An achievable rate region for distributed source coding and dispersive information routing

Abstract

This paper considers the problem of optimal multi-hop routing of correlated sources over a network with multiple sinks and arbitrary network demands. We recently introduced a new routing paradigm in [10] called `dispersive information routing' (DIR), wherein the intermediate nodes are allowed to split a packet and forward a subset of the received bits on each forward path. DIR ensures that each sink receives just the information it requires to decode the sources it intends to reconstruct, and thereby outperforms conventional routing techniques in the literature. We proposed an encoding scheme called `power binning' which achieves complete rate region and the minimum cost under this paradigm when each sink is allowed to receive packets only from the sources it wants to reconstruct. This paper considers the optimum encoding scheme when every source can (possibly) communicate with every sink irrespective of what the sinks reconstruct. This generalization happens to be considerably more complex and we derive an achievable rate region and an associated achievable cost using principles from distributed source coding and multiple descriptions encoding.

Full text

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An Achievable Rate Region for Distributed Source
Coding and Dispersive Information Routing
Kumar Viswanatha, Emrah Akyol and Kenneth Rose
ECE Department, University of California - Santa Barbara
{kumar,eakyol,rose}@ece.ucsb.edu
Abstract—This paper considers the problem of optimal multi-
hop routing of correlated sources over a network with multiple
sinks and arbitrary network demands. We recently introduced
a new routing paradigm in [10] called ‘dispersive information
routing’ (DIR), wherein the intermediate nodes are allowed to
split a packet and forward a subset of the received bits on
each forward path. DIR ensures that each sink receives just
the information it requires to decode the sources it intends
to reconstruct, and thereby outperforms conventional routing
techniques in the literature. We proposed an encoding scheme
called ‘power binning’ which achieves complete rate region and
the minimum cost under this paradigm when each sink is allowed
to receive packets only from the sources it wants to reconstruct.
This paper considers the optimum encoding scheme when every
source can (possibly) communicate with every sink irrespective
of what the sinks reconstruct. This generalization happens to
be considerably more complex and we derive an achievable rate
region and an associated achievable cost using principles from
distributed source coding and multiple descriptions encoding.
Index Terms—Distributed source coding, joint compression
and routing
I. INTRODUCTION
The problem of minimum cost routing of correlated sources
over a multi-hop network has recently attracted researchers
due to its direct applicability in sensor networks. What makes
this problem particularly interesting is the the challenge of
designing encoders which address the interplay between joint
compression and routing. Broadly, the research in this field
can be grouped into two camps. The first approach performs
compression at intermediate nodes [1] and the second resorts
to distributed source coding [2], [3], [4]. This paper focuses
on the latter category.
The field of distributed source coding (multi-terminal source
coding) began with the seminal work of Slepian and Wolf
[5] and Wyner and Ziv [6]. Several publications followed
considering different multi-terminal scenarios and obtaining
achievability bounds for them [7]. Han and Kobayashi [8]
(see also [9]) derived an achievable rate region for a general
multi-terminal source coding problem with multiple sources
and sinks, with each source being reconstructed at a subset of
sinks losslessly.
One of the first attempts to unify Slepian-Wolf compression
and routing in a network was by Cristescu et.al in [2]. Here,
we call the routing mechanism they considered as ‘Broad-
The work was supported by the NSF under grants CCF-0728986 and CCF
- 1016861
casting’1, wherein each source broadcasts its information to
all sinks which intend to reconstruct it (which is motivated
by routing mechanisms for independent sources). [3] showed
that Slepian-Wolf compression followed by ‘Broadcasting’ is
optimum for single sink networks and cannot be outperformed
by any other joint compression-routing scheme. However, [4]
demonstrated the extent of suboptimality of ‘Broadcasting’ in
case of multi-sink networks.
In a precursor work [10], we introduced a new rout-
ing paradigm called ‘dispersive information routing’ (DIR),
wherein the intermediate nodes are allowed to ‘split’ a packet
and forward different subsets of the packets on each forward
path. It was demonstrated using simple examples that DIR
outperforms broadcasting for multi-sink networks. In [10],
we considered the scenario where sinks receive packets only
from sources which they intend to reconstruct and derived
the complete rate region and the minimum cost achievable
under DIR. This scenario is called ‘no helpers’ case in the
literature [9]. In this paper, we consider the more general case
wherein each sink can (possibly) receive packets from any
source immaterial of what they reconstruct. This scenario was
recently addressed for broadcasting in [4] and was shown to
have substantial gains over the ‘no helper case’, albeit being
considerably more complex. Here, we derive an achievable
rate region for the general DIR problem with helpers using
principles from Multiple-Descriptions encoding [11] and Han
and Kobayashi decoding.
In the remaining part of this section, we motivate DIR using
an example where there are no helpers. In section II, we
illustrate the new coding scheme using one of the simplest
scenarios with helpers and extend it to a general network in
section III.
A. Motivating example
Consider the network shown in Figure 1a. A source X0
is to be reconstructed at two sinks S1 and S2 which have
access to side information X1 and X2 respectively. Source
X0communicates with the two sinks through an intermediate
node (we term the ’collector’) which is functionally a simple
router. The edge weights on each path in the network is as
shown in the Figure. Assume that the cost of communication
through a link is a simple product of the rate and the edge
1Note that we loosely use the term broadcasting instead of multi-casting
to stress the fact that all the bits sent by a source are routed to all the sinks
which reconstruct it

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(a) Broadcast Routing(b) DIR(c) 2 Source - 2 Sink example
Figure 1: (a) Rates and weights for ‘Broadcasting’ (b) Rates and effective weights for ‘DIR’ (c) The 2 Source - 2 Sink example.
Each source acts as the principle source for one sink and as a helper for the other.
weight, i.e. f(r,c) = rc. The objective is to find the minimum
communication cost for lossless reconstruction of X0 at the
two sinks.
Under broadcast routing [2], the collector forwards all the
bits it receives to both the sinks. The minimum rate at which
the source sends for lossless reconstruction at both the sinks is
R = max(H(X0|X1),H(X0|X2)) and hence, the minimum
communication cost under broadcast routing is CB= (C0+
C1+C2)max(H(X0|X1),H(X0|X2)). However, observe that
the minimum rates required on the branches connecting the
collector to sinks S1and S2are H(X0|X1) and H(X0|X2)
respectively. Clearly, broadcast routing leads to suboptimality
on one of these two branches if H(X0|X1) ?= H(X0|X2).
We introduced a new routing technique in [10], called
dispersive information routing (DIR), which achieves optimal
cost for this network. Under this new paradigm, the interme-
diate nodes (collector in this example) are allowed to ‘split’
a packet and forward different subsets of the received bits on
the forward paths. We could equivalently think of the source
transmitting 3 smaller packets to the collector. The first two
packets, at rates R1and R2, are destined to sinks S1and S2
respectively and the third packet at rate R12is destined to both
the sinks, as shown in Figure 1b. We demonstrated using a
simple variant of the random binning paradigm, called ‘Power
binning’, that the complete achievable rate region for the tuple
(R1,R2,R12) is:
R1+ R12
R2+ R12
≥
≥
H(X0|X1)
H(X0|X2)
(1)
The minimum cost operating point satisfies equations 1
and minimizes the cost function CDIR = {(C0+ C1)R1+
(C0+ C2)R2+ (C0+ C1+ C2)R12}. The solution is either
of the two points (R1,R2,R12)={0,H(X0|X1),H(X0|X2)−
H(X0|X1)} or {H(X0|X1)−H(X0|X2),H(X0|X2),0} and
both achieve a lower cost compared to broadcast routing if
H(X0|X1) ?= H(X0|X2). This example clearly demonstrates
the gains of DIR over broadcast routing to communicate
correlated sources over a network.
For a general network with K sinks, each source transmits
2Kpackets and each packet is routed to a subset of sinks.
In [10], we showed that, when only the requested sources
(sources being reconstructed) are allowed to communicate with
the sinks, power binning achieves the complete rate region
and hence, achieves the minimum communication cost under
DIR. The extent of suboptimality due to this restriction was
investigated in [4], where examples were presented to show
potentially unbounded gains by allowing unrequested sources
to send information to a sink in the context of ‘broadcasting’.
In this paper, we find an achievable rate region and an
associated cost for dispersive information routing with helpers.
We note that there has been considerable volume of work
related to minimum cost network coding for correlated sources
[12]. Note that unlike network coding [12], DIR does not
require possibly expensive coders at intermediate nodes, but
rather can always be realized using conventional routers with
each source transmitting multiple packets into the network
intended to different subsets of sinks. Therefore, hereafter, we
interchangeably use the ideas of “packet splitting” at inter-
mediate nodes and conventional routing of smaller packets,
noting that either can be realized using the other. The potential
implications of DIR on network coding are beyond the scope
of this paper and will be considered as part of future work.
II. 2 SOURCE - 2 SINK EXAMPLE
To keep the notations and understanding simple, we begin
with one of the simplest setups which illustrates the underlying
ideas. We will provide intuitive description for the encoding
scheme here and defer the formal proofs for the general case as
part of Theorem 1 in section III. Consider the network shown
in Figure 1c. Two sources s1and s2observe correlated random
variables X1 and X2. Two sinks S1 and S2 require lossless
reconstructions of X1 and X2 respectively. The sources can
communicate with the sinks only through a collector node. The
edge weights are as shown in the figure. Observe that, while
each source is requested by one sink, they act as helpers for
the other.
Under dispersive information routing, each source transmits
a packet to every subset of sinks. In this example, source s1
sends 3 packets to the collector at rates (R1,1,R1,2,R1,12)
respectively. The collector forwards the first packet to sink
S1, the second to S2 and the third to both S1 and S2.

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Similarly, source s2sends 3 packets to the collector at rates
(R2,1,R2,2,R2,12) which are forwarded to the corresponding
sinks. Note that, the rates of some of these packets could
be 0. Our objective is to determine the set of achievable
rate tuples (R1,1,R1,2,R1,12,R2,1,R2,2,R2,12) which allow
for lossless reconstruction of respective sources at the two
sinks. The minimum cost then follows by finding the point
in the achievable rate region which minimizes the effective
communication cost:
?
2
?
A non-single letter characterization of the complete rate region
is possible using the results of Han and Kobayashi in [8].
They also provide a single-letter partial achievable rate region.
However, their result assumes 3 independent encoders at each
source, which is an unnecessary restriction. We present a more
general rate region, which maintains the dependencies between
the messages at each encoder.
Suppose we are given
(U1,12,U1,1,U1,2,U2,12,U2,1,U2,2) jointly distributed with
(X1,X2) such that the following Markov chain conditions
hold:
CDIR
=
2
i=1
(Cic+ Cci)Ri,i+ (C1c+ Cc2)R1,2+
i=1
(Cic+ Cc1+ Cc2)Ri,12+ (C2c+ Cc1)R2,1 (2)
random variables
(U1,12,U1,1)
(U1,12,U1,2)
↔
↔
X1↔ X2↔ (U2,12,U2,1)
X1↔ X2↔ (U2,12,U2,2)
(3)
where X ↔ Y ↔ Z denotes that (X,Y,Z) form a Markov
chain in that order. Note that Ui,S is sent in the packet from
source i to sinks j : j ∈ S. The encoding is divided into 3
stages.
Encoding : We first focus on the encoding at s1. In the
first stage, 2nR
generated independently, with elements drawn according to
the marginal PMF P(U1,12). Conditioned on each of these
codewords, 2nR
generated according to the conditional PMFs P(U1,1|U1,12)
and P(U1,2|U1,12) respectively. Codebooks for U2,12,U2,1and
U2,2 are generated at s2 in a similar fashion. On observing
a sequence Xn
the codebooks of (U1,12,U1,1,U1,2) which are jointly typical
with Xn
proaches 1 if R
and R
1,2
≥
lected be denoted by (u1,12,u1,1,u1,2). Similar constraints
on (R
2,1,R
Denote the codewords selected at s2 by (u2,12,u2,1,u2,2).
It follows from (3) and the ‘Conditional Markov Lemma’
in [14] that (xn
2,u1,12,u1,1,u2,12,u2,1)
(xn
?-typical set of length n sequences.
In the second stage of encoding, each encoder uniformly
divides the 2nR
?
1,12codewords of U1,12, each of length n are
?
1,1and 2nR
?
1,2codewords of U1,1and U1,2are
1, s1 first tries to find a codeword tuple from
1. The probability of finding such a codeword tuple ap-
?
1,12≥ I(X1;U1,12), R
?
I(X1;U1,2|U1,12). Let the codewords se-
?
1,1≥ I(X1;U1,1|U1,12)
??
2,2,R
?
2,12) must be satisfied for encoding at s2.
1,xn
∈Tn
?
and
1,xn
2,u1,12,u1,2,u2,12,u2,2) ∈ Tn
?, where Tn
? denotes the
?
i,Scodewords of Ui,S into 2nR
??
i,Sbins ∀i ∈
{1,2}, S ∈ {1,2,12}. All the codewords which have the
same bin index m are said to fall in the bin Ci,S(m)
∀m ∈ (1...2nR
in bin Ci,S(m) is 2n(R
(u1,12,u1,1,u1,2) in the first stage and if the bin indices as-
sociated with (u1,12,u1,1,u1,2) are (m1,12,m1,1,m1,2), then
m1,1is routed to sink S1, m1,2to sink S2and m1,12to both
the sinks S1and S2. Similarly, bin indices (m2,12,m2,1,m2,2)
are routed from s2to the corresponding sinks.
The third stage of encoding, resembles the ‘Power Binning’
scheme of [10]. Every typical sequence of Xn
random bin index uniformly chosen over [1 : 2n˜ R1,1]. All
sequences with the same index, l1, form a bin B1,1(l1,1)
∀l1,1∈ {1...2n˜ R1,1}. Upon observing a sequence Xn
with bin index l1,1, in addition to m1,1, s1also routes index
l1,1 to S1. Similarly bin index l2,2 is routed from s2 to S2
in addition to m2,2. These bin indices are used to reconstruct
Xn
i is to be reconstructed at a subset of sinks Πi ⊆ Π, the
source assigns 2|Πi|− 1 independently generated indices to
each sequence and each index is routed to a subset of Πi.
Decoding : We again focus on sink S1. S1 receives the
indices (m1,12,m1,1,m2,12,m2,1,l1,1,l1,12,l2,1,l2,12). It first
looks for a pair of unique codewords from C1,12(m1,12) and
C2,12(m2,12) which are jointly typical. Obviously, there is at
least one pair, (u1,12,u2,12), which is jointly typical. It can
be easily shown using standard typicality arguments that, the
probability that no other pair of codewords are jointly typical
approaches 1 if:
??
i,S). Note that the number of codewords
?
i,S−R
??
i,S). If s1selects the codewords
1is assigned a
1∈ Tn
?
1and Xn
2losslessly at the decoders. Note that, if source
R
??
1,12
??
2,12
??
2,12
≥
≥
≥
I(X1;U1,12|U2,12)
I(X2;U2,12|U1,12)
I(X1,X2;U1,12,U2,12)
R
R
??
1,12+ R
(4)
The decoder at S1 next looks at the codebooks of U1,1 and
U2,1which were generated conditioned on u1,12and u2,12re-
spectively to find a unique pair of codewords from C1,1(m1,1)
and C2,1(m2,1) which are jointly typical with (u1,12,u2,12).
We again have one pair, (u1,1,u2,1), which is jointly typical
with (u1,12,u2,12). It can be shown using arguments similar
to [8] that, the probability of finding no other jointly typical
pair approaches 1 if :
(R
?
1,1− R
??
1,1) ≤ −H(U1,1|U2,1,U1,12,U2,12)
+H(U1,1|U1,12)
(5)
(R
?
2,1− R
??
2,1) ≤ −H(U2,1|U1,1,U1,12,U2,12)
+H(U2,1|U2,12)
(6)
?
(R
?
1,1− R
(R
2,1− R
??
1,1)+
≤
H(U1,1|U1,12) + H(U2,1|U2,12)
−H(U1,1,U2,1|U1,12,U2,12)
???
2,1)
?
(7)

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On substituting the constraints R
and R
condition in (3) we get:
?
1,1≥ I(X1;U1,1|U1,12)
?
2,1≥ I(X2;U2,1|U2,12), and using the Markov chain
R
??
1,1
??
2,1
??
2,1
≥
≥
≥
I(X1;U1,1|U1,12,U2,12,U2,1)
I(X2;U2,1|U2,12,U1,12,U1,1)
I(X1,X2;U1,1,U2,1|U1,12,U2,12) (8)
decoding
R
R
??
1,1+ R
After
(u1,12,u1,1,u2,12,u2,1), the decoder at S1 looks for a
unique sequence from B1,1(l1,1) which is jointly typical
with (u1,12,u1,1,u2,12,u2,1). We again have xn
this property. It can be shown that the probability of
finding no other sequence which is jointly typical with
(u1,12,u1,1,u2,12,u2,1) approaches 1 if:
successfully thecodewords
1satisfying
˜R1,1≥ H(X1|U1,12,U1,1,U2,12,U2,1)
Similar constraints on rates can be obtained for lossless
decoding at S2. The first packet from s1, destined to only S1,
carries indices (m1,1,l1,1) at rate R1,1 = R
second and third packets carry m1,2and m1,12at rates R
and R
sinks. Similarly, 3 packets are transmitted from s2 carrying
indices {m2,1,m2,12,(m2,2,l2,2)} at rates (R
˜R2,2)
tosinks
{S1,S2,(S1,S2)}
able rates for (R1,1,R1,2,R1,12,R2,1,R2,2,R2,12) can now
be obtained using (4), (8) and (9). The convex clo-
sure of achievable rates over all such random variables
(U1,12,U1,1,U1,2,U2,12,U2,1,U2,2) gives the achievable rate
region for the 2 source - 2 sinks DIR problem.
(9)
??
1,1+˜R1,1. The
??
1,2
??
1,12respectively and are routed to the corresponding
??
2,1,R
??
2,12,R
??
2,2+
respectively. Achiev-
III. DISPERSIVE INFORMATION ROUTING WITH HELPERS -
GENERAL SETUP
Let a network be represented by an undirected connected
graph G = (V,E). Each edge e ∈ E is associated with
an edge weight, we. The communication cost is assumed to
be a simple product of the edge rate and edge weight, i.e.
Ce= rewe2. The nodes V consist of N source nodes (denote
by s1,s2...sN), M sinks (denote by S1,S2...SN), and
|V |−N−M intermediate nodes. We denote by Σ = {1...N}
and Π = {1...M}. Source node siobserves random variable
Xidistributed over a finite alphabet Xi. Sink Sjreconstructs
(requests) a subset of the sources denoted by Σj ⊆ Σ.
Conversely, source si is reconstructed at a subset of sinks
denoted by Πi ⊆ Π. The objective is to find the minimum
communication cost achievable by dispersive information rout-
ing for lossless reconstruction of the requested sources at each
sink when every source can (possibly) communicate with every
sink.
In what follows, 2Sdenotes the set of all subsets (power
set) of any set S and |S| denotes the set cardinality. Note that
|2S| = 2|S|. Scdenotes the set complement and φ denotes the
null set. For two sets S1and S2, we denote the set difference
2The approach is applicable to more general cost functions.
by S1−S2= {K : K ∈ S1,K / ∈ S2}. We denote by 2S−φ, the
set of all non-empty subsets of S. We use the shorthand {Ui}S
for {Ui,K: K ∈ S} and {UΓ}Sfor {Ui,K: i ∈ Γ, K ∈ S}3.
A. Obtaining the effective costs
Under DIR each source transmits 2M− 1 packets into the
network, each meant for a different subset of sinks. Let the
packet from source si to the subset of sinks S ∈ 2Πbe
denoted by Pi,Sand let it carry information at rate Ri,S. The
optimum route for packet Pi,Sfrom the source to these sinks is
determined by a spanning tree optimization (minimum Steiner
tree) [13]. The minimum cost of transmitting packet Pi,Swith
Ri,S bits from source i to the subset of sinks S, denoted by
di(S) is :
di(s) = Ri,S× min
Q∈Ei,S
?
e∈Q
we
(10)
where Ei,S denotes the set of all paths from source i to the
subset of sinks S. Our next objective is to find an achievable
rate region for the tuple (Ri,S∀i ∈ Σ,S ∈ 2Π). The minimum
communication cost then follows directly from a simple linear
programming formulation.
B. An achievable rate region
We extend the coding scheme described in section II to
derive an achievable rate region for the tuple (Ri,S∀i ∈
Σ,S ∈ 2Π− φ) using principles from Multiple Descriptions
encoding [11], albeit with more complex notation. Without
loss of generality, we assume that, every source can send
packets to every sink. The corresponding costs can be set to
∞ if some paths do not exist.
Before stating the achievable rate region in Theorem 1, we
define the following subsets of 2Π:
IW
IW+
=
{S : S ∈ 2Π, |S| = W}
{S : S ∈ 2Π, |S| > W}
=
Let B be any subset of Π with |B| ≤ W. We define the
following subsets of IW and IW+:
IW(B)
IW+(B)
We also define J(K) = {S : S ∈ 2Π, |K?S| > 0}
defined on arbitrary finite sets, jointly distributed with {X}Σ,
such that ∀j ∈ Π:
P({X}Σ,{UΣ}J(j)) = P({X}Σ)
=
{S : S ∈ IW, B ⊆ S}
{S : S ∈ IW+, B ⊆ S}
=
Let {UΣ}2Π−φbe any set of N(2M− 1) random variables
?
i∈Σ
P({Ui}J(j)|Xi) (11)
The above Markov condition ensures that all the codewords
which reach a sink are jointly typical with {X}Σj. We define
βj(W,Q1,...,QN) ∀j ∈ Π,W ∈ Π,Qi⊆ IW(j) as:
βj(W,Qi∀i)
i∈Σ
+
=
−
H?{Ui}Qi∀i|{Ui}Qc
?
H?{Ui}Qi|{Ui}IW+,Xi
?
(12)
i,{Ui}IW+(j)∀i?
3Note the difference between {Ui}Sand Ui,S. {Ui}Sis a set of variables,
whereas Ui,Sis a single variable.

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Figure 2: Example of a 2-sink, 1 Helper DIR
where Qc
as:
i= IW−Qi. We also define γj(Γ) ∀j ∈ Π,Γ ⊆ Σj
γj(Γ) = H?{X}Γ|{X}Γc,{UΣ}IW+(j)
where Γc= Σj−Γ. We state our main result in the following
theorem.
?
(13)
Theorem 1. Let {UΣ}2Π−φbe any set of random variables
satisfying (11). Let R
rate tuples such that, ∀j ∈ Π,W ∈ {1,...,M},Qi⊆ IW(j):
?
and let˜Ri,S∀i ∈ Σ,S ∈ 2Πj− φ satisfy:
?
∀j ∈ Π,Γ ∈ 2Σj−φ. Then, the achievable rate region for the
tuple (Ri,S∀i ∈ Σ,S ∈ 2Π− φ) contains all rates such that,
?
R
i,S
The convex closure of the achievable tuples over all such
N(2M−1) random variables satisfying (11) is the achievable
rate region for DIR and is denoted by RDIR.
Proof: Omitted due to space constraints.
We note that the converse to this achievability region does
not hold in general. However, we can prove that the converse
holds for the following two important non-trivial special cases:
(1) When there are no helpers : When there are no helpers,
setting Ui,S= Φ ∀i ∈ Σ,S ∈ 2Π− φ, where Φ is a constant,
leads to the rate region in [10]. The converse to this rate region
follows directly from arguments similar to the converse of
Slepian-Wolf Theorem [5].
(2) A 2 sink network with a single helper : The converse
can be proven in general for any 2 sink network with a
single helper. We omit the details here due to space con-
straints. However, we just give a simple example of a 2 sink
network with a single helper. Consider the network shown
in Figure 2, with 3 sources and 2 sinks. Note that s0 acts
as a helper to both the sinks. The rate region of Theorem
1 for the tuple (R1,1,R2,2,R0,12,R0,1,R0,2) simplifies to
the following. The set of achievable rate tuple for each
??
i,S∀i ∈ Σ,S ∈ 2Π− φ be any set of
i∈Σ
?
S∈Qi
R
??
i,S
≥
max(βj(W,Q1,...,QN),0) (14)
i∈Γ
?
S:j∈S
˜Ri,S≥ γj(Γ)
(15)
Ri,S≥
R
??
i,S+˜Ri,S
??
if S ⊆ 2Πi− φ
if S ? 2Πi− φ
(16)
(U0,U1,U2) jointly distributed with (X1,X0,X2) such that
X1 ↔ X0 ↔ (U0,U1,U2) and X2 ↔ X0 ↔ (U0,U1,U2)
is given by R0,12 ≥ I(X0;U0), R0,1 ≥ I(X0;U1|U0),
R0,2 ≥ I(X0;U2|U0), R1,1 ≥ H(X1|U0,U1) and R2,2 ≥
H(X2|U0,U2). The closure over all such (U0,U1,U2) is the
complete rate region for this problem. The converse follows
in similar lines to the derivation of the outer bound in [8].
IV. CONCLUSION
This paper considers a new routing paradigm called disper-
sive information routing, wherein each intermediate node is
allowed to split a packet and forward subsets of the packet on
each forward path. In our prior work, we considered a special
case of the problem when each sink is allowed to receive
packets only from the sources it intends to reconstruct, and
derived the complete rate region. In this paper, considered a
more general framework wherein each sink can (possibly) re-
ceive packets from all the sources. Unfortunately, the problem
becomes considerably more complex. We derived an achiev-
able rate region using principles from multiple descriptions
encoding and Han and Kobayashi decoding which is complete
only for certain special cases of the setup.
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