String Indexing for Patterns with Wildcards

TL;DR

This paper introduces new indexing methods for string pattern matching with wildcards, significantly improving query times and space efficiency over previous approaches, and extends these methods to handle variable length gaps.

Contribution

The paper presents novel indexing algorithms for wildcard pattern matching that achieve faster query times and better space bounds, including a linear space index and a time-space trade-off.

Findings

Linear space index with $O(m+\sigma^j \log \log n + occ)$ query time

Index with $O(m+j+occ)$ query time using $O(\sigma^{k^2} n \log^k \log n)$ space

Generalization to patterns with variable length gaps

Abstract

We consider the problem of indexing a string $t$ of length $n$ to report the occurrences of a query pattern $p$ containing $m$ characters and $j$ wildcards. Let $occ$ be the number of occurrences of $p$ in $t$, and $\sigma$ the size of the alphabet. We obtain the following results. - A linear space index with query time $O(m+\sigma^j \log \log n + occ)$. This significantly improves the previously best known linear space index by Lam et al. [ISAAC 2007], which requires query time $\Theta(jn)$ in the worst case. - An index with query time $O(m+j+occ)$ using space $O(\sigma^{k^2} n \log^k \log n)$, where $k$ is the maximum number of wildcards allowed in the pattern. This is the first non-trivial bound with this query time. - A time-space trade-off, generalizing the index by Cole et al. [STOC 2004]. We also show that these indexes can be generalized to allow variable length gaps in the pattern. Our results are obtained using a novel combination of well-known and new techniques, which could be of independent interest.