TY - GEN

T1 - Broadcasting with side information

AU - Alon, Noga

AU - Hassidim, Avinatan

AU - Lubetzky, Eyal

AU - Stav, Uri

AU - Weinstein, Amit

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2008

Y1 - 2008

N2 - A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt be the minimum number of bits that has to be transmitted when each block is of length t, and let β be the limit β= lim →Infin; βt/t. Informally, β is the average communication cost per bit in each block (for long blocks). Finding the coding rate β, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on βt, and prove that for any constant C there is an explicit broadcast setting in which β= 2 but β1 > C. One of these examples answers a question of [15]. In addition, we provide examples with the following counterintuitive direct-sum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the combined setting may yield a significant saving in communication over concatenating optimal encodings for the individual settings. This result also provides new non-linear coding schemes which improve upon the largest known gap between linear and non-linear Network Coding, thus improving the results of [8]. The proofs are based on a relation between this problem and results in the study of Witsenhausen's rate, OR graph products, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs.

AB - A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt be the minimum number of bits that has to be transmitted when each block is of length t, and let β be the limit β= lim →Infin; βt/t. Informally, β is the average communication cost per bit in each block (for long blocks). Finding the coding rate β, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on βt, and prove that for any constant C there is an explicit broadcast setting in which β= 2 but β1 > C. One of these examples answers a question of [15]. In addition, we provide examples with the following counterintuitive direct-sum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the combined setting may yield a significant saving in communication over concatenating optimal encodings for the individual settings. This result also provides new non-linear coding schemes which improve upon the largest known gap between linear and non-linear Network Coding, thus improving the results of [8]. The proofs are based on a relation between this problem and results in the study of Witsenhausen's rate, OR graph products, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs.

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U2 - 10.1109/FOCS.2008.41

DO - 10.1109/FOCS.2008.41

M3 - Conference contribution

AN - SCOPUS:57949107168

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 823

EP - 832

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -