Matching Patterns with Variables under Hamming Distance

TL;DR

This paper investigates approximate pattern matching with variables under Hamming distance, providing efficient algorithms for certain classes of patterns and establishing intractability results for more complex cases.

Contribution

It introduces algorithms for approximate matching of variable patterns with Hamming distance and characterizes the complexity boundaries based on pattern structure.

Findings

Efficient algorithms for regular patterns with no repeated variables.

Matching algorithms for patterns with restricted variable interleaving.

Intractability results for patterns with arbitrary variable interleaving.

Abstract

A pattern $\alpha$ is a string of variables and terminal letters. We say that $\alpha$ matches a word $w$, consisting only of terminal letters, if $w$ can be obtained by replacing the variables of $\alpha$ by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. In this paper, we approach this problem in a generalized setting, by considering approximate pattern matching under Hamming distance. More precisely, we are interested in what is the minimum Hamming distance between $w$ and any word $u$ obtained by replacing the variables of $\alpha$ by terminal words. Firstly, we address the class of regular patterns (in which no variable occurs twice) and propose efficient algorithms for this problem, as well as matching conditional lower bounds. We show that the problem can still be solved efficiently if we allow repeated variables, but restrict the way the different variables can be interleaved according to a locality parameter. However, as soon as we allow a variable to occur more than once and its occurrences can be interleaved arbitrarily with those of other variables, even if none of them occurs more than once, the problem becomes intractable.