Robust Network Coding in the Presence of Link Failure Using a Multicast Rate-Diversity Trade off

Abstract

We introduce a centralized joint network-channel coding scheme for maximized throughput and increased robustness against link failure in a communication network. We introduce RNC1(h,k) to multicast k independent data in a directed and acyclic network with capacity h, klesh . The redundancy through path diversity increases the resistance of the network against link failure. An improved version of RNC1, RNC2(h,k), is introduced to guarantee the rate k for h-k flow failures. The proposed schemes exploit the network structure for code design, which results in a manageable design complexity.

Full text

Page 1

Robust Network Coding in the Presence of
Link Failure Using a Multicast Rate-Diversity
Trade off

Hossein Bahramgiri, Mohammad Jabbari, Farshad Lahouti
Wireless Multimedia Communications Laboratory
School of ECE, College of Engineering
University of Tehran
[h.bahramgiri, m.jabari]@ece.ut.ac.ir, lahouti@ut.ac.ir



Abstract- We introduce a centralized joint network-
channel coding scheme for maximized throughput and
increased robustness against
communication network. We introduce RNC1(h,k) to
multicast k independent data in a directed and acyclic
network with capacity h,
k ≤ ≤
path diversity increases the resistance of the network
against link failure. An improved version of RNC1,
RNC2(h,k), is introduced to guarantee the rate k for h-k
flow failures. The proposed schemes exploit the network
structure for code design, which results in a manageable
design complexity.

Index terms- Network coding, joint network-channel
coding, link failure, multicast

I. INTRODUCTION
Network coding is a coding scheme in the network layer and
improves upon routing algorithms in communication
networks. Ahlswede et al. in [1] demonstrated that the
multicast capacity, deducted from the max-flow min-cut
theorem, can be achieved through network coding. Next in [2],
Li et al. showed that capacity is achieved by linear network
coding even with finite alphabet size. Koetter and Medard in
[3] introduced an algebraic approach to network coding and
studied a variety of information flows. Next in [4], Ho et al.
introduced a simple, distributed and random algorithm to
design linear network codes. Jaggi et al. in [5] introduced
polynomial time centralized algorithms that guarantee the true
solution. They presented two algorithms: deterministic linear
information flow (DLIF), a completely deterministic scheme,
and random linear information flow (RLIF), which is random
in the middle stages. Fraguoli et al. in [6], develop a
distributed and deterministic method for network code design
by translating the problem as a graph coloring problem.
Considering errors in the network channels, the above
algorithms may fail or degrade. One approach is to combat the
error in other layers, specifically physical and data link layers.
A cross layer approach is to consider a joint network-channel
coding scheme. Chou et al. in [7], introduced a practical
distributed network coding approach based on the random
network coding algorithm of [4]. The approach shows
resistance against errors by transmitting some extra random
combinations of data. In [8,9], Cai and Yeung derive network
generalization of Hamming, Singleton and Gilbert-Varshamov
bounds. In [10], Matsumoto suggested a centralized
link failure in a
h
. The redundancy through
deterministic algorithm for construction of linear network
error-correcting codes that attain the Singleton bound.
Koetter and Medard in [3] defined link failure pattern as
an error pattern, in which a subset of links fail to transmit
data. They demonstrated that in a network with multicast
rate of h, there exists a receiver-based coding scheme,
which provides rate
hk ≤
for all link failure patterns that
do not reduce the capacity below k. In [5] a method to find
such a solution is introduced, where it is required that
prior to the network code design, all the corresponding
link failure patterns are determined. In this scheme, the
complexity and the field size grow linearly with the
number of failure patterns, which is in general a large
number in practical situations. Ho et al in [11], showed the
benefits of random network coding towards robustness in
the presence of network changes and link failures. In [12],
the two types of receiver-based and network-wide
recovery schemes are analyzed and results for quantifying
network management are presented. They show that there
is not a single coding solution for all recoverable single-
link failures and at least the receivers must react to the
failure pattern. In [13], El Rouayheb et al. investigate
network codes that enable instantaneous recovery from
single edge failure for unicast connections
In this work, we introduce robust network coding
RNC(h,k), as a centralized joint network-channel coding
scheme, which increases the resistance of network against
link failure. The idea is to trade-off multicast rate of h,
creating a diversity of size h-k to increase the robustness.
The design is based on the network structure rather than
specific failure patterns, which leads to a reduction of the
design complexity.
Two robust network coding algorithms, referred to as
RNC1 and RNC2, are proposed. The former is a simple
and polynomial-time algorithm,
increased robustness in the presence of link failures,
independent of the failure patterns. The latter, RNC2(h,k),
guarantees the rate k for h-k flow failures of each of the
receivers. In this scheme, failure patterns are considered
as they affect the flows, and the code is checked only for
flow failures, resulting in a manageable complexity.
The rest of this paper is organized as follows: In section
II, the problem is defined in detail. In section III, we
introduce robust network coding in the presence of link
failures in a multicast session. The field size and the
complexity of the algorithm are studied. In section IV, an
which facilitates
1-4244-1199-8/07/$25.00 ©2007 IEEE.

IEEE Information Theory Workshop, Bergen, Norway, 2007.

Page 2

improvement to the proposed scheme is introduced, which
guarantees the rate k for h-k flow failures. Finally, the results
are presented in section V.

II. PROBLEM DEFINITION
A network can be represented by a graph G=(V,E), which is
assumed to be directed and acyclic. V represents the set of
nodes and E is the set of links or edges. We assume that each
edge has unit capacity. If an edge between two nodes has a
positive integer capacity of c, we consider c parallel edges of
unit capacity between two nodes. Thus, an edge
represented by
),,(
ivue =
, where
i indicates the edge number. Each node
edges represented by the set
session in the network with a transmitter
receivers
VT ⊂
. We assume that the transmission in the
network is synchronous.
The source generates a set of k independent data symbols
} ,...,{
1
xX =
, which must be transmitted to all receivers.
The local encoding vector of the edge e, denoted by
represents the combination coefficients of the input edges of
the tail(e). The symbol on the edge e is constructed from the
global encoding vector of e, b(e), which is a vector containing
the coefficients of data units in the set X for the edge e:
∑
Γ ∈ ′
))((
etaile
I
The network encounters link failure, a non-ergodic error
model, which may damage the data transmission process. The
error position and duration is not known a priori. With a link
failure, the link becomes completely unavailable. The purpose
is to combat this type of error.

III. ROBUST NETWORK CODING
The network capacity is given by the max-flow min-cut
theorem. According to the fundamental result of [1],

{
Maxflow Minc
Tt
∈
and applying network coding,
h =
transmitted to all the receivers in a multicast session. Any link
failure may reduce the capacity of the network in some
durations of time. In these durations, the algorithms for
transmitting rate h collapses and as the coding techniques
combine the data symbols, it may lead to missing all the data.
Here, we introduce an algorithm designed based on the
network structure, which increases the overall resistance of the
network against link failure. We refer to this algorithm as
Robust Network Coding 1 (RNC1).
In RNC1(h,k), the purpose is to transmit k units of independent
data in a network with error-free capacity of h. This scheme
provides a diversity (redundancy) of size h-k, which
determines the robustness of the network code against link
failure. To exploit this redundancy effectively, we propose the
following algorithm. The algorithm is based on a centralized
approach, and is inspired by the work of Jaggi et al. [5].
However, the ideas and techniques presented for robust
network code design could also be cast into other design
algorithms.
The design of RNC1(h,k) involves two steps. The first step is
to find h flows from the source to each of the receivers. For
every receiver t, we find the set
edge-disjoint flows from the source to the receiver t. Note that
Ee∈
head
can be
)(
e
and
),
V
(
v
, has some input
. Consider a multicast
Vs∈
and a set of
etail
v∈
u
==
)(v
I Γ
kx
(.)
e
m
,
′′
=
)()()(
e
eee
bmb
. (1)
)}(
t
multicast
=
, (2)
multicast
c
units of data can be
} ,...,{1
t
h
tt
ff
=Φ
, containing h
the flows of any two receivers may have common edges.
An edge that is present in at least one flow is referred to as
an active edge.
In the second step, fixing the flows, RNC1 determines the
transmission strategy in the network. There are h edge-
disjoint flows from the source to each of the receivers.
The transmitter sends h symbols to each of the receivers
as combinations of the k independent symbols in X .
Receiving any k-element subset of h symbols is enough to
extract all data.
As in [5], the set
h edges transmitting h symbols to this receiver. The set of
the global encoding vectors of these edges, corresponding
to the transmitting symbols, is denoted by
must have the property that each of its k-element subsets
forms an independent set of vectors. We refer to this
property as k-independence. To initialize the algorithm,
we add a new node s′ to the network and connect it to s
with h directed edges from s′ to s. Therefore, we have the
new graph
),(
EVG
′′
=
′
with
()(){}
hssssEE
,, ,...,1 ,,
′′
∪=
′
the set of added edges,
algorithm considers the active edges, one by one. The
edge e is considered, only after all the edges in
))((
etail
have been processed. The graph is acyclic, so
all the links are considered by this strategy. The edge e
replaces
)(e
←
Γ
in
Q(e) denotes the set of all receivers, for which the edge e
is in one of their flows. After replacement, k-
independence is verified for each updated
algorithm concludes when the edges ending at the
receivers are processed.
To process edge e, me is selected randomly and b(e) is
computed using equation 1. Next, the resulting
verified for k-independence for each
The following theorem determines a bound for the
probability of success in designing me as a function of the
field size.
Theorem 1: In RNC1(h,k), if the coefficients of me are
selected randomly from the finite field F, the probability
of k-independence of the sets
T
k
⎠⎝
−
1
!
bab
b
−
⎠⎝
Proof- The flows of a receiver are disjoint, thus
( )
TeQ
≤
. Therefore, the coefficient candidates for me
t C is defined for receiver t, consisting of
tB . The set
tB
{ }
sVV
′
∪=
′
, and
. The set
({
s
t C is initialized by
)(
hss
,, ,...,1 ,
′
)}
s
,
′
. The
I Γ
t
t C for each receiver
)(eQt∈
, where
tB . The
t B is
( ) eQt ∈
.
t C ,
( ) eQt ∈
, is given by
F
h
p
⎟⎟
⎞
⎜⎜
⎛
−
−≥
1
1
, (3)
where
)! ( !
a
a
=
⎟⎟
⎞
⎜⎜
⎛
.
are at most checked for T receivers. Consider the
receiver
( ) eQt ∈
, and
for the edge e. In
)(e
←
Γ
is replaced by e. The vector
m is selected randomly. Using equation 1, we examine a
candidate of b(e) for k-independence for the updated
Specifically, the vector b(e) with every k-1 element subset
of the global encoding vectors of h-1 other edges in
must construct an independent set. The probability that
t C and
tB that are to be updated
t C ,
t
e
tB .
t
C

Page 3

b(e) fails each test for k-independence is
resulted using Lemma 4 in [5]. The number of k-1 element
Fp
1
1=
, which is
subsets under consideration is
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
1
1
k
h
. Applying union
bound, the probability that me fails at least one of the tests for
h
p
1
⎠⎝
−
in
)(eQ
, which counts to at most T , the probability that
fails at least one of the verifications for k-independence is
T
k
⎠⎝
−
1
k-independence is
F
k
1
1
2
⎟⎟
⎞
⎜⎜
⎛
−
≤
. Considering the receivers
e
m
F
h
p
⎟⎟
⎞
⎜⎜
⎛
−
≤
1
3
. Therefore, the probability of k-independence
of
t C ,
( ) eQt ∈
, is given by
F
T
k
h
pp
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
−≥−=
1
1
11
3
.

If we set the success probability of the design of
⎞
⎜⎜
⎝
−
F
k
h
F
2 .
1
⎠
⎝
−
Theorem 2: The expected running time of RNC1(h,k) is
1
(
⎟⎟
⎠
⎝
−
k
Proof- The algorithm is executed edge by edge. There are at
most E active edges. Every edge is at most in |T| flows. For
≤
)(
verification of k-independence is to
?
e
m for edge e
as
2 / 1
≥
1
1
1
⎟⎟
⎠
⎛
−
−≥
T
h
p
, then we have
T
k
1
⎟⎟
⎞
⎜⎜
⎛
−
≥
. (4)
)
1
2
⎞
⎜⎜
⎛
−
h
kTEO
.
edge e, at most
TeQ
be performed, each of which consists of
⎟⎟
⎠
. Thus, the
1
⎟⎟
⎠
⎝
−
k
algorithm
⎞
⎜⎜
⎝
O
⎛
−
−
1
1
k
h
tests. The
expected running time of each test is )(
2
k
expected running time for each edge is
)
1
(
2
⎞
⎜⎜
⎛
−
h
kTO
and
the expected
⎛
kT
running time of the is
)
1
1
(
2
⎟⎟
⎠
⎞
⎜⎜
⎝
−
−
k
h
EO
.
?
RNC1 is independent of link failure patterns and increases the
overall resistance of the network against link failure. As
discussed in theorems 1 and 2, both the complexity and the
field size of RNC1(h,k) grow linearly with
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
1
1
k
h
.
Figure 1 demonstrates an example. We define a multicast
session from the node a, to the two nodes m and n. Assuming
all the edges have unit capacities, the capacity of the network
is 3. We apply RNC1(3,2) to increase the robustness of the
network against link failure. Due to equation 4, we can
consider
11||
=
F
, for example. The global encoding vectors
are denoted on each edge. In this example
2
1
1
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
k
h
.
Table 1 presents the performance (rate of successful decoding)
of RNC1(8,k) in the presence of failure patterns, which reduce
the capacity down to k. The results are obtained by evaluating
a sufficiently large number of multi-layer networks with
capacity 8 in the presence of a sufficiently large number of
random failure patterns. We observe that the rate of
successful decoding approaches 100% by increasing the
field size.
In the next section, we introduce an algorithm which
guarantees maximum throughput for h-k flow failures for
each receiver.



Figure 1- A network in which data is multicast from node a to nodes m
and n. The multicast capacity is 3. Three flows of the nodes m and n are
shown by directed solid and dashed lines, respectively. Global encoding
vectors of the edges for RNC1(3,2) with
11
=
F
are denoted.

Field Size
h
⎢⎢
⎢
⎝
⎡
⎜⎜
⎝
−
k=3
k=4
K=5
k=6
k=7
p
T
k
F
⎥⎥
⎥
⎤
⎤
⎡
⎟⎟
⎠
⎞
2 .
⎜⎜
⎛
−
−
=
1
1

97.8 99.4 99.5 98.7 91.5
p
T
k
h
F
⎥⎥
⎥
⎢⎢
⎢
⎟⎟
⎠
⎞
8 .
⎛
−
=
1
1

99.5 99.9 99.9 99.7 98.4

Table 1- The performance (rate of successful decoding (%)) of
RNC1(8,k) for failure patterns that reduce the capacity down to k for
various k. ⎡ ⎤p
a
denotes the smallest prime number greater than a.

IV. IMPROVED ROBUST NETWORK CODING
In RNC1(h,k), the receivers can extract all data by
receiving the symbols of any k flows out of all h edge-
disjoint flows. A flow fails if at least one link of the flow
fails. Using RNC1, if there is only one receiver, failing
any set of h-k flows, the receiver can extract all the data
using the remaining flows. However, in the scenario with
more than one receiver, receiving k flows for each
receiver is not sufficient for extracting all the data, since
the failed flows of other receivers may have affected the
contents of the received symbols. For example, in figure
1, the flow
,(),, [(
1
bbaf
,( ),,( ),,( ),, [(
2
jjffccaf
link (b, f) destroys
f1, but does not directly fails any
flows of m. However, this still affects the flow
sense that it may lead to collapsing the k-independence of
the
guarantee k-independence in the presence of h-k flow
failures for each receiver, we must examine these joint
flows. A joint-flow is a flow that has at least a common
edge with at least one flow of another receiver. A joint-
failure is a failure pattern in which at least one joint-flow
fails. Here, we present an improvement to RNC1, referred
to as RNC2, to overcome joint-failures. In RNC2, in each
edge e, the local encoding vector, me, is verified for k-
independence for all
( ) eQt ∈
)] ,(),,( ),
njjff
n=
joins into
)]
m
m=
in one edge. Failure of
n
m
f2, in the
t B at receiver m (decoding failure). Therefore, to
for failure-free and all joint-

Page 4

failure patterns that do not reduce the capacity below k.
Therefore, the constructed code resists all such joint-failure
patterns.
Now, assume
tn flows of h flows of the receiver t are joint-
flows.
flows of receiver t that do not reduce the capacity below k.
t R, denotes the set of joint-failure patterns for receiver t in
k
t R denotes the set of all joint-failure patterns for the
i
which i joint-flows fail. We have
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
i
n
R
t
it,
, and
∑
=
i
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
kh
t
k
t
i
n
R
0
. (5)
Note that
0
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
b
a
if
ab >
. Rk is the set of all joint-failure
patterns that do not reduce the capacity below k. Thus,
∏ ∑∏
∈∈
Tt
Tt
−
=
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
==
kh
i
t
k
t
k
i
n
RR
0
. (6)
In RNC2, the local encoding vector of an edge is examined for
all joint-failure patterns one by one.

Theorem 3: In RNC2(h,k), if the coefficients of me are
selected randomly from the finite field F, the probability that
the set
)(eQt∈∀
, is k-independent for all the specified
failure patterns is as follows,
T
k
⎠
⎝
−
1
Proof- The argument follows from the proof of theorem 1. The
details are omitted for brevity.

If we set the success probability for each edge as
1
1
≥⋅
⎟⎟
⎠
⎝
−
F
k
h
F
⋅
⎟⎟
⎠⎝
−
1

Theorem 4: The expected running time of RNC2(h,k) is
1
..(
⎟⎟
⎠⎝
−
k
Proof- In each edge e, k-independence is to be verified for all
the specified failure patterns. Thus, the number of tests is
multiplied by
R
. The argument follows from the proof of
t C ,
k
R
F
h
p
⋅
⎟⎟
⎞
⎜⎜
⎛
−
−≥
1
1
. (7)

?
2 / 1
1
⎞
⎜⎜
⎛
−
−≥
k
R
T
h
p
, then we have,
k
RT
k
⎞
2 .
⎜⎜
⎛
−
≥
1
(8)
)
1
2
⎞
⎜⎜
⎛
−
h
RkTEO
k
.
k
theorem 2. The details are omitted for brevity.
A pseudo-code for RNC2 is presented in figure 2. The
proposed RNC2 algorithm using a field of size given in
equation 8, with the expected running time presented in
theorem 4, resists against joint-failure patterns and thus
guarantees the resistance against h-k flow failures for each
receiver. Our simulations validated the mentioned design
objective.
Comparing to the RNC1 algorithm, the expected running time
and the field size of RNC2 algorithm increases by a factor of
R
. We can compute an upper bound for
?
kk
R
as follows,
T
kh
∑
=
i
Tt
kh
i
k
i
h
i
h
R
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
≤
∏ ∑
∈
−
−
=
00
(9)
Note that the algorithm introduced in [5], guarantees the
rate k for all link failure patterns, which do not reduce the
capacity below k. This advantage is achieved at the cost of
a substantial increase in the complexity and field size.
Specifically, the complexity is
)(
2HhTEO
and the field
size is
considered. In a common network,
and grows exponentially with the number of edges. In the
example of figure 1,
3
n
and
HTF
2
≥
, where H is the set of failure patterns
| H is a large number
|
1=
3
2=
n
, and
16
=
k
R
.
Therefore, the complexity of RNC1 and RNC2 is
proportional to
2
1
1
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
k
h
and
32.
1
1
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
k
R
k
h
,
respectively, which is much less than the number of all
failure patterns that do not reduce the capacity below 2,
1761
=
H
. RNC2 outperforms RNC1 in terms of
robustness; while RNC1 already provides acceptable
performance as depicted in table I.


RNC2(h,k):
h is calculated from Max-flow Min-cut theorem
F is a finite field satisfying equation 4

for each t∈T do
t
Φ :the set of h disjoint flows from s to t
t
joint
Φ
: the set of joint flows of the receiver t

find
find
find
capacity below k

for each t∈T do
:
k
R
the set of all joint-failure patterns that do not reduce the
initialize Ct
for each
initialize Bf,t
k
Rf ∈
do


for each edge e in topological order do
for each
Qt ∈
←
select me randomly

f ∈
Qf
joint-failure pattern f)
(v
the failure pattern f
for all


( ) e
{ \


}{ )}(
eeCC
t
tt
∪Γ
←

for each
k
R
do
(
)(e
: the set of receivers using the edge e in the
(
: )
f
I Γ
the set of active input edge of the node v in
( ) e
)
e
Qt
f
∈
b

∑
(
Γ∈
′
′′
=
))
Γ
(
)()((
etaile
fef
f
I
ee
bm

)}({ )}({\
,,
eebBB
f
t
ftftf
′
B,′
b
∪=
←



if

tf
is not k-independent
Go to select

tf
′
tf
BB
,,
=



Figure 2- The pseudo-code for RNC2(h,k)

Page 5

REFERENCES
[1] R. Ahlswede, N. Cai, S. R. Li, and R. W. Yeung,
“Network information flow,” IEEE Transactions on
Information Theory, July 2000.
[2] S-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network
coding,” IEEE Trans. Inform. Theory, 49:371–381, Feb.
2003.
[3] R. Koetter and M. Medard, “An algebraic approach to
network coding,” IEEE/ACM Trans. Networking, 2003.
[4] T. Ho, R. Koetter, M. Medard, D. R. Karger, and M.
Effros, “The benefits of coding over routing in a
randomized setting,” International Symposium on
Information Theory (ISIT), 2003.
[5] S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, and
L. Tolhuizen, “Polynomial time algorithms for multicast
network code construction,” IEEE Trans. Inf. Theory,
Vol. 51, No.6, 2005.
[6] C. Fragouli, E. Soljanin, and A. Shokrollahi, "Network
Coding as a Coloring Problem," Proceeding of CISS,
2004.
[7] P. A. Chou, Y. Wu, and K. Jain. “Practical network
coding,” In Proc. 41st Allerton Conf. Communication,
Control and Computing, Monticello, IL, Oct. 2003.
Invited paper.
[8] R. W. Yeung and N. Cai, “Network error correction, part
I: Basic concepts and upper bounds,” Comm. in Inform.
and Systems, vol. 6, no. 1, pp. 19–36, 2006.
[9] R. W. Yeung and N. Cai, “Network error correction, part
II: Lower bounds,” Comm. in Inform. and Systems, vol. 6,
no. 1, pp. 37–54, 2006.
[10] R. Matsumoto, "Construction Algorithm for Network
Error-Correcting Codes Attaining the Singleton Bound,"
IEICE Trans. Fundamentals, VOL.E90–A, NO.9 Sept.
2007.
[11] T. Ho, M. Medard, R. Koetter, D. R. Karger, M. Effros, J.
Shi and B. Leong, " A random linear network coding
approach to multicast," IEEE Trans. Inf. Theory, Vol. 52,
No. 10, pp. 4413- 4430, 2006.
[12] T. Ho, M. Medard and R. Koetter, "An Information-
Theoretic View of Network Management," IEEE Trans.
Inf. Theory, Vol. 51, No. 4, 2005.
[13] S. Y. El Rouayheb, A. Sprintson, and C. Georghiades.
"Simple Network Codes for Instantaneous Recovery from
Edge Failures in Unicast Connections," Technical Report
WCL-TR-06-101, Texas A&M University, College
Staion, Texas, January 2006.