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Visibility Queries in Simple Polygons

Authors Sujoy Bhore , Chih-Hung Liu , Anurag Murty Naredla , Yakov Nekrich , Eunjin Oh , André van Renssen , Frank Staals , Haitao Wang , Jie Xue

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LIPIcs.ICALP.2026.33.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • simple polygons
  • visibility polygons
  • visibility queries
  • polygon decompositions

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Abstract

Given a simple polygon P with n vertices, we consider the problem of constructing a data structure for visibility queries: for any query point q ∈ P, compute the visibility polygon of q in P. To obtain O(log n + k) query time, where k is the size of the visibility polygon of q, the previous best result requires O(n³) space. In this paper, we propose a new data structure that uses O(n^{2+ε}) space, for any ε > 0, while achieving the same query time. If only O(n²) space is available, the best known result provides O(log² n + k) query time. We improve this to O(log n log log n + k) time. When restricted to o(n²) space, the only previously known approach, aside from the O(n)-time algorithm that computes the visibility polygon without preprocessing, is an O(n)-space data structure that supports O(k log n)-time queries. We construct a data structure using O(n log n) space that answers visibility queries in O(n^{1/2+ε} + k) time. In addition, for the special case in which q lies on the boundary of P, we build a data structure of O(n log n) space supporting O(log² n + k) query time; alternatively, we achieve O(log n + k) query time using O(n^{1+ε}) space. To achieve our results, we propose a new method for decomposing simple polygons, which may be of independent interest.

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Sujoy Bhore, Chih-Hung Liu, Anurag Murty Naredla, Yakov Nekrich, Eunjin Oh, André van Renssen, Frank Staals, Haitao Wang, and Jie Xue. Visibility Queries in Simple Polygons. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 33:1-33:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.33

Author Details

Sujoy Bhore
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India
Chih-Hung Liu
  • Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
Anurag Murty Naredla
  • Department of Computer Science, University of Manitoba, Winnipeg, Canada
Yakov Nekrich
  • Michigan Technological University, Houghton, MI, USA
Eunjin Oh
  • POSTECH, Pohang, Republic of Korea
André van Renssen
  • The University of Sydney, Australia
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Jie Xue
  • New York University Shanghai, China

Funding

  • Bhore, Sujoy: Work supported in part by ANRF ARG-MATRICS, Grant 002465.
  • Liu, Chih-Hung: Supported by Ministry of Education, Taiwan under Yushan Fellow Program with the grant number MOE-111-YSFEE-0003-006-P1.
  • Nekrich, Yakov: Supported by the National Science Foundation under NSF grant 2203278.
  • Oh, Eunjin: Supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.RS-2024-00440239, Sublinear Scalable Algorithms for Large-Scale Data Analysis) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2024-00358505).
  • van Renssen, André: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).

Acknowledgements

This work originated at Shonan Meeting No. 234, "Path Planning and Routing in Geometric Environments." We thank the participants for their valuable discussions, and we are also grateful to the organizers and the National Institute of Informatics for hosting the event.

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