3,866 research outputs found
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
A New Index Coding Scheme Exploiting Interlinked Cycles
We study the index coding problem in the unicast message setting, i.e., where
each message is requested by one unique receiver. This problem can be modeled
by a directed graph. We propose a new scheme called interlinked cycle cover,
which exploits interlinked cycles in the directed graph, for designing index
codes. This new scheme generalizes the existing clique cover and cycle cover
schemes. We prove that for a class of infinitely many digraphs with messages of
any length, interlinked cycle cover provides an optimal index code.
Furthermore, the index code is linear with linear time encoding complexity.Comment: To be presented at the 2015 IEEE International Symposium on
Information Theory (ISIT 2015), Hong Kon
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
Generalized Interlinked Cycle Cover for Index Coding
A source coding problem over a noiseless broadcast channel where the source
is pre-informed about the contents of the cache of all receivers, is an index
coding problem. Furthermore, if each message is requested by one receiver, then
we call this an index coding problem with a unicast message setting. This
problem can be represented by a directed graph. In this paper, we first define
a structure (we call generalized interlinked cycles (GIC)) in directed graphs.
A GIC consists of cycles which are interlinked in some manner (i.e., not
disjoint), and it turns out that the GIC is a generalization of cliques and
cycles. We then propose a simple scalar linear encoding scheme with linear time
encoding complexity. This scheme exploits GICs in the digraph. We prove that
our scheme is optimal for a class of digraphs with message packets of any
length. Moreover, we show that our scheme can outperform existing techniques,
e.g., partial clique cover, local chromatic number, composite-coding, and
interlinked cycle cover.Comment: Extended version of the paper which is to be presented at the IEEE
Information Theory Workshop (ITW), 2015 Jej
- …