2026-08-13T18:53:09Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20565 2024-08-23T05:50:28Z ddc:004 open_access

Quantum Algorithms for Hopcroft’s Problem Andrejevs, Vladimirs Belovs, Aleksandrs Vihrovs, Jevgēnijs Quantum algorithms Quantum walks Computational Geometry 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 Õ(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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Vladimirs Andrejevs and Aleksandrs Belovs and Jevgēnijs Vihrovs 2024 Is Part Of LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2024.9 urn:nbn:de:0030-drops-205653 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9 eng https://creativecommons.org/licenses/by/4.0/legalcode