A Relation Between Network Computation and Functional Index Coding Problems

TL;DR

This paper establishes a fundamental equivalence between network computation problems and a specific class of functional index coding problems, showing that solutions in one domain correspond directly to solutions in the other.

Contribution

It introduces a novel correspondence between network computation and functional index coding problems, extending known transformations between network coding and index coding.

Findings

Network computation problems can be transformed into functional index coding problems.

Existence of solutions in network computation problems is equivalent to solutions in the corresponding index coding problems.

The length of the index code corresponds to the feasibility of the network computation solution.

Abstract

In contrast to the network coding problem wherein the sinks in a network demand subsets of the source messages, in a network computation problem the sinks demand functions of the source messages. Similarly, in the functional index coding problem, the side information and demands of the clients include disjoint sets of functions of the information messages held by the transmitter instead of disjoint subsets of the messages, as is the case in the conventional index coding problem. It is known that any network coding problem can be transformed into an index coding problem and vice versa. In this work, we establish a similar relationship between network computation problems and a class of functional index coding problems, viz., those in which only the demands of the clients include functions of messages. We show that any network computation problem can be converted into a functional index coding problem wherein some clients demand functions of messages and vice versa. We prove that a solution for a network computation problem exists if and only if a functional index code (of a specific length determined by the network computation problem) for a suitably constructed functional index coding problem exists. And, that a functional index coding problem admits a solution of a specified length if and only if a suitably constructed network computation problem admits a solution.