Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0,01l}nand wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Rihas prior side information about x, induced by a directed graph Grain nodes; Ri knows the bits of a; in the positions {j | (i,j) is an edge of G}.G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEXcodes for {0,1}nwith side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight.

Motivated by a problem of transmitting data over broadcast channels (BirkandKol, INFOCOM1998), we study the following coding problem: a sender communicates with n receivers Rl,.., Rn. He holds an input x isin {0, 1}nand wishes to broadcast a single message so that each receiver Rican recover the bit xi. Each Rihas prior side information about x, induced by a directed graph G on n nodes; Riknows the bits of x in the positions {j | (i, j) is anedge of G}. We call encoding schemes that achieve this goal INDEX codes for {0, 1}nwith side information graph G. In this paper we identify a measure on graphs, the minrank, which we conjecture to exactly characterize the minimum length of INDEX codes. We resolve the conjecture for certain natural classes of graphs. For arbitrary graphs, we show that the minrank bound is tight for both linear codes and certain classes of non-linear codes. For the general problem, we obtain a (weaker) lower bound that the length of an INDEX code for any graph G is at least the size of the maximum acyclic induced subgraph of G

The Informed-Source Coding On Demand (ISCOD) approach for efficiently supplying nonidentical data from a central server to multiple caching clients over a broadcast channel is presented. The key idea underlying ISCOD is the joint exploitation of the data blocks already cached by each client, the server's full knowledge of client-cache contents and client requests, and the fact that each client only needs to be able to derive the blocks requested by it rather than all the blocks ever transmitted or even the union of the blocks requested by the different clients. We present two-phase ISCOD algorithms: the server first creates ad-hoc error-correction sets based on its knowledge of client states; next, it uses erasure-correction codes to construct the data for transmission. Each client uses its cached data and the received supplemental data to derive its requested blocks. The result is up to a several-fold reduction in the amount of transmitted supplemental data. Also, we define k-partial cliques in a directed graph and cast ISCOD in terms of partial-clique covers.

We present the informed source coding on-demand (ISCOD) approach for efficiently supplying non-identical data from a central server to multiple caching clients through a broadcast channel. The key idea underlying ISCOD is the joint exploitation of the data already cached by each client, the server's full awareness of client cache contents and client requests, and the fact that each client only needs to be able to derive the items requested by it rather than all the items ever transmitted or even the union of the items requested by the different clients. We present a set of two-phase ISCOD algorithms. The server uses these algorithms to assemble ad-hoc error correction sets based its knowledge of every client's cache content and of the items requested by it; next, it uses error-correction codes to construct the data that is actually transmitted. Each client uses its cached data and the received supplemental data to derive the items that it has requested. This technique achieves a reduction of up to tens of percents in the amount of data that must be transmitted in order for every client to be able to derive the data requested by it. Finally, we define k-partial cliques in a directed graph, and cast the two phase approach in terms of partial clique covers. As a byproduct of this work, bounds and a close approximation for the expected cardinality of the maximum matching in a random graph have been derived and are outlined.