Parameterized Pattern Matching -- Succinctly

TL;DR

This paper introduces space-efficient indexes for parameterized pattern matching that improve upon previous solutions, enabling faster query times with significantly reduced space requirements.

Contribution

It presents the first succinct index for parameterized pattern matching, reducing space from previous solutions and extending techniques to related matching problems.

Findings

First succinct index with $n \log \sigma + ext{O}(n)$ bits

Query time improved to $ ext{O}((|P|+ occ ext{·} ext{log} n) ext{·} ext{log} \sigma ext{·} ext{log} ext{log} \sigma)$

Extended techniques to structural and dictionary matching indices

Abstract

We consider the $Parameterized$ $Pattern$ $Matching$ problem, where a pattern $P$ matches some location in a text $\mathsf{T}$ iff there is a one-to-one correspondence between the alphabet symbols of the pattern to those of the text. More specifically, assume that the text $\mathsf{T}$ contains $n$ characters from a static alphabet $\Sigma_s$ and a parameterized alphabet $\Sigma_p$, where $\Sigma_s \cap \Sigma_p = \varnothing$ and $|\Sigma_s \cup \Sigma_p|=\sigma$. A pattern $P$ matches a substring $S$ of $\mathsf{T}$ iff the static characters match exactly, and there exists a one-to-one function that renames the parameterized characters in $S$ to that in $P$. Previous indexing solution [Baker, STOC 1993], known as $Parameterized$ $Suffix$ $Tree$, requires $\Theta(n\log n)$ bits of space, and can find all $occ$ occurrences of $P$ in $\mathcal{O}(|P|\log \sigma+ occ)$ time. In this paper, we present the first succinct index that occupies $n \log \sigma + \mathcal{O}(n)$ bits and answers queries in $\mathcal{O}((|P|+ occ\cdot \log n) \log\sigma\log \log \sigma)$ time. We also present a compact index that occupies $\mathcal{O}(n\log\sigma)$ bits and answers queries in $\mathcal{O}(|P|\log \sigma+ occ\cdot \log n)$ time. Furthermore, the techniques are extended to obtain the first succinct representation of the index of Shibuya for $Structural$ $Matching$ [SWAT, 2000], and of Idury and Sch\"{a}ffer for $Parameterized$ $Dictionary$ $Matching$ [CPM, 1994].