Encoding Complexity of Network Coding with Two Simple Multicast Sessions

TL;DR

This paper provides efficient algorithms and bounds for solving network coding problems with two multicast sessions, significantly improving understanding of their complexity and resource requirements.

Contribution

It proves that solvability, solution construction, and optimal solutions for 2-URMS networks can be achieved in linear or polynomial time, with tight bounds on encoding links and field size.

Findings

Solvability can be decided in O(|E|) time.

A solution can be constructed in O(|E|) time.

The number of encoding links is bounded by max{3, 2N-2}.

Abstract

The encoding complexity of network coding for single multicast networks has been intensively studied from several aspects: e.g., the time complexity, the required number of encoding links, and the required field size for a linear code solution. However, these issues as well as the solvability are less understood for networks with multiple multicast sessions. Recently, Wang and Shroff showed that the solvability of networks with two unit-rate multicast sessions (2-URMS) can be decided in polynomial time. In this paper, we prove that for the 2-URMS networks: $1)$ the solvability can be determined with time $O(|E|)$; $2)$ a solution can be constructed with time $O(|E|)$; $3)$ an optimal solution can be obtained in polynomial time; $4)$ the number of encoding links required to achieve a solution is upper-bounded by $\max\{3,2N-2\}$; and $5)$ the field size required to achieve a linear solution is upper-bounded by $\max\{2,\lfloor\sqrt{2N-7/4}+1/2\rfloor\}$, where $|E|$ is the number of links and $N$ is the number of sinks of the underlying network. Both bounds are shown to be tight.