Submitted to the IEEE Transactions on Information Theory,
52(1):155-170, 2006.
Abstract
Consider the following network communication setup, originating
in a sensor networking application we refer to as the "sensor
reachback" problem. We have a directed graph G=(V,E), where
V = {v_0,v_1,...,v_n} and E \subseteq V \times V. If
(v_i,v_j) \in E, then node i can send messages to
node j over a discrete memoryless channel
(\mathcal{X}_{ij}, p_{ij}(y|x), \mathcal{Y}_{ij}), of capacity
C_{ij}. The channels are independent. Each node v_i
gets to observe a source of information U_i (i=0...M),
with joint distribution p(U_0,U_1,...,U_M). Our goal is to
solve an incast problem in G: nodes exchange messages with their
neighbors, and after a finite number of communication rounds, one of
the M+1 nodes (v_0 by convention) must have received
enough information to reproduce the entire field of observations
(U_0,U_1,...,U_M), with arbitrarily small probability of error.
In this paper, we prove that such perfect reconstruction is possible
if and only if
H(U_S|U_{S^c}) < \sum_{i\in S,j\in S^c} C_{ij},
for all S \subseteq {0...M}, S \neq \emptyset,
0 \in S^c. Our main finding is that in this setup a general
source/channel separation theorem holds, and that Shannon information
behaves as a classical network flow, identical in nature to the flow
of water in pipes. At first glance, it might seem surprising that
separation holds in a fairly general network situation like the one
we study. A closer look, however, reveals that the reason for this
is that our model allows only for independent point-to-point channels
between pairs of nodes, and not multiple-access and/or broadcast
channels, for which separation is well known not to
hold~\cite[pp.\ 448-49]{CoverT:91}. This "information as flow" view
provides an algorithmic interpretation for our results, among which
perhaps the most important one is the optimality of implementing codes
using a layered protocol stack.
J. Barros, S. D. Servetto.
Coding Theorems for Network Information Flow
with Correlated Sources. In the Proceedings of the IEEE
International Symposium on Information Theory (ISIT), Adelaide,
Australia, September 2005.
J. Barros, C. Peraki, S. D. Servetto.
Efficient Network
Architectures for Sensor Reachback. In the Proceedings
of the IEEE International Zuerich Seminar on Communications, Zuerich,
Switzerland, February 2004. Invited paper.
J. Barros, S. D. Servetto.
Reachback Capacity
with Non-Interfering Nodes. In the Proceedings of the IEEE
International Symposium on Information Theory (ISIT), Yokohama, Japan,
June-July 2003.
J. Barros, S. D. Servetto.
On the Capacity of
the Reachback Channel in Wireless Sensor Networks. In the
Proceedings of the IEEE Workshop on Multimedia Signal Processing (special
session on "Signal Processing for Wireless Networks"), US Virgin Islands,
December 2002. Invited paper.