An Algorithmic Bridge Between Hamming and Levenshtein Distances

Elazar Goldenberg, Tomasz Kociumaka, Robert Krauthgamer, Barna Saha

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The edit distance between strings classically assigns unit cost to every character insertion, deletion, and substitution, whereas the Hamming distance only allows substitutions. In many real-life scenarios, insertions and deletions (abbreviated indels) appear frequently but significantly less so than substitutions. To model this, we consider substitutions being cheaper than indels, with cost a1 for a parameter a ≥ 1. This basic variant, denoted EDa, bridges classical edit distance (a = 1) with Hamming distance (a → ∞), leading to interesting algorithmic challenges: Does the time complexity of computing EDa interpolate between that of Hamming distance (linear time) and edit distance (quadratic time)? What about approximating EDa? We first present a simple deterministic exact algorithm for EDa and further prove that it is near-optimal assuming the Orthogonal Vectors Conjecture. Our main result is a randomized algorithm computing a (1 + ϵ)-approximation of EDa(X,Y), given strings X,Y of total length n and a bound k ≥ EDa(X,Y). For simplicity, let us focus on k ≥ 1 and a constant ϵ > 0; then, our algorithm takes Õ(na + ak3) time. Unless a = Õ(1), in which case EDa resembles the standard edit distance, and for the most interesting regime of small enough k, this running time is sublinear in n. We also consider a very natural version that asks to find a (kI,kS)-alignment, i.e., an alignment with at most kI indels and kS substitutions. In this setting, we give an exact algorithm and, more importantly, an Õ(nkkSI +kSkI3)-time (1,1+ϵ)-bicriteria approximation algorithm. The latter solution is based on the techniques we develop for EDa for a = Θ(kkSI ), and its running time is again sublinear in n whenever kI ≪ kS and the overall distance is small enough. These bounds are in stark contrast to unit-cost edit distance, where state-of-the-art algorithms are far from achieving (1 + ϵ)-approximation in sublinear time, even for a favorable choice of k.

Original languageEnglish
Title of host publication14th Innovations in Theoretical Computer Science Conference, ITCS 2023
EditorsYael Tauman Kalai
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772631
DOIs
StatePublished - 1 Jan 2023
Event14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States
Duration: 10 Jan 202313 Jan 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume251
ISSN (Print)1868-8969

Conference

Conference14th Innovations in Theoretical Computer Science Conference, ITCS 2023
Country/TerritoryUnited States
CityCambridge
Period10/01/2313/01/23

All Science Journal Classification (ASJC) codes

  • Software

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