Quantum Algorithm for Lexicographically Minimal String Rotation

TL;DR

This paper introduces a quantum algorithm that efficiently finds the lexicographically minimal string rotation, outperforming classical algorithms in both worst and average cases, with applications in graph and automata analysis.

Contribution

A novel quantum query algorithm for LMSR with optimal average-case complexity, improving over classical methods and applicable to various structural identification problems.

Findings

Quantum algorithm has $O(n^{3/4})$ query complexity.

Average-case complexity is $O(\sqrt n \log n)$, asymptotically optimal.

Outperforms classical randomized algorithms in worst and average cases.

Abstract

Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an $O(n^{3/4})$ quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity $O(\sqrt n \log n)$, which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its $\Omega\left(\sqrt{n/\log n}\right)$ lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.