Graphs can be succinctly indexed for pattern matching in $ O(|E|^2 + |V|^{5 / 2}) $ time

TL;DR

This paper introduces a polynomial-time constructible, space-efficient index for arbitrary graphs that enables faster pattern matching, with applications in automata theory and automaton acceptance testing.

Contribution

It presents the first succinct pattern matching index for arbitrary graphs with improved space and query efficiency over previous methods.

Findings

Index can be built in $O(|E|^2 + |V|^{5/2})$ time.

Supports pattern matching in $O(|P| imes q^2 imes ext{log}(q imes | ext{Sigma}|))$ time.

Applications include automaton acceptance testing and automaton nondeterminism reduction.

Abstract

For the first time we provide a succinct pattern matching index for arbitrary graphs that can be built in polynomial time, which requires less space and answers queries more efficiently than the one in [SODA 2021]. We show that, given an edge-labeled graph $ G = (V, E) $, there exists a data structures of $|E /_{\le_G}|(\lceil \log|\Sigma|\rceil + \lceil\log q\rceil + 2)\cdot (1+o(1)) + |V /_{\le_G}|\cdot (1+o(1))$ bits which can be built in $ O(|E|^2 + |V /_{\le_G}|^{5 / 2}) $ time and supports pattern matching on $ G $ in $O(|P| \cdot q^2 \cdot \log(q\cdot |\Sigma|))$ time, where $ G /_{\le_G} = (V /_{\le_G}, E /_{\le_G}) $ is a quotient graph obtained by collapsing some nodes in $ G $ (so $ |V /_{\le_G}| \le |V| $ and $ |E /_{\le_G}| \le |E| $) and $ q $ is the width of the maximum co-lex relation on $ G $. Our results have relevant applications in automata theory. First, we can build a succinct data structure to decide whether a string is accepted by a given automaton. Second, starting from an automaton $ \mathcal{A} $, one can define a relation $ \preceq_\mathcal{A} $ and a quotient automaton that capture the nondeterminism of $ \mathcal{A} $, improving the results in [SODA 2021].