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Quantum Algorithms for Hopcroft’s Problem

Authors Vladimirs Andrejevs , Aleksandrs Belovs, Jevgēnijs Vihrovs

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Author Details

Vladimirs Andrejevs
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Aleksandrs Belovs
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Jevgēnijs Vihrovs
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia

Funding

  • Belovs, Aleksandrs: Supported by QOPT (QuantERA ERA-NET Cofund).
  • Vihrovs, Jevgēnijs: Supported by Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1/22/I/CFLA/001 and QOPT (QuantERA ERA-NET Cofund).

Cite AsGet BibTex

Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.9

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Quantum algorithms
  • Quantum walks
  • Computational Geometry

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