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Computing the k-Visibility Region of a Point in a Polygon

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Abstract

Two points p and q in a simple polygon P are k-visible when the line segment pq crosses the boundary of P at most k times. Given a query point q, a positive integer k, and a polygon P, we design an algorithm that computes the region of P that is k-visible from q in O(nk) time, where n denotes the number of vertices of P. This region is called the k-visibility region of q. This is the first algorithm parameterized in terms of k, resulting in an asymptotically faster worst-case running time compared to previous algorithms when k is \(o(\log {n})\), and bridging the gap between the O(n)-time algorithm for computing the 0-visibility region of q in P and the \(O(n\log n)\)-time algorithm for computing the k-visibility region of q in P for any k.

We also design a data structure of size O(n5) that supports visibility queries, returning the k-visible region of P for any arbitrary query point q in \(O(\log {n}+m)\) time, where m denotes the number of vertices on the boundary of the output visibility region.

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Notes

  1. All indices are computed modulo the size of the corresponding vertex set: m + 1 in this case.

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Authors and Affiliations

  1. University of Windsor and CAMufacturing Solutions, Windsor, Canada

    Yeganeh Bahoo

  2. Carleton University, Ottawa, Canada

    Prosenjit Bose

  3. University of Manitoba, Winnipeg, Canada

    Stephane Durocher

  4. Inria, CNRS, Grenoble Institute of Engineering, LJK, Université Grenoble Alpes, Grenoble, France

    Stephane Durocher

  5. Simon Fraser University, Vancouver, Canada

    Thomas C. Shermer

Authors
  1. Yeganeh Bahoo
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  2. Prosenjit Bose
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  3. Stephane Durocher
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  4. Thomas C. Shermer
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Correspondence to Yeganeh Bahoo.

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This article belongs to the Topical Collection: Special Issue on International Workshop on Combinatorial Algorithms (IWOCA 2019)

Guest Editors: Charles Colbourn, Roberto Grossi, Nadia Pisanti

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Bahoo, Y., Bose, P., Durocher, S. et al. Computing the k-Visibility Region of a Point in a Polygon. Theory Comput Syst 64, 1292–1306 (2020). https://doi.org/10.1007/s00224-020-09999-0

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  • DOI: https://doi.org/10.1007/s00224-020-09999-0

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