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arXiv · 2405.01160

Quantum algorithms for Hopcroft's problem

Vladimirs Andrejevs·Aleksandrs Belovs·Jevgēnijs Vihrovs
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Abstract

In this work we study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. Given $n$ points and $n$ lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n^{4/3})$ time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity $\widetilde O(n^{5/6})$. The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.

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Vladimirs Andrejevs, Aleksandrs Belovs, Jevgēnijs Vihrovs. 2024-05-02. Quantum algorithms for Hopcroft's problem. https://arxiv.org/abs/2405.01160

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