Comparison communication protocols

TL;DR

This paper studies a restricted communication protocol where Alice and Bob can only use comparison functions, introducing a geometric tiling number measure that characterizes complexity and can be computed efficiently.

Contribution

It defines the geometric tiling number as a new complexity measure for comparison-based communication protocols and establishes its relation to function complexity.

Findings

Complexity is characterized by the geometric tiling number.

The geometric tiling number can be computed in polynomial time.

Analogous results hold for a restricted decision tree model.

Abstract

We introduce a restriction of the classical 2-party deterministic communication protocol where Alice and Bob are restricted to using only comparison functions. We show that the complexity of a function in the model is, up to a constant factor, determined by a complexity measure analogous to Yao's tiling number, which we call the geometric tiling number which can be computed in polynomial time. As a warm-up, we consider an analogous restricted decision tree model and observe a 1-dimensional analog of the above results.