Quantum jumbled pattern matching

TL;DR

This paper introduces a quantum algorithm that significantly improves the efficiency of online jumbled pattern matching, reducing the query time to square root of the text size, surpassing classical limitations.

Contribution

The paper presents the first quantum algorithm for online jumbled pattern matching with $O(\sqrt{N})$ query time, outperforming classical bounds.

Findings

Quantum algorithm achieves $O(\sqrt{N})$ query time for JPM.

Classical JPM conjecture suggests higher complexity for similar tasks.

Quantum approach surpasses classical lower bounds for online JPM.

Abstract

Let $S_1, S_2 \in \Sigma^*$ strings, we say that $S_1$ {\em jumble match} $S_2$ if they are permutations of each other. Given a text $T$ of size $N$ and a string $S \in \Sigma^*$, the problem of \emph{Jumbled Pattern Matching} (JPM) is to determine all substrings in $T$ jumbled matching $S$. In classical computing, a widespread conjecture is that JPM requires $\Omega(N^{2-\epsilon})$ preprocessing time and space for $O(1)$ query time, or $\Omega(N^{1-\delta})$ query time in the online version, with $\epsilon, \delta >0$. In this paper, we present a quantum algorithm for the online JPM in $O(\sqrt{N})$ time.