Quantum Algorithm for Searching for the Longest Segment and the Largest Empty Rectangle

TL;DR

This work studies the Largest Empty Segment and Largest Empty Rectangle problems under a quantum query model. It develops Grover-based quantum algorithms delivering a near-quadratic speed-up in 1D and polynomial speed-ups for restricted 2D variants (LSQR, LREC2, LRECW), with corresponding lower bounds discussed. The results clarify the potential and limits of quantum speed-ups for geometric maximum-empty problems, and they outline open challenges for tightening bounds and extending speed-ups to the general 2D LREC problem. Overall, the paper advances understanding of quantum advantages in structured search over geometric configurations and provides a roadmap for future improvements.

Abstract

In the paper, we consider the problem of searching for the Largest empty rectangle in a 2D map, and the one-dimensional version of the problem is the problem of searching for the largest empty segment. We present a quantum algorithm for the Largest Empty Square problem and the Largest Empty Rectangle of a fixed width $d$ for $n\times n$-rectangular map. Query complexity of the algorithm is $\tilde{O}(n^{1.5})$ for the square case, and $\tilde{O}(n\sqrt{d})$ for the rectangle with a fixed width $d$ case, respectively. At the same time, the lower bounds for the classical case are $Ω(n^2)$, and $Ω(nd)$, respectively. The Quantum algorithm for the one-dimensional version of the problem has $O(\sqrt{n}\log n\log\log n)$ query complexity. The quantum lower bound for the problem is $Ω(\sqrt{n})$ which is almost equal to the upper bound up to a log factor. The classical lower bound is $Ω(n)$. So, we obtain the quadratic speed-up for the problem.

Theorems & Definitions (15)

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