On network coding for sum-networks

TL;DR

This paper explores the theoretical foundations of network coding for sum-networks, establishing equivalences with other network types, and analyzing the conditions for linear and non-linear solvability over finite fields.

Contribution

It demonstrates the equivalence between sum-networks and multiple-unicast networks, and characterizes solvability conditions based on polynomial roots and field characteristics.

Findings

Existence of solvably equivalent sum-networks for any multiple-unicast network

Linear solvability depends on common roots of polynomials over finite fields

Non-linear coding can solve sum-networks where linear coding fails

Abstract

A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear network coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under non-linear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.