Multi-User Linearly Separable Computation: A Coding Theoretic Approach

Abstract

In this work, we investigate the problem of multiuser linearly separable function computation, where N servers help compute the desired functions (jobs) of K users. In this setting each desired function can be written as a linear combination of up to L (generally non-linear) sub-functions. Each server computes some of the sub-tasks, and communicates a linear combination of its computed outputs (files) in a singleshot to some of the users, then each user linearly combines its received data in order to recover its desired function. We explore the range of the optimal computation cost via establishing a novel relationship between our problem, syndrome decoding and covering codes. The work reveals that in the limit of large N , the optimal computation cost -in the form of the maximum fraction of all servers that must compute any subfunction -is lower bounded as γ ≥ H −1 q ( log q (L) N), for any fixed log q (L)/N . The result reveals the role of the computational rate log q (L)/N , which cannot exceed what one might call the computational capacity Hq(γ) of the system.

Multi-User Linearly-Separable Distributed Computing

“…Then, in terms of the construction of schemes that efficiently resolve our distributed computing problem, our work provides a never-before-seen connection between distributed computing and the powerful structure of perfect codes. Deviating from the approach in [32] which uses covering codes to reduce the computation cost in asymptotic settings, we here show how perfect codes -which optimize both the covering radius and packing density of codesyield an improved solution to our distributed computing problem, both in terms of cumulative as well as worst-case costs, and do so for finite dimensions. To the best of our understanding, this is the first time that perfect codes (and the closely related quasi-perfect codes) have been associated with distributed computing and the equivalent problem of matrix factorization.…”

“…Motivated by the same need to efficiently parallelize multiple computational tasks, our work here studies the known multi-user linearly decomposable scenario introduced in [32], [33], which can be seen as a multi-user extension to [34], and which entails a master node that acts as a total trusted authority in managing N server nodes, serving K users that each demand their function to be computed. Under the linearly-decomposable assumption, where each function is a linear combination of L basis subfunctions, it was recently shown (cf.…”

“…Under the linearly-decomposable assumption, where each function is a linear combination of L basis subfunctions, it was recently shown (cf. [32], [33]) that the multi-user distributed computing problem is mathematically equivalent to a sparse matrix factorization problem of the form F = DE, where F ∈ GF(q) K×L describes the linear coefficients of the demands of the users, where the so-called computation and encoding matrix E ∈ GF(q) N ×L describes which subfunctions each server evaluates and how each server combines the subfunctions' outputs, while the so-called communication and decoding matrix D ∈ GF(q) K×N describes the server-to-user activated connectivity and the manner with which each user combines its received data.…”