eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
9:1
9:16
10.4230/LIPIcs.MFCS.2024.9
article
Quantum Algorithms for Hopcroft’s Problem
Andrejevs, Vladimirs
1
https://orcid.org/0009-0009-7265-9203
Belovs, Aleksandrs
1
Vihrovs, Jevgēnijs
1
https://orcid.org/0000-0002-3143-2610
Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.9/LIPIcs.MFCS.2024.9.pdf
Quantum algorithms
Quantum walks
Computational Geometry