On the Complexity of Finding Approximate LCS of Multiple Strings

TL;DR

This paper investigates the computational complexity of finding approximate longest common substrings among multiple strings, proposing efficient algorithms for certain cases and establishing lower bounds under complexity hypotheses.

Contribution

It introduces algorithms for restricted ALCS variants using advanced data structures and analyzes their complexity, extending the study to indeterminate strings.

Findings

Algorithms with quadratic and near-linear run times for specific ALCS variants

Conditional lower bounds based on the Strong Exponential Time Hypothesis

Extension of methods to indeterminate strings

Abstract

Finding an Approximate Longest Common Substring (ALCS) within a given set $S=\{s_1,s_2,\ldots,s_m\}$ of $m \ge 2$ strings is a key problem in computational biology, such as identifying related mutations across multiple genetic sequences. We study several variants of ALCS problems that, given integers $k$ and $t \le m$, seek the longest string $u$ -- or the longest substring $u$ of any string in $S$ -- that lies within distance $k$ of at least one substring in $t$ distinct strings from $S$. While the general problems are NP-hard, we present efficient algorithms for restricted cases under Hamming and edit distances using the $LCP_k$ and $k$-errata tree data structures. Our methods achieve run times of $\mathcal{O}(N^2)$, $\mathcal{O}(k\ell N^2)$, and $\mathcal{O}(mN\log^k \ell)$, where $\ell$ is the length of the longest string and $N$ is the sum of the lengths of all the strings in $S$. We also establish conditional lower bounds under the Strong Exponential Time Hypothesis and extend our study to indeterminate strings.