Visibility Queries in Simple Polygons

Authors

• Sujoy Bhore, Indian Institute of Technology Bombay
• Chih Hung Liu, National Taiwan University
• Anurag Murty Naredla, University of Manitoba
• Yakov Nekrich, Michigan Technological UniversityFollow
• Eunjin Oh, Pohang University of Science and Technology
• André van Renssen, The University of Sydney
• Frank Staals, Universiteit Utrecht
• Haitao Wang, John and Marcia Price College of Engineering
• Jie Xue, NYU Shanghai

Document Type

Conference Proceeding

Publication Date

7-1-2026

Department

Department of Computer Science

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 nlog 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(klog n)-time queries. We construct a data structure using O(nlog 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(nlog 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.

Publisher's Statement

© Sujoy Bhore, Chih-Hung Liu, Anurag Murty Naredla, Yakov Nekrich, Eunjin Oh, André van Renssen, Frank Staals, Haitao Wang, and Jie Xue. Publisher’s version of record: 10.4230/LIPIcs.ICALP.2026.33

Publication Title

Leibniz International Proceedings in Informatics Lipics

ISBN

9783959774284

Recommended Citation

Bhore, S., Liu, C., Naredla, A., Nekrich, Y., Oh, E., van Renssen, A., Staals, F., Wang, H., & Xue, J. (2026). Visibility Queries in Simple Polygons. Leibniz International Proceedings in Informatics Lipics, 374. http://doi.org/10.4230/LIPIcs.ICALP.2026.33
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p2/2826

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Version

Publisher's PDF