A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences

TL;DR

This paper establishes a multi-round communication lower bound for the Gap Hamming problem, leading to new space complexity bounds for data stream algorithms and resolving open questions in the field.

Contribution

It proves an rac{n}{2} lower bound for multi-round randomized communication protocols for Gap Hamming, extending previous one-way bounds and impacting data stream complexity.

Findings

Multi-round rac{n}{2} lower bound for Gap Hamming

Space lower bounds for approximate counting and entropy in data streams

Tight bounds on one-way deterministic communication complexity

Abstract

The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their $n$-bit input strings is large (i.e., at least $n/2 + \sqrt n$) or small (i.e., at most $n/2 - \sqrt n$); they do not care if it is neither large nor small. This $\Theta(\sqrt n)$ gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an $\Omega(n)$ lower bound on this problem was known only in the one-way setting. We prove an $\Omega(n)$ lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that $\epsilon$-approximately counting the number of distinct elements in a data stream requires $\Omega(1/\epsilon^2)$ space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight $n - \Theta(\sqrt{n}\log n)$ lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an $\Omega(n)$ lower bound on the one-way randomized communication complexity.