# Optimal strategies for computing symmetric Boolean functions in collocated networks

## TL;DR

This paper develops optimal communication strategies for computing symmetric Boolean functions in collocated networks, minimizing data exchange while ensuring zero-error computation, with exact solutions for some classes and near-optimal for others.

## Contribution

It introduces exact optimal strategies for threshold and delta functions, and an order-optimal approach for interval functions, advancing understanding of data fusion efficiency in collocated networks.

## Key findings

- Optimal strategies for threshold functions derived
- Order-optimal strategies with preconstant for interval functions
- Characterization of complexity for percentile functions

## Abstract

We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements with zero error. We allow for block computation to enhance data fusion efficiency, and determine the minimum worst-case total bits to be communicated to perform the desired computation. We restrict attention to the class of symmetric Boolean functions, which only depend on the number of 1s among the n measurements. We define three classes of functions, namely threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and an order-optimal strategy with optimal preconstant for interval functions. Using these results, we can characterize the complexity of computing percentile type functions, which is of great interest. In our analysis, we use lower bounds from communication complexity theory, and provide an achievable scheme using information theoretic tools.