Maximizing Diversity in (near-)Median String Selection

TL;DR

This paper develops algorithms to find multiple diverse median strings under Hamming distance, enhancing robustness in data aggregation tasks like bioinformatics and pattern recognition.

Contribution

It introduces exact and approximation algorithms for generating diverse near-optimal median strings, utilizing structural insights and error-correcting code techniques.

Findings

Exact algorithm for diameter variant identifies maximally diverse median pairs.

$(1-\, ext{epsilon})$-approximation algorithm for sum dispersion.

Bi-criteria approximation for min dispersion enables multiple diverse medians.

Abstract

Given a set of strings over a specified alphabet, identifying a median or consensus string that minimizes the total distance to all input strings is a fundamental data aggregation problem. When the Hamming distance is considered as the underlying metric, this problem has extensive applications, ranging from bioinformatics to pattern recognition. However, modern applications often require the generation of multiple (near-)optimal yet diverse median strings to enhance flexibility and robustness in decision-making. In this study, we address this need by focusing on two prominent diversity measures: sum dispersion and min dispersion. We first introduce an exact algorithm for the diameter variant of the problem, which identifies pairs of near-optimal medians that are maximally diverse. Subsequently, we propose a $(1-\epsilon)$-approximation algorithm (for any $\epsilon >0$) for sum dispersion, as well as a bi-criteria approximation algorithm for the more challenging min dispersion case, allowing the generation of multiple (more than two) diverse near-optimal Hamming medians. Our approach primarily leverages structural insights into the Hamming median space and also draws on techniques from error-correcting code construction to establish these results.