8 research outputs found
Caching and Coded Multicasting: Multiple Groupcast Index Coding
The capacity of caching networks has received considerable attention in the
past few years. A particularly studied setting is the case of a single server
(e.g., a base station) and multiple users, each of which caches segments of
files in a finite library. Each user requests one (whole) file in the library
and the server sends a common coded multicast message to satisfy all users at
once. The problem consists of finding the smallest possible codeword length to
satisfy such requests. In this paper we consider the generalization to the case
where each user places requests. The obvious naive scheme consists
of applying times the order-optimal scheme for a single request, obtaining
a linear in scaling of the multicast codeword length. We propose a new
achievable scheme based on multiple groupcast index coding that achieves a
significant gain over the naive scheme. Furthermore, through an information
theoretic converse we find that the proposed scheme is approximately optimal
within a constant factor of (at most) .Comment: 5 pages, 1 figure, to appear in GlobalSIP14, Dec. 201
Linear Codes are Optimal for Index-Coding Instances with Five or Fewer Receivers
We study zero-error unicast index-coding instances, where each receiver must
perfectly decode its requested message set, and the message sets requested by
any two receivers do not overlap. We show that for all these instances with up
to five receivers, linear index codes are optimal. Although this class contains
9847 non-isomorphic instances, by using our recent results and by properly
categorizing the instances based on their graphical representations, we need to
consider only 13 non-trivial instances to solve the entire class. This work
complements the result by Arbabjolfaei et al. (ISIT 2013), who derived the
capacity region of all unicast index-coding problems with up to five receivers
in the diminishing-error setup. They employed random-coding arguments, which
require infinitely-long messages. We consider the zero-error setup; our
approach uses graph theory and combinatorics, and does not require long
messages.Comment: submitted to the 2014 IEEE International Symposium on Information
Theory (ISIT
A New Class of Index Coding Instances Where Linear Coding is Optimal
We study index-coding problems (one sender broadcasting messages to multiple
receivers) where each message is requested by one receiver, and each receiver
may know some messages a priori. This type of index-coding problems can be
fully described by directed graphs. The aim is to find the minimum codelength
that the sender needs to transmit in order to simultaneously satisfy all
receivers' requests. For any directed graph, we show that if a maximum acyclic
induced subgraph (MAIS) is obtained by removing two or fewer vertices from the
graph, then the minimum codelength (i.e., the solution to the index-coding
problem) equals the number of vertices in the MAIS, and linear codes are
optimal for this index-coding problem. Our result increases the set of
index-coding problems for which linear index codes are proven to be optimal.Comment: accepted and to be presented at the 2014 International Symposium on
Network Coding (NetCod
The Single-Uniprior Index-Coding Problem: The Single-Sender Case and The Multi-Sender Extension
Index coding studies multiterminal source-coding problems where a set of
receivers are required to decode multiple (possibly different) messages from a
common broadcast, and they each know some messages a priori. In this paper, at
the receiver end, we consider a special setting where each receiver knows only
one message a priori, and each message is known to only one receiver. At the
broadcasting end, we consider a generalized setting where there could be
multiple senders, and each sender knows a subset of the messages. The senders
collaborate to transmit an index code. This work looks at minimizing the number
of total coded bits the senders are required to transmit. When there is only
one sender, we propose a pruning algorithm to find a lower bound on the optimal
(i.e., the shortest) index codelength, and show that it is achievable by linear
index codes. When there are two or more senders, we propose an appending
technique to be used in conjunction with the pruning technique to give a lower
bound on the optimal index codelength; we also derive an upper bound based on
cyclic codes. While the two bounds do not match in general, for the special
case where no two distinct senders know any message in common, the bounds
match, giving the optimal index codelength. The results are expressed in terms
of strongly connected components in directed graphs that represent the
index-coding problems.Comment: Author final manuscrip
Duality Codes and the Integrality Gap Bound for Index Coding
Abstract—This paper considers a base station that delivers packets to multiple receivers through a sequence of coded transmissions. All receivers overhear the same transmissions. Each receiver may already have some of the packets as side information, and requests another subset of the packets. This problem is known as the index coding problem and can be represented by a bipartite digraph. An integer linear program is developed that provides a lower bound on the minimum number of transmissions required for any coding algorithm. Conversely, its linear programming relaxation is shown to provide an upper bound that is achievable by a simple form of vector linear coding. Thus, the information theoretic optimum is bounded by the integrality gap between the integer program and its linear relaxation. In the special case when the digraph has a planar structure, the integrality gap is shown to be zero, so that exact optimality is achieved. This work illuminates the relationship between index coding, duality, and integrality gaps between integer programs and their linear relaxations. I