Quantum Algorithm for the Multiple String Matching Problem

TL;DR

This paper introduces a quantum algorithm for the multiple string matching problem that significantly improves upon classical methods, especially for long dictionary words, achieving near-optimal quantum complexity.

Contribution

It presents a quantum algorithm that matches the quantum lower bound for multiple string matching, extending the classical Aho-Corasick approach into the quantum domain.

Findings

Quantum algorithm has $O(n+\sqrt{mL ext{log} n}+m ext{log} n)$ query complexity.

Time complexity is $O^*(n+\sqrt{mL})$, matching the quantum lower bound.

Significant improvement for dictionaries with long words.

Abstract

Let us consider the Multiple String Matching Problem. In this problem, we consider a long string, denoted by $t$, of length $n$. This string is referred to as a text. We also consider a sequence of $m$ strings, denoted by $S$, which we refer to as a dictionary. The total length of all strings from the dictionary is represented by the variable L. The objective is to identify all instances of strings from the dictionary within the text. The standard classical solution to this problem is Aho-Corasick Algorithm that has $O(n+L)$ query and time complexity. At the same time, the classical lower bound for the problem is the same $\Omega(n+L)$. We propose a quantum algorithm with $O(n+\sqrt{mL\log n}+m\log n)$ query complexity and $O(n+\sqrt{mL\log n}\log b+m\log n)=O^*(n+\sqrt{mL})$ time complexity, where $b$ is the maximal length of strings from the dictionary. This improvement is particularly significant in the case of dictionaries comprising long words. Our algorithm's complexity is equal to the quantum lower bound $O(n + \sqrt{mL})$, up to a log factor. In some sense, our algorithm can be viewed as a quantum analogue of the Aho-Corasick algorithm.