Conference Paper

Index Coding with Side Information

Dept. of Electr. Eng., Technion-Israel Inst. of Technol., Haifa
DOI: 10.1109/FOCS.2006.42 Conference: Foundations of Computer Science, 2006. FOCS '06. 47th Annual IEEE Symposium on
Source: IEEE Xplore

ABSTRACT

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}n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G on n nodes; Ri knows 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} n with 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

Full-text preview

Available from: psu.edu
  • Source
    • "The unique packets which each client has requested is known as the " want set " of c i , c i ∈ C, and given as W i ⊂ P. The server has the side information knowledge, which is the set of symbols each client c i has cached and known as the " has set " H i ⊆ P\W i . The objective of the ICSI problem is to minimize the total number of transmissions ℓ, while satisfying the requests of all the clients [8]. For the index coding problem it is assumed that the unicast data flow is represented by a singleton want set, W i = {p i }, this as shown by Lemma 1 does not affect the result for the unicast transmission data flow assuming lossless broadcast channel. "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we consider the scenario of multiple unicast flows intersecting a common router in an opportunistic wireless network. Instead of forwarding packets in each of the flows independently, the router can perform inter-session network coding and transmit codewords to improve the network throughput. Unlike coding for multicast data flow for which an optimal code can be constructed in polynomial time, coding for unicast data flows is a more complicated coding problem and has been shown to be an NP-hard problem. Opportunities for inter-session network coding have also been shown to exist in single-hop wireless data dissemination network such as Wi-Fi and WiMAX networks. In this paper we propose an efficient coding scheme for unicast flows and demonstrate its higher coding gain over previously proposed coding schemes, validated using simulation results and TinyOS based wireless sensor testbed packet reception status data. We also show that our proposed algorithm is optimal for all 238 non-isomorphic coding instances for n ≤ 4, and for 9500 of the 9608 non-isomorphic coding instances for n = 5, where n is the number of unicast flows.
    Full-text · Article · Dec 2015
  • Source
    • "Our main result is based on the following observation. When the users' demands and cache contents are fixed, the delivery phase can be seen as an index coding problem [7], [9] and [10]. For the index coding problem, an outer bound based on the sub-modularity of entropy is proposed in [10, Theorem 1] and loosened in [10, Corollary 1]. "
    [Show abstract] [Hide abstract]
    ABSTRACT: Caching is an efficient way to reduce peak-hour network traffic congestion by storing some contents at user's local cache without knowledge of later demands. Maddah-Ali and Niesen initiated a fundamental study of caching systems; they proposed a scheme (with uncoded cache placement and linear network coding delivery) that is provably optimal to within a factor 12. In this paper, by noticing that when the cache contents and the demands are fixed, the caching problem can be seen as an index coding problem, we show the optimality of Maddah-Ali and Niesen's scheme assuming that cache placement is restricted to be uncoded and the number of users is not less than the number of files. Furthermore, this result states that further improvement to the Maddah-Ali and Niesen's scheme in this regimes can be obtained only by coded cache placement.
    Full-text · Article · Nov 2015
  • Source
    • "Any unicast index problem can be equivalently reduced to an single unicast problem discussed in [3]. For this canonical unicast index coding problem, it was shown that the length of the optimal linear index code is equal to the minrank of the side information graph of the index coding problem but finding the minrank is NP hard. "
    [Show abstract] [Hide abstract]
    ABSTRACT: This paper deals with vector linear index codes for multiple unicast index coding problems where there is a source with K messages and there are K receivers each wanting a unique message and having symmetric (with respect to the receiver index) two-sided antidotes (side information). Optimal scalar linear index codes for several such instances of this class of problems for one-sided antidotes(not necessarily adjacent) have already been reported. These codes can be viewed as special cases of the symmetric unicast index coding problems discussed by Maleki, Cadambe and Jafar with one sided adjacent antidotes. In this paper, starting from a given multiple unicast index coding problem with with K messages and one-sided adjacent antidotes for which a scalar linear index code $\mathfrak{C}$ is known, we give a construction procedure which constructs a sequence (indexed by m) of multiple unicast index problems with two-sided adjacent antidotes (for the same source) for all of which a vector linear code $\mathfrak{C}^{(m)}$ is obtained from $\mathfrak{C}.$ Also, it is shown that if $\mathfrak{C}$ is optimal then $\mathfrak{C}^{(m)}$ is also optimal for all $m.$ We illustrate our construction for some of the known optimal scalar linear codes.
    Full-text · Article · Nov 2015
Show more