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

TL;DR

This paper introduces quantum algorithms for finding the largest empty rectangle and segment in 2D maps, achieving quadratic speed-ups over classical methods with specific query complexities.

Contribution

The paper presents novel quantum algorithms for geometric search problems, significantly improving query complexity compared to classical algorithms.

Findings

Quantum algorithms achieve $ ilde{O}(n^{1.5})$ and $ ilde{O}(n\sqrt{d})$ complexity.

Quantum algorithms for 1D problems have $O(\sqrt{n}\log n\log\log n)$ query complexity.

Quantum lower bounds are close to upper bounds, indicating near-optimal performance.

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 $\Omega(n^2)$, and $\Omega(nd)$, 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 $\Omega(\sqrt{n})$ which is almost equal to the upper bound up to a log factor. The classical lower bound is $\Omega(n)$. So, we obtain the quadratic speed-up for the problem.