A Faster Algorithm for Finding Closest Pairs in Hamming Metric

TL;DR

This paper introduces a new randomized algorithm for the Closest Pair Problem in Hamming metric that outperforms previous methods for small dimensions, matches existing complexity for larger dimensions, and simplifies analysis while being practically effective.

Contribution

The paper presents a novel randomized algorithm for the Hamming closest pair problem that improves theoretical complexity and offers a modular, simplified analysis adaptable to various input distributions.

Findings

Outperforms previous algorithms in small dimensions

Matches best-known complexity in moderate to large dimensions

Shows promising practical performance in initial implementation

Abstract

We study the Closest Pair Problem in Hamming metric, which asks to find the pair with the smallest Hamming distance in a collection of binary vectors. We give a new randomized algorithm for the problem on uniformly random input outperforming previous approaches whenever the dimension of input points is small compared to the dataset size. For moderate to large dimensions, our algorithm matches the time complexity of the previously best-known locality sensitive hashing based algorithms. Technically our algorithm follows similar design principles as Dubiner (IEEE Trans. Inf. Theory 2010) and May-Ozerov (Eurocrypt 2015). Besides improving the time complexity in the aforementioned areas, we significantly simplify the analysis of these previous works. We give a modular analysis, which allows us to investigate the performance of the algorithm also on non-uniform input distributions. Furthermore, we give a proof of concept implementation of our algorithm which performs well in comparison to a quadratic search baseline. This is the first step towards answering an open question raised by May and Ozerov regarding the practicability of algorithms following these design principles.