Some bounds on the capacity of communicating the sum of sources

Abstract

We consider directed acyclic networks with multiple sources and multiple terminals where each source generates one i.i.d. random process over an abelian group and all the terminals want to recover the sum of these random processes. The different source processes are assumed to be independent. The solvability of such networks has been considered in some previous works. In this paper we investigate on the capacity of such networks, referred as sum-networks, and present some bounds in terms of min-cut, and the numbers of sources and terminals.

Multi-Session Function Computation and Multicasting in Undirected Graphs

“…4) Linear Coding: Another popular achievable strategy is linear coding where each intermediate node forwards a linear combination of the incoming symbols. For general directed graphs, when the function to be computed is linear, random linear coding is known to be optimal [33]. In [17], it is shown that linear coding is insufficient when the function to be computed is nonlinear and the potential loss due to employing linear coding is quantified.…”

“…For linear function computation with a single terminal, linear coding is known to be optimal [33], but the case of multiple terminals has not been studied.…”

“…The single-session linear-function-computation problem has been well studied, and for directed graphs (potentially cyclic), it is known that the cut-set bound is achievable [33]. The achievable strategy is given by random linear coding and is inspired by the strategy used for the dual multicasting problem.…”

“…The achievable strategy is given by random linear coding and is inspired by the strategy used for the dual multicasting problem. The duality between linear coding for single-session linearfunction computation and linear coding for multicasting was observed in [33]. However, there is no natural way to extend this duality (or even the achievable strategy) to the computation of general functions.…”

Linear coding for network computing

“…However, it is known that for single-receiver networks, linear coding is sufficient for solvability when computing a scalar linear target function [4], [16]. Analogous to the coding capacity for network coding, the notion of computing capacity was defined for network computing in [8] and is the supremum of achievable rates of computing the network's target function.…”