A Data-Structure for Approximate Longest Common Subsequence of A Set of Strings

TL;DR

This paper introduces a novel data-structure for efficiently approximating the longest common subsequence among a set of strings, using a tree structure to enable sublinear query times with controlled approximation error.

Contribution

It presents a new data-structure that preprocesses multiple strings for faster approximate LCS queries, incorporating error tolerance and extending to LIS problems.

Findings

Achieves sublinear-time approximate LCS queries

Handles error via character replacements in the approximation

Extends methodology to the longest increasing subsequence problem

Abstract

Given a set of $k$ strings $I$, their longest common subsequence (LCS) is the string with the maximum length that is a subset of all the strings in $I$. A data-structure for this problem preprocesses $I$ into a data-structure such that the LCS of a set of query strings $Q$ with the strings of $I$ can be computed faster. Since the problem is NP-hard for arbitrary $k$, we allow an error that allows some characters to be replaced by other characters. We define the approximation version of the problem with an extra input $m$, which is the length of the regular expression (regex) that describes the input, and the approximation factor is the logarithm of the number of possibilities in the regex returned by the algorithm, divided by the logarithm regex with the minimum number of possibilities. Then, we use a tree data-structure to achieve sublinear-time LCS queries. We also explain how the idea can be extended to the longest increasing subsequence (LIS) problem.