The Influence of Placement on Transmission in Distributed Computing of Boolean Functions
Ahmad Tanha, Derya Malak
TL;DR
The paper addresses distributed computing of linearly-separable Boolean functions under a placement-transmission tradeoff, proposing a framework based on joint sensitivity and influence to quantify how dataset placement impacts communication. It defines $Inf$ and $as$ to capture the effect of placement on the function outcome and derives a lower bound $as_S(f) \ge \frac{N}{2^{M-1}}$ with corresponding transmission bound $T^{(S)} \ge N$. An explicit optimal placement $\mathcal{S}^*$ achieves $Inf_{\mathcal{S}_n^*}(f) = 2^{-(M-1)}$, $as_{\mathcal{S}^*}(f) = \frac{N}{2^{M-1}}$, and $T^{(\mathcal{S}^*)} = N$, demonstrating that careful placement can minimize communication costs. The results are illustrated with an example showing substantial gains from function-aware placement, and the work points to extending the influence-based approach to nonlinear Boolean functions to broaden applicability.
Abstract
In this paper, we explore a distributed setting, where a user seeks to compute a linearly-separable Boolean function of degree $M$ from $N$ servers, each with a cache size $M$. Exploiting the fundamental concepts of sensitivity and influences of Boolean functions, we devise a novel approach to capture the interplay between dataset placement across servers and server transmissions and to determine the optimal solution for dataset placement that minimizes the communication cost. In particular, we showcase the achievability of the minimum average joint sensitivity, $\frac{N}{2^{M-1}}$, as a measure for the communication cost.