Lower Bounds for Interactive Function Computation via Wyner Common Information

TL;DR

This paper establishes lower bounds on the information complexity of distributed function computation using Wyner common information, providing exact calculations in some cases and highlighting limitations in others.

Contribution

It introduces a novel lower bound technique for information complexity based on Wyner common information and analyzes its tightness in specific scenarios.

Findings

Lower bounds are derived for independent inputs using Wyner common information.

The bounds are tight for the equality function on a ternary alphabet.

In general, the bounds may not always be tight, as shown by counterexamples.

Abstract

The question of how much communication is required between collaborating parties to compute a function of their data is of fundamental importance in the fields of theoretical computer science and information theory. In this work, the focus is on coming up with lower bounds on this. The information cost of a protocol is the amount of information the protocol reveals to Alice and Bob about each others inputs, and the information complexity of a function is the infimum of information costs over all valid protocols. For the amortized case, it is known that the optimal rate for the computation is equal to the information complexity. Exactly computing this information complexity is not straight forward however. In this work we lower bound information complexity for independent inputs in terms of the Wyner common information of a certain pair of random variables. We show a structural property for the optimal auxiliary random variable of Wyner common information and exploit this to exactly compute the Wyner common information in certain cases. The lower bound obtained through this technique is shown to be tight for a non-trivial example - equality (EQ) for the ternary alphabet. We also give an example to show that the lower bound may, in general, not be tight.