On Distributed Computing for Functions with Certain Structures

TL;DR

This paper develops a new method to derive bounds on the rate of distributed function computation, characterizes the optimal rate for certain classes of functions and sources, and simplifies proofs of existing results.

Contribution

It introduces a novel converse bound based on function structure, defines informative functions, and characterizes optimal rates for smooth and certain non-full support sources.

Findings

Optimal rate characterized for smooth sources and informative functions.

Hypergraph entropy used to determine rates for composition functions.

Simplified proof for the rate requirement of Boolean function computation.

Abstract

The problem of distributed function computation is studied, where functions to be computed is not necessarily symbol-wise. A new method to derive a converse bound for distributed computing is proposed; from the structure of functions to be computed, information that is inevitably conveyed to the decoder is identified, and the bound is derived in terms of the optimal rate needed to send that information. The class of informative functions is introduced, and, for the class of smooth sources, the optimal rate for computing those functions is characterized. Furthermore, for i.i.d. sources with joint distribution that may not be full support, functions that are composition of symbol-wise function and the type of a sequence are considered, and the optimal rate for computing those functions is characterized in terms of the hypergraph entropy. As a byproduct, our method also provides a conceptually simple proof of the known fact that computing a Boolean function may require as large rate as reproducing the entire source.