the corresponding decoded packet
min+rnVI=min+gRn−1VI
is consistent with probability at most k+1
q.
Conditioned on rnnot being in the row space of Rn−1, we can parti-
tion the set of possible values for rninto cosets
r+gRn−1:g∈Fn−1
q
where ris not in the row space of Rn−1; the corresponding values of the
inth decoded packet are given by
min+rVI+gRn−1VI:g∈Fn−1
q.
Noting as before that Rn−1VI6=0and applying Corollary 6.2, the inth
decoded packet is consistent with probability at most k+1
q.
Proof of Corollary 6.1: Suppose two or more different sets of packets
are used for decoding. If not all of them contain at least one non-genuine
packet, the decoded values obtained from different decoding sets will
differ, and indicate an error. Otherwise, suppose all the decoding sets
contain at least one non-genuine packet. Consider the sets in turn,
denoting by s′
ithe number of unmodified packets in the ith set that are
not in any set j < i. For any particular values of packets in sets j < i,
we have from Theorem 6.5 that at most a fraction k+1
qs′
iof decoding
outcomes for set ihave consistent data and hash values. Thus, the
overall fraction of consistent decoding outcomes is at most k+1
qPis′
i=
k+1
qs.
Bibliography
S. Aceda´nski, S. Deb, M. M´edard, and R. Koetter, “How good is random
linear coding based distributed networked storage?” in Proc. WINMEE,
RAWNET and NETCOD 2005 Workshops, Riva del Garda, Italy, Apr.
2005.
M. Adler, N. J. A. Harvey, K. Jain, R. Kleinberg, and A. R. Lehman, “On the
capacity of information networks,” IEEE Transactions on Information
Theory, vol. 52, no. 6, June 2006.
A. Agarwal and M. Charikar, “On the advantage of network coding for im-
proving network throughput,” in Information Theory Workshop, 2004.
R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information
flow,” IEEE Trans. Inform. Theory, vol. 46, no. 4, pp. 1204–1216, July
2000.
R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows: Theory, Algo-
rithms, and Applications. Upper Saddle River, NJ: Prentice Hall, 1993.
B. Awerbuch and T. Leighton, “A simple local-control approximation algo-
rithm for multicommodity flow,” in Proc. 34th Annu. IEEE Symp. Foun-
dations of Computer Science, Palo Alto, CA, Nov. 1993, pp. 459–468.
——, “Improved approximation algorithms for the multicommodity flow prob-
lem and local competitive routing in dynamic networks,” in STOC ’94:
Proc. 26th Ann. ACM Symp. Theory of Computing, Montreal, QC,
Canada, May 1994, pp. 487–496.
D. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex analysis and optimization.
Athena Scientific, 2003.
D. P. Bertsekas, “A class of optimal routing algorithms for communication
networks,” in Proc. 5th Int. Conf. Computer Communication (ICCC ’80),
Atlanta, GA, Oct. 1980, pp. 71–76.
——, Nonlinear Programming. Belmont, MA: Athena Scientific, 1995.
——, Network Optimization: Continuous and Discrete Models. Belmont,
MA: Athena Scientific, 1998.
D. P. Bertsekas, E. M. Gafni, and R. G. Gallager, “Second derivative algo-
rithms for minimum delay distributed routing in networks,” IEEE Trans.
Commun., vol. 32, no. 8, pp. 911–919, Aug. 1984.
D. P. Bertsekas and R. Gallager, Data Networks, 2nd ed. Upper Saddle River,
NJ: Prentice Hall, 1992.
D. Bertsimas, I. C. Paschalidis, and J. Tsitsiklis, “On the large deviations be-
havior of acyclic networks of G/G/1 queues,” Ann. Appl. Probab., vol. 8,
179
180 Bibliography
no. 4, pp. 1027–1069, Nov. 1998.
S. Bhadra, S. Shakkottai, and P. Gupta, “Min-cost selfish multicast with net-
work coding,” IEEE Trans. Inform. Theory, vol. 52, no. 11, pp. 5077–
5087, Nov. 2006.
K. Bhattad and K. R. Nayayanan, “Weakly secure network coding,” in Proc.
WINMEE, RAWNET and NETCOD 2005 Workshops, Riva del Garda,
Italy, Apr. 2005.
K. Bhattad, N. Ratnakar, R. Koetter, and K. R. Narayanan, “Minimal net-
work coding for multicast,” in Proc. 2005 IEEE Int. Symp. Information
Theory (ISIT 2005), Adelaide, Australia, Sept. 2005, pp. 1730–1734.
J. W. Byers, M. Luby, and M. Mitzenmacher, “A digital fountain approach
to asynchronous reliable multicast,” IEEE J. Select. Areas Commun.,
vol. 20, no. 8, pp. 1528–1540, Oct. 2002.
N. Cai and R. W. Yeung, “Secure network coding,” in Proc. 2002 IEEE
Int. Symp. Information Theory (ISIT 2002), Lausanne, Switzerland,
June/July 2002, p. 323.
——, “Network error correction, part II: Lower bounds,” Commun. Inf. Syst.,
vol. 6, no. 1, pp. 37–54, 2006.
T. H. Chan, “On the optimality of group network codes,” in Proc. 2005 IEEE
Int. Symp. Information Theory (ISIT 2005), Adelaide, Australia, Sept.
2005, pp. 1992–1996.
D. Charles, K. Jain, and K. Lauter, “Signatures for network coding,” in Proc.
2006 Conf. Information Sciences and Systems (CISS 2006), Princeton,
NJ, Mar. 2006, invited paper.
C. Chekuri, C. Fragouli, and E. Soljanin, “on average throughput and alpha-
bet size in network coding,” IEEE Transactions on Information Theory,
vol. 52, no. 6, June 2006.
H. Chen and A. Mandelbaum, “Discrete flow networks: Bottleneck analysis
and fluid approximations,” Math. Oper. Res, vol. 16, no. 2, pp. 408–446,
May 1991.
H. Chen and D. D. Yao, Fundamentals of Queueing Networks: Performance,
Asymptotics, and Optimization, ser. Applications of Mathematics. New
York, NY: Springer, 2001, vol. 46.
P. A. Chou, Y. Wu, and K. Jain, “Practical network coding,” in
Proceedings of 41st Annual Allerton Conference on Communica-
tion, Control, and Computing, October 2003. [Online]. Available:
http://research.microsoft.com/ pachou/pubs/ChouWJ03.pdf
——, “Practical network coding,” in Proc. 41st Annu. Allerton Conf. Com-
munication, Control, and Computing, Monticello, IL, Oct. 2003.
T. M. Cover and J. A. Thomas, Elements of Information Theory. New York,
NY: John Wiley & Sons, 1991.
R. L. Cruz and A. V. Santhanam, “Optimal routing, link scheduling and power
control in multi-hop wireless networks,” in Proc. IEEE INFOCOM 2003,
vol. 1, San Francisco, CA, Mar./Apr. 2003, pp. 702–711.
I. Csisz´ar, “Linear codes for sources and source networks: Error exponents,
universal coding,” IEEE Trans. Inform. Theory, vol. 28, no. 4, pp. 585–
592, July 1982.
Y. Cui, Y. Xue, and K. Nahrstedt, “Optimal distributed multicast routing us-
ing network coding: Theory and applications,” SIGMETRICS Perform.
Eval. Rev., vol. 32, no. 2, pp. 47–49, Sept. 2004.
A. F. Dana, R. Gowaikar, R. Palanki, B. Hassibi, and M. Effros, “Capacity of
Bibliography 181
wireless erasure networks,” IEEE Trans. Inform. Theory, vol. 52, no. 3,
pp. 789–804, Mar. 2006.
S. Deb and M. Medard, “Algebraic gossip: A network coding approach to
optimal multiple rumor mongering,” IEEE Transactions on Information
Theory, submitted, 2004.
S. Deb, M. M´edard, and C. Choute, “Algebraic gossip: A network coding
approach to optimal multiple rumor mongering,” IEEE Trans. Inform.
Theory, vol. 52, no. 6, pp. 2486–2507, June 2006.
S. Deb and R. Srikant, “Congestion control for fair resource allocation in
networks with multicast flows,” IEEE/ACM Trans. Networking, vol. 12,
no. 2, pp. 274–285, Apr. 2004.
R. Dougherty, C. Freiling, and K. Zeger, “Insufficiency of linear coding in
network information flow,” IEEE Transactions on Information Theory,
vol. 51(8):, pp. 2745–2759, August 2005.
——, “Networks, matroids, and non-Shannon information inequalities,” IEEE
Trans. Inform. Theory, vol. 53, no. 6, June 2007.
——, “Insufficiency of linear coding in network information flow,” IEEE Trans.
Inform. Theory, vol. 51, no. 8, pp. 2745–2759, Aug. 2005.
M. Effros, T. Ho, and S. Kim, “A tiling approach to network code design for
wireless networks,” in Proc. 2006 IEEE Information Theory Workshop
(ITW 2006), Punta del Este, Uruguay, Mar. 2006, pp. 62–66.
E. Erez and M. Feder, “Convolutional network codes,” in Proc. 2004 IEEE Int.
Symp. Information Theory (ISIT 2004), Chicago, IL, June/July 2004, p.
146.
——, “Convolutional network codes for cyclic networks,” in Proc. WINMEE,
RAWNET and NETCOD 2005 Workshops, Riva del Garda, Italy, Apr.
2005.
A. Eryilmaz and D. S. Lun, “Control for inter-session network coding,” in
Proc. 2007 Information Theory and Applications Workshop (ITA 2007),
San Diego, CA, Jan./Feb. 2007.
M. Feder, D. Ron, and A. Tavory, “Bounds on linear codes for network mul-
ticast,” Electronic Colloquium on Computational Complexity, Report
TR03-033, May 2003.
J. Feldman, T. Malkin, C. Stein, and R. A. Servedio, “On the capacity of
secure network coding,” in Proc. 42nd Annu. Allerton Conf. Communi-
cation, Control, and Computing, Monticello, IL, Sept./Oct. 2004.
C. Fragouli and E. Soljanin, “A connection between network coding and con-
volutional codes,” in Proc. 2004 IEEE Int. Conf. Communications (ICC
2004), vol. 2, Paris, France, June 2004, pp. 661–666.
——, “Information flow decomposition for network coding,” IEEE Transac-
tions on Information Theory, submitted, 2004.
——, “Information flow decomposition for network coding,” IEEE Trans. In-
form. Theory, vol. 52, no. 3, pp. 829–848, Mar. 2006.
S. Ghez, S. Verd´u, and S. C. Schwartz, “Stability properties of slotted Aloha
with multipacket reception capability,” IEEE Trans. Automat. Contr.,
vol. 33, no. 7, pp. 640–649, July 1988.
E. N. Gilbert, “A comparison of signaling alphabets,” Bell Syst. Tech. J.,
vol. 31, pp. 504–522, May 1952.
C. Gkantsidis and P. R. Rodriguez, “Network coding for large scale content
distribution,” in Proc. IEEE INFOCOM 2005, vol. 4, Miami, FL, Mar.
2005, pp. 2235–2245.
182 Bibliography
R. Gowaikar, A. F. Dana, R. Palanki, B. Hassibi, and M. Effros, “On the
capacity of wireless erasure networks,” in Proc. 2004 IEEE Int. Symp.
Information Theory (ISIT 2004), Chicago, IL, June/July 2004, p. 401.
T. S. Han, “Slepian-Wolf-Cover theorem for networks of channels,” Inf. Con-
trol, vol. 47, pp. 67–83, 1980.
N. J. A. Harvey, R. Kleinberg, C. Nair, and Y. Wu, “A ’chicken & egg’ network
coding problem,” in Proc. 2007 IEEE Int. Symp. Information Theory
(ISIT 2007), Nice, France, June 2007.
N. J. A. Harvey, D. R. Karger, and K. Murota, “Deterministic network coding
by matrix completion,” in SODA ’05: Proc. 16th Annu. ACM-SIAM
Symp. Discrete Algorithms, Jan. 2005, pp. 489–498.
T. Ho, D. R. Karger, M. M´edard, and R. Koetter, “Network coding from
a network flow perspective,” in Proceedings of the IEEE International
Symposium on Information Theory, 2003.
T. Ho, M. M´edard, R. Koetter, D. Karger, M. Effros, J. Shi, and B. Leong, “A
random linear network coding approach to multicast,” IEEE Transactions
on Information Theory, vol. 52, no. 10, pp. 4413–4430, October 2006.
T. Ho, M. M´edard, J. Shi, M. Effros, and D. R. Karger, “On randomized
network coding,” in Proceedings of 41st Annual Allerton Conference
on Communication, Control, and Computing, October 2003. [Online].
Available: http://web.mit.edu/trace/www/allerton.pdf
T. Ho and H. Viswanathan, “Dynamic algorithms for multicast with intra-
session network coding,” in Proc. 43rd Annual Allerton Conference on
Communication, Control, and Computing, 2005.
——, “Dynamic algorithms for multicast with intra-session network coding,”
IEEE Transactions on Information Theory, submitted, 2006.
T. Ho, “Networking from a network coding perspective,” Ph.D. dissertation,
Massachusetts Institute of Technology, May 2004.
T. Ho, Y.-H. Chang, and K. J. Han, “On constructive network coding for
multiple unicasts,” in Proc. 44th Annu. Allerton Conf. Communication,
Control, and Computing, Monticello, IL, Sept. 2006, invited paper.
T. Ho, R. Koetter, M. M´edard, D. R. Karger, and M. Effros, “The benefits
of coding over routing in a randomized setting,” in Proc. 2003 IEEE Int.
Symp. Information Theory (ISIT 2003), Yokohama, Japan, June/July
2003, p. 442.
T. Ho, B. Leong, R. Koetter, M. M´edard, M. Effros, and D. R. Karger, “Byzan-
tine modification detection for multicast networks using randomized net-
work coding,” in Proc. 2004 IEEE Int. Symp. Information Theory (ISIT
2004), Chicago, IL, June/July 2004, p. 144.
T. Ho, B. Leong, R. Koetter, and M. Medard, “Distributed asynchronous al-
gorithms for multicast network coding,” in Proceedings of First Workshop
on Network Coding, Theory and Applications, 2005.
T. Ho, M. M´edard, and M. Effros, “Network coding for correlated sources,” in
Proc. 2004 Conf. Information Sciences and Systems (CISS 2004), Prince-
ton, NJ, Mar. 2004, invited paper.
T. Ho, M. M´edard, J. Shi, M. Effros, and D. R. Karger, “On randomized
network coding,” in Proc. 41st Annu. Allerton Conf. Communication,
Control, and Computing, Monticello, IL, Oct. 2003.
T. Ho and H. Viswanathan, “Dynamic algorithms for multicast with intra-
session network coding,” in Proc. 43rd Annu. Allerton Conf. Communi-
cation, Control, and Computing, Monticello, IL, Sept. 2005.
Bibliography 183
S. Jaggi, P. Chou, and K. Jain, “Low complexity algebraic network codes,” in
Proc. 2003 IEEE Int. Symp. Information Theory (ISIT 2003), Yokohama,
Japan, June/July 2003, p. 368.
S. Jaggi and M. Langberg, “Resilient network coding in the presence of eaves-
dropping byzantine adversaries,” in Proc. IEEE International Symposium
on Information Theory, Nice, France, June 2007.
S. Jaggi, M. Langberg, T. Ho, and M. Effros, “Correction of adversarial errors
in networks,” in Proc. 2005 IEEE International Symposium on Informa-
tion Theory (ISIT 2005), Adelaide, Australia, Sept. 2005, pp. 1455–1459.
S. Jaggi, M. Langberg, S. Katti, T. Ho, D. Katabi, and M. M´edard, “Resilient
network coding in the presence of byzantine adversaries,” in Proc. IEEE
INFOCOM 2007, Anchorage, AK, May 2007.
S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain, and L. M.
G. M. Tolhuizen, “Polynomial time algorithms for multicast network code
construction,” IEEE Trans. Inform. Theory, vol. 51, no. 6, pp. 1973–1982,
June 2005.
——, “Polynomial time algorithms for multicast network code construction,”
IEEE Trans. Inform. Theory, vol. 51, no. 6, pp. 1973–1982, June 2005.
K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, “Impact of interference
on multi-hop wireless network performance,” in MobiCom ’03: Proc. 9th
Annu. Int. Conf. Mobile Computing and Networking, San Diego, CA,
2003, pp. 66–80.
M. Johansson, L. Xiao, and S. Boyd, “Simultaneous routing and power allo-
cation in CDMA wireless data networks,” in Proc. 2003 IEEE Int. Conf.
Communications (ICC 2003), vol. 1, Anchorage, AK, May 2003, pp. 51–
55.
S. Katti, D. Katabi, W. Hu, H. Rahul, and M. Mdard, “Practical network cod-
ing for wireless environments,” in Proceedings of 43rd Allerton Conference
on Communication, Control, and Computing, Monticello, IL, September
2005.
S. Katti, D. Katabi, W. Hu, H. Rahul, and M. M´edard, “The importance of
being opportunistic: Practical network coding for wireless environments,”
in Proc. 43rd Annu. Allerton Conf. Communication, Control, and Com-
puting, Monticello, IL, Sept. 2005.
S. Katti, H. Rahul, W. Hu, D. Katabi, M. M´edard, and J. Crowcroft, “XORs
in the air: Practical wireless network coding,” SIGCOMM Comput. Com-
mun. Rev., vol. 36, no. 4, pp. 243–254, 2006.
F. P. Kelly, Reversibility and Stochastic Networks. Chichester: John Wiley
& Sons, 1979.
R. Khalili and K. Salamatian, “On the capacity of erasure relay channel:
Multi-relay case,” in Proc. 2005 IEEE Information Theory Workshop
(ITW 2005), Rotorua, New Zealand, Aug./Sept. 2005.
M. Kim, M. M´edard, V. Aggarwal, U.-M. O’Reilly, W. Kim, C. W. Ahn,
and M. Effros, “Evolutionary approaches to minimizing network coding
resources,” in Proc. IEEE INFOCOM 2007, Anchorage, AK, May 2007.
M. Kodialam and T. Nandagopal, “Characterizing achievable rates in multi-
hop wireless mesh networks with orthogonal channels,” IEEE/ACM
Trans. Networking, vol. 13, no. 4, pp. 868–880, Aug. 2005.
R. Koetter and F. R. Kschischang, “Coding for errors and erasures in ran-
dom network coding,” IEEE Transactions on Information Theory, 2007,
submitted.
184 Bibliography
R. Koetter, M. Effros, T. Ho, and M. M´edard, “Network codes as codes on
graphs,” in Proc. 2004 Conf. Information Sciences and Systems (CISS
2004), Princeton, NJ, Mar. 2004.
R. Koetter and M. M´edard, “An algebraic approach to network coding,”
IEEE/ACM Trans. Networking, vol. 11, no. 5, pp. 782–795, Oct. 2003.
G. Kramer and S. Savari, “Progressive d-separating edge set bounds on net-
work coding rates,” in Proceedings of the International Symposium on
Information Theory, 2005.
S.-Y. R. Li and R. W. Yeung, “On convolutional network coding,” 2006.
S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans.
Inform. Theory, vol. 49, no. 2, pp. 371–381, Feb. 2003.
Z. Li and B. Li, “Network coding in undirected networks,” in Proc. 2004
Conf. Information Sciences and Systems (CISS 2004), Princeton, NJ,
Mar. 2004.
——, “Network coding: The case of multiple unicast sessions,” in Proc. 42nd
Annu. Allerton Conf. Communication, Control, and Computing, Monti-
cello, IL, Sept./Oct. 2004.
——, “Efficient and distributed computation of maximum multicast rates,”
in Proc. IEEE INFOCOM 2005, vol. 3, Miami, FL, Mar. 2005, pp. 1618–
1628.
M. Luby, “LT codes,” in Proc. 34th Annu. IEEE Symp. Foundations of Com-
puter Science, Vancouver, Canada, Nov. 2002, pp. 271–280.
D. S. Lun, “Efficient operation of coded packet networks,” Ph.D. dissertation,
Massachusetts Institute of Technology, June 2006.
D. S. Lun, M. M´edard, and M. Effros, “On coding for reliable communication
over packet networks,” in Proc. 42nd Annu. Allerton Conf. Communica-
tion, Control, and Computing, Monticello, IL, Sept./Oct. 2004, invited
paper.
D. S. Lun, M. M´edard, T. Ho, and R. Koetter, “Network coding with a cost
criterion,” in Proc. 2004 Int. Symp. Information Theory and its Applica-
tions (ISITA 2004), Parma, Italy, Oct. 2004, pp. 1232–1237.
D. S. Lun, M. M´edard, and D. R. Karger, “On the dynamic multicast problem
for coded networks,” in Proc. WINMEE, RAWNET and NETCOD 2005
Workshops, Riva del Garda, Italy, Apr. 2005.
D. S. Lun, M. M´edard, and R. Koetter, “Efficient operation of wireless packet
networks using network coding,” in Proc. Int. Workshop Convergent Tech-
nologies (IWCT) 2005, Oulu, Finland, June 2005, invited paper.
D. S. Lun, M. M´edard, R. Koetter, and M. Effros, “Further results on coding
for reliable communication over packet networks,” in Proc. 2005 IEEE
Int. Symp. Information Theory (ISIT 2005), Adelaide, Australia, Sept.
2005, pp. 1848–1852.
D. S. Lun, P. Pakzad, C. Fragouli, M. M´edard, and R. Koetter, “An analysis
of finite-memory random linear coding on packet streams,” in Proc. 4th
Int. Symp. Modeling and Optimization in Mobile, Ad Hoc and Wireless
Networks (WiOpt ’06), Boston, MA, Apr. 2006.
D. S. Lun, N. Ratnakar, R. Koetter, M. M´edard, E. Ahmed, and H. Lee,
“Achieving minimum-cost multicast: A decentralized approach based on
network coding,” in Proc. IEEE INFOCOM 2005, vol. 3, Miami, FL, Mar.
2005, pp. 1608–1617.
D. S. Lun, N. Ratnakar, M. M´edard, R. Koetter, D. R. Karger, T. Ho,
E. Ahmed, and F. Zhao, “Minimum-cost multicast over coded packet
Bibliography 185
networks,” IEEE Trans. Inform. Theory, vol. 52, no. 6, pp. 2608–2623,
June 2006.
P. Maymounkov, “Online codes,” NYU, Technical Report TR2002-833, Nov.
2002.
P. Maymounkov, N. J. A. Harvey, and D. S. Lun, “Methods for efficient net-
work coding,” in Proc. 44th Annu. Allerton Conf. Communication, Con-
trol, and Computing, Monticello, IL, Sept. 2006.
M. M´edard, M. Effros, T. Ho, and D. R. Karger, “On coding for non-multicast
networks,” in Proc. 41st Annu. Allerton Conf. Communication, Control,
and Computing, Monticello, IL, Oct. 2003.
R. Motwani and P. Raghavan, Randomized Algorithms. Cambridge: Cam-
bridge University Press, 1995.
M. Neely, E. Modiano, and C. Rohrs, “Dynamic power allocation and routing
for time varying wireless networks,” IEEE J. Select. Areas Commun.,
vol. 23, no. 1, pp. 89–103, Jan. 2005.
M. J. Neely, “Energy optimal control for time-varying wireless networks,”
IEEE Trans. Inform. Theory, vol. 52, no. 7, pp. 2915–2934, July 2006.
G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization.
New York, NY: John Wiley & Sons, 1999.
A. Ouorou, P. Mahey, and J.-P. Vial, “A survey of algorithms for convex
multicommodity flow problems,” Manage. Sci., vol. 46, no. 1, pp. 126–
147, Jan. 2000.
J. G. Oxley, Matroid Theory. Oxford Univ. Press, 1992.
P. Pakzad, C. Fragouli, and A. Shokrollahi, “Coding schemes for line net-
works,” in Proc. 2005 IEEE Int. Symp. Information Theory (ISIT 2005),
Adelaide, Australia, Sept. 2005, pp. 1853–1857.
J.-S. Park, M. Gerla, D. S. Lun, Y. Yi, and M. M´edard, “Codecast: A network-
coding-based ad hoc multicast protocol,” IEEE Wireless Commun. Mag.,
vol. 13, no. 5, pp. 76–81, Oct. 2006.
I. C. Paschalidis and Y. Liu, “Large deviations-based asymptotics for inventory
control in supply chains,” Oper. Res., vol. 51, no. 3, pp. 437–460, May–
June 2003.
A. Ramamoorthy, K. Jain, P. A. Chou, and M. Effros, “Separating distributed
source coding from network coding,” IEEE Transactions on Information
Theory, vol. 52, no. 6, June 2006.
S. Ramanathan, “Multicast tree generation in networks with asymmetric
links,” IEEE/ACM Trans. Networking, vol. 4, no. 4, pp. 558–568, Aug.
1996.
A. Rasala Lehman and E. Lehman, “Complexity classification of network in-
formation flow problems,” in SODA ’04: Proc. 15th Annu. ACM-SIAM
Symp. Discrete algorithms, New Orleans, LA, Jan. 2004, pp. 142–150.
N. Ratnakar, R. Koetter, and T. Ho, “Linear flow equations for network coding
in the multiple unicast case,” in Proc. DIMACS Working Group Network
Coding, Piscataway, NJ, Jan. 2005.
N. Ratnakar, D. Traskov, and R. Koetter, “Approaches to network coding
for multiple unicasts,” in Proc. 2006 International Zurich Seminar on
Communications (IZS 2006), Zurich, Switzerland, Feb. 2006, pp. 70–73,
invited paper.
S. Riis, “Linear versus non-linear Boolean functions in network flow,” in Proc.
2004 Conf. Information Sciences and Systems (CISS 2004), Princeton,
NJ, Mar. 2004.
186 Bibliography
J. R. Roche, R. W. Yeung, and K. P. Hau, “Symmetrical multilevel diversity
coding,” IEEE Trans. Inform. Theory, vol. 43, no. 3, pp. 1059–1064, May
1997.
Y. E. Sagduyu and A. Ephremides, “Crosslayer design and distributed MAC
and network coding in wireless ad hoc networks,” in Proc. 2005 IEEE
Int. Symp. Information Theory (ISIT 2005), Adelaide, Australia, Sept.
2005, pp. 1863–1867.
——, “Joint scheduling and wireless network coding,” in Proc. WINMEE,
RAWNET and NETCOD 2005 Workshops, Riva del Garda, Italy, Apr.
2005.
P. Sanders, S. Egner, and L. Tolhuizen, “Polynomial time algorithms for net-
work information flow,” in SPAA ’03: Proc. 15th Annu. ACM Symp.
Parallel Algorithms and Architectures, San Diego, CA, June 2003, pp.
286–294.
S. Sengupta, S. Rayanchu, and S. Banerjee, “An analysis of wireless network
coding for unicast sessions: The case for coding-aware routing,” in Proc.
IEEE INFOCOM 2007, Anchorage, AK, May 2007.
H. D. Sherali and G. Choi, “Recovery of primal solutions when using subgradi-
ent optimization methods to solve Lagrangian duals of linear programs,”
Oper. Res. Lett., vol. 19, pp. 105–113, 1996.
A. Shokrollahi, “Raptor codes,” IEEE Trans. Inform. Theory, vol. 52, no. 6,
pp. 2551–2567, June 2006.
N. Shulman, “Communication over an unknown channel via common broad-
casting,” Ph.D. dissertation, Tel Aviv University, July 2003.
N. Shulman and M. Feder, “Static broadcasting,” in Proc. 2000 IEEE Int.
Symp. Information Theory (ISIT 2000), Sorrento, Italy, June 2000, p. 23.
R. Srikant, The Mathematics of Internet Congestion Control. Boston, MA:
Brikh¨auser, 2004.
J. Tan and M. M´edard, “Secure network coding with a cost criterion,” in
Proc. 4th Int. Symp. Modeling and Optimization in Mobile, Ad Hoc and
Wireless Networks (WiOpt ’06), Boston, MA, Apr. 2006.
L. Tassiulas and A. F. Ephremides, “Stability properties of constrained queue-
ing systems and scheduling policies for maximum throughput in multihop
radio networks,” IEEE Trans. Automat. Contr., vol. 37, no. 12, pp. 1936–
1948, Dec. 1992.
D. Traskov, N. Ratnakar, D. S. Lun, R. Koetter, and M. M´edard, “Network
coding for multiple unicasts: An approach based on linear optimization,”
in Proc. 2006 IEEE Int. Symp. Information Theory (ISIT 2006), Seattle,
WA, July 2006, pp. 1758–1762.
J. N. Tsitsiklis, “Decentralized detection,” in Advances in Statistical Signal
Processing. Greenwich, CT: JAI Press, 1993, vol. 2, pp. 297–344.
E. C. van der Meulen, “Three-terminal communication channels,” Adv. Appl.
Probab., vol. 3, pp. 120–154, 1971.
R. R. Varshamov, “Estimate of the number of signals in error-correcting
codes,” Dokl. Acad. Nauk SSSR, vol. 117, pp. 739–741, 1957.
Y. Wu, “A trellis connectivity analysis of random linear network coding with
buffering,” in Proc. 2006 IEEE Int. Symp. Information Theory (ISIT
2006), Seattle, WA, July 2006, pp. 768–772.
Y. Wu, M. Chiang, and S.-Y. Kung, “Distributed utility maximization for
network coding based multicasting: A critical cut approach,” in Proc. 4th
Int. Symp. Modeling and Optimization in Mobile, Ad Hoc and Wireless
Bibliography 187
Networks (WiOpt ’06), Boston, MA, Apr. 2006.
Y. Wu, P. A. Chou, and S. Y. Kung, “Information exchange in wireless net-
works with network coding and physical-layer broadcast,” in CISS, 2005.
Y. Wu, P. A. Chou, and S.-Y. Kung, “Information exchange in wireless net-
works with network coding and physical-layer broadcast,” in Proc. 2005
Conf. Information Sciences and Systems (CISS 2005), Baltimore, MD,
Mar. 2005.
——, “Minimum-energy multicast in mobile ad hoc networks using network
coding,” IEEE Trans. Commun., vol. 53, no. 11, pp. 1906–1918, Nov.
2005.
Y. Wu, P. A. Chou, Q. Zhang, K. Jain, W. Zhu, and S.-Y. Kung, “Network
planning in wireless ad hoc networks: A cross-layer approach,” IEEE J.
Select. Areas Commun., vol. 23, no. 1, pp. 136–150, Jan. 2005.
Y. Xi and E. M. Yeh, “Distributed algorithms for minimum cost multicast with
network coding,” in Proc. 43rd Annu. Allerton Conf. Communication,
Control, and Computing, Monticello, IL, Sept. 2005.
——, “Distributed algorithms for minimum cost multicast with network cod-
ing in wireless networks,” in Proc. 4th Int. Symp. Modeling and Opti-
mization in Mobile, Ad Hoc and Wireless Networks (WiOpt ’06), Boston,
MA, Apr. 2006.
L. Xiao, M. Johansson, and S. Boyd, “Simultaneous routing and resource
allocation via dual decomposition,” IEEE Trans. Commun., vol. 52, no. 7,
pp. 1136–1144, July 2004.
X. Yan, R. W. Yeung, and Z. Zhang, “The capacity region for multi-source
multi-sink network coding,” in Proc. 2007 IEEE International Symposium
on Information Theory (ISIT 2007), 2007.
R. W. Yeung, “Multilevel diversity coding with distortion,” IEEE Trans. In-
form. Theory, vol. 41, no. 2, pp. 412–422, Mar. 1995.
——, A First Course in Information Theory. Kluwer Academic/Plenum,
2002.
R. W. Yeung and N. Cai, “Network error correction, part I: Basic concepts
and upper bounds,” Commun. Inf. Syst., vol. 6, no. 1, pp. 19–36, 2006.
R. W. Yeung, S.-Y. R. Li, N. Cai, and Z. Zhang, “Network coding theory:
Part I: Single source,” Foundations and Trends in Communications and
Information Theory, vol. 2, no. 4, 2005.
——, “Network coding theory: Part II: Multiple sources,” Foundations and
Trends in Communications and Information Theory, vol. 2, no. 5, 2005.
R. W. Yeung and Z. Zhang, “Distributed source coding for satellite commu-
nications,” IEEE Trans. Inform. Theory, vol. 45, no. 4, pp. 1111–1120,
May 1999.
——, “On symmetrical multilevel diversity coding,” IEEE Trans. Inform.
Theory, vol. 45, no. 2, pp. 609–621, Mar. 1999.
F. Zhao, T. Kalker, M. Medard, and K. J. Han, “Signatures for content dis-
tribution with network coding,” in Proc. 2007 IEEE International Sym-
posium on Information Theory (ISIT 2007), 2007.
L. Zosin and S. Khuller, “On directed Steiner trees,” in SODA ’02: Proc. 13th
Annu. ACM-SIAM Symp. Discrete Algorithms, San Francisco, CA, Jan.
2002, pp. 59–63.
Index
q-Johnson scheme, 35
adversarial errors, 148
arc, 11
batch coding, 34
Bellman-Ford algorithm, 99, 117, 123
bipartite matching, 30
bursty networks, 34
coding advantage
undirected, multiple unicast, 67
coding subgraph, 11
coding vector, 33
coding vector, global, 19
Completely Opportunistic Coding
(cope), 70
Completely Opportunistic Coding
(cope) protocol, 147
compression, distributed, 40
convolutional network coding, 36
correlated sources, 40
cyclic networks, 36
delay-free network coding, 17
distributed random linear coding
vector space approach, 34, 167
distributed random linear network
coding
coding vector approach, 33
Edmonds matrix, 30
encoding vector
auxiliary, 81
global, 78
erasure code, 7
Fano matroid, 60
fractional coding problem formulation,
58
generation, 32
generic linear network code, 153
Gilbert-Varshamov Bound, generalized,
157
graph, 11
Grassmann graph, 35
Hamming Bound, generalized, 151
hyperarc, 5, 11
hypergraph, 11
information exchange, 68
information inequalities, 64
integrality gap, Steiner tree , 24
inter-session network coding, 56
line graph, 33, 36
linear network code
generic, 153
link, 11
matroidal network, 60
max-flow/min-cut, 20, 56, 91, 94, 100
multicast, 21
networks with cycles/delays, 39
minimum entropy decoding, 41
mobile ad-hoc network (manet), 15
multicast, 3
multicast network code construction, 25
polynomial-time, 25
random, 28, 33
multicommodity flow, 110
multiple source multicast, 22
network
butterfly, 3
modified butterfly, 4
modified wireless butterfly, 5
slotted Aloha relay, 106
wireless butterfly, 5
188
Index 189
network error correction, 149
bounds, 149
distributed random network coding,
161
coding vector approach, 162
vector space approach, 167
polynomial-complexity, 161
network error detection, 167
non-Fano matroid, 60
non-Shannon-type information
inequalities, 64
nonlinear network coding, 60
one-hop xor coding, 68
operator channel, 167
packet networks, 32
path intersection scenario, 68
poison-antidote, 67, 128
poison-antidote scenario
wireless, 68
random linear network coding, 28, 33,
78
realizable, 39
scalar linear network coding
multicast, 17
non-multicast, 57
security, 148
separation, source and network coding,
43
Singleton Bound, generalized, 152
Slepian-Wolf, 40
solvability, 20
multicast, 21
Steiner tree
integrality gap, 24
minimum weight, 24
packing, 23
Steiner tree problem, 102
subgraph
coding, 11
static, 12, 17
time-expanded, 12, 75
throughput advantage
multicast, 23
time-expanded graph, 36
transfer matrix, 19
transfer matrix determinant polynomial,
31
V´amos matroid, 65
Varshamov Bound, generalized, 157
Varshamov Bound, strengthened
generalized, 159
vector linear network coding, 41, 58
virtual source node, 23
