On the Average-case Complexity of Pattern Matching with Wildcards

TL;DR

This paper introduces new average-case algorithms for pattern matching with wildcards, providing the first non-trivial time bounds and demonstrating a complexity separation from exact pattern matching.

Contribution

It presents the first average-case complexity bounds for pattern matching with wildcards and adapts these results for wildcards in both pattern and text.

Findings

First non-trivial average-case time bounds for wildcard pattern matching.

Provable complexity separation between exact and wildcard pattern matching.

Algorithms applicable when wildcards are restricted to pattern or text.

Abstract

Pattern matching with wildcards is the problem of finding all factors of a text $t$ of length $n$ that match a pattern $x$ of length $m$, where wildcards (characters that match everything) may be present. In this paper we present a number of fast average-case algorithms for pattern matching where wildcards are restricted to either the pattern or the text, however, the results are easily adapted to the case where wildcards are allowed in both. We analyse the \textit{average-case} complexity of these algorithms and show the first non-trivial time bounds. These are the first results on the average-case complexity of pattern matching with wildcards which, as a by product, provide with first provable separation in complexity between exact pattern matching and pattern matching with wildcards in the word RAM model.