Approximating Approximate Pattern Matching

TL;DR

This paper introduces efficient algorithms for approximate pattern matching under  distance, providing deterministic and randomized solutions with near-linear runtime for all non-negative p, improving upon prior work for  distances.

Contribution

It presents new -approximate algorithms with  runtime for all non-negative p, including deterministic methods for p  1 and randomized methods for 0  p  1, extending previous results.

Findings

Deterministic -approximation for all p  1 with  runtime.

Randomized -approximation for 0  p  1, offering a tradeoff.

Improved algorithms over previous work for approximate pattern matching.

Abstract

Given a text $T$ of length $n$ and a pattern $P$ of length $m$, the approximate pattern matching problem asks for computation of a particular \emph{distance} function between $P$ and every $m$-substring of $T$. We consider a $(1\pm\varepsilon)$ multiplicative approximation variant of this problem, for $\ell_p$ distance function. In this paper, we describe two $(1+\varepsilon)$-approximate algorithms with a runtime of $\widetilde{O}(\frac{n}{\varepsilon})$ for all (constant) non-negative values of $p$. For constant $p \ge 1$ we show a deterministic $(1+\varepsilon)$-approximation algorithm. Previously, such run time was known only for the case of $\ell_1$ distance, by Gawrychowski and Uzna\'nski [ICALP 2018] and only with a randomized algorithm. For constant $0 \le p \le 1$ we show a randomized algorithm for the $\ell_p$, thereby providing a smooth tradeoff between algorithms of Kopelowitz and Porat [FOCS~2015, SOSA~2018] for Hamming distance (case of $p=0$) and of Gawrychowski and Uzna\'nski for $\ell_1$ distance.