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Dynamic Distribution-Sensitive Point Location

Authors Siu-Wing Cheng , Man-Kit Lau

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Author Details

Siu-Wing Cheng
  • Department of Computer Science and Engineering, HKUST, Hong Kong, China
Man-Kit Lau
  • Department of Computer Science and Engineering, HKUST, Hong Kong, China

Funding

Supported by Research Grants Council, Hong Kong, China (project no. 16201116).

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Siu-Wing Cheng and Man-Kit Lau. Dynamic Distribution-Sensitive Point Location. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.30

Abstract

We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in ℝ², and we are given an oracle that can return in O(1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision S with n vertices such that each query is answered in O(OPT) expected time, where OPT is the minimum expected time of the best linear decision tree for point location in S. The space and construction time are O(n log² n). An update of S as a mixed sequence of k edge insertions and deletions takes O(k log⁵ n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(n log⁵ n) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • dynamic planar point location
  • convex subdivision
  • linear decision tree

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