Network Coding for Correlated Sources

We consider the ability of a distributed randomized network coding approach to multicast, to one or more receivers, correlated sources over a network where compression may be required. We give, for two arbitrarily correlated sources in a general network, upper bounds on the probability of decoding error at each receiver, in terms of network parameters. In the special case of a Slepian-Wolf source network consisting of a link from each source to the receiver, our error exponents reduce to known error exponents for linear Slepian-Wolf coding. I. I NTRODUCTION The achievable capacity of multicast networks with independent sources employing network coding was given in [1]. Reference [3] showed how to achieve this capacity in a distributed setting using randomized linear codes, and gave error bounds for transmission of independent and linearly correlated sources. This paper considers linear network coding in the context of a distributed source coding problem, where compression may be required to transmit information from correlated sources over a network to one or more receivers. An example of such a problem is given in Figure 1. We demonstrate that a distributed randomized network coding approach, an adaptation of that in [3], performs compression when necessary in a multicast network, generalizing known error exponents for linear Slepian-Wolf coding [2] in a natural way. Specifically, for two arbitrarily correlated discrete memoryless sources in a general network, we give error bounds for minimum entropy and maximum a posteriori probability decoding at each receiver. In the special case of a Slepian-Wolf source network consisting of one receiver connected directly by a capacitated link to each source, our error exponents reduce to the corresponding results for linear Slepian-Wolf coding. The latter scenario may thus be considered a degenerate case of network coding. In this approach, all nodes other than the receiver nodes independently select random linear mappings from vectors of input bits onto vectors of output bits. An illustration is given in Figure 2. The receivers need only know the overall linear combination of source processes in each of their incoming signals. This information can be transmitted to the receivers by sending a canonical basis through the network. The overhead of transmitting these coefficients decreases with increasing length of blocks over which the codes and network state are constant. Tracey Ho and Muriel Ḿedard are with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, e-mail:{trace, medard }@mit.edu Michelle Effros is with the Data Compression Laboratory, California Institute of Technology, Pasadena, CA 91125, e-mail: effros@caltech.edu Ralf Koetter is with the Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801, e-mail: koetter@csl.uiuc. edu This research is supported in part by NSF Grants CCR-0325324 and CCR0220039, NSF ITR on network coding, University of Illinois subaward #0325673, Hewlett-Packard 008542-008, and Caltechs Lee Center for Advanced Networking. Rcv 1 Rcv 2 X1 h h H(X1, X2)− h H(X1, X2)− h h ≥ H(X1|X2),H(X2|X1) h