# Better Gap-Hamming Lower Bounds via Better Round Elimination

## TL;DR

This paper establishes a significantly improved lower bound on the communication complexity of the Gap Hamming Distance problem, demonstrating that fewer rounds require exponentially larger messages, with implications for data stream algorithms.

## Contribution

The paper introduces a new round elimination technique that yields a lower bound of Omega(n/(k^2 log k)) bits for k-round protocols, improving previous bounds exponentially.

## Key findings

- Lower bound of Omega(n/(k^2 log k)) bits for k-round protocols
- Implications for space complexity in data stream algorithms
- Further improvements to Omega(n) bound using different techniques

## Abstract

Gap Hamming Distance is a well-studied problem in communication complexity, in which Alice and Bob have to decide whether the Hamming distance between their respective n-bit inputs is less than n/2-sqrt(n) or greater than n/2+sqrt(n). We show that every k-round bounded-error communication protocol for this problem sends a message of at least Omega(n/(k^2\log k)) bits. This lower bound has an exponentially better dependence on the number of rounds than the previous best bound, due to Brody and Chakrabarti. Our communication lower bound implies strong space lower bounds on algorithms for a number of data stream computations, such as approximating the number of distinct elements in a stream. Subsequent to this result, the bound has been improved by some of us to the optimal Omega(n), independent of k, by using different techniques.