Optimal computation of symmetric Boolean functions in Tree networks

TL;DR

This paper develops optimal communication strategies for computing symmetric Boolean functions in tree networks, achieving minimal data exchange and extending results to general graphs with near-optimal complexity.

Contribution

It introduces a novel information-theoretic scheme that attains lower bounds for in-network symmetric Boolean function computation in tree networks.

Findings

Optimal in-network computation protocol for sum-threshold functions.

Lower bounds established using fooling sets and cut-set arguments.

Scheme extended to non-binary alphabets and general graph topologies.

Abstract

In this paper, we address the scenario where nodes with sensor data are connected in a tree network, and every node wants to compute a given symmetric Boolean function of the sensor data. We first consider the problem of computing a function of two nodes with integer measurements. We allow for block computation to enhance data fusion efficiency, and determine the minimum worst-case total number of bits to be exchanged to perform the desired computation. We establish lower bounds using fooling sets, and provide a novel scheme which attains the lower bounds, using information theoretic tools. For a class of functions called sum-threshold functions, this scheme is shown to be optimal. We then turn to tree networks and derive a lower bound for the number of bits exchanged on each link by viewing it as a two node problem. We show that the protocol of recursive innetwork aggregation achieves this lower bound in the case of sumthreshold functions. Thus we have provided a communication and in-network computation strategy that is optimal for each link. All the results can be extended to the case of non-binary alphabets. In the case of general graphs, we present a cut-set lower bound, and an achievable scheme based on aggregation along trees. For complete graphs, the complexity of this scheme is no more than twice that of the optimal scheme.