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Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique

Authors Timothy M. Chan , Zhengcheng Huang

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric optimization problems
  • parametric search
  • shortest path
  • disk graphs
  • Fréchet distance
  • visibility
  • distance selection
  • randomized algorithms

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Abstract

In a series of papers, Avraham, Filtser, Kaplan, Katz, and Sharir (SoCG'14), Kaplan, Katz, Saban, and Sharir (ESA'23), and Katz, Saban, and Sharir (ESA'24) studied a class of geometric optimization problems - including reverse shortest path in unweighted and weighted unit-disk graphs, discrete Fréchet distance with one-sided shortcuts, and reverse shortest path in visibility graphs on 1.5-dimensional terrains - for which standard parametric search does not work well due to a lack of efficient parallel algorithms for the corresponding decision problems. The best currently known algorithms for all the above problems run in O^*(n^{6/5}) = O^*(n^{1.2}) time (ignoring subpolynomial factors), and they were obtained using a technique called shrink-and-bifurcate. We improve the running time to Õ(n^{8/7}) ≈ O(n^{1.143}) for these problems. Furthermore, specifically for reverse shortest path in unweighted unit-disk graphs, we improve the running time further to Õ(n^{9/8}) = Õ(n^{1.125}).

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Timothy M. Chan and Zhengcheng Huang. Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.32

Author Details

Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign, IL, USA
Zhengcheng Huang
  • Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

  • Chan, Timothy M.: Work supported by NSF Grant CCF-2224271.

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