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Computing the Visibility Polygon of an Island in a Polygonal Domain

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Abstract

Given a set \({\mathcal {P}}\) of h pairwise-disjoint polygonal obstacles with a total of n vertices in the plane, we study the problem of computing the (weak) visibility polygon from a polygonal obstacle \(P^*\) (an island) in \({\mathcal {P}}\). The problem was previously solved in \(O(n^4)\) time, which has been proved worst-case optimal. However, since h may be much smaller than n, it is desirable to have an algorithm whose running time is also a function of h. In this paper, we present such an algorithm of \(O(n^2h^2)\) time, and our algorithm improves the previous result when \(h=o(n)\). In addition, when all obstacles in \({\mathcal {P}}\) (including \(P^*\)) are convex, our algorithm runs in \(O(n+h^4)\) time.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their comments that help improve the presentation of the paper. D. Z. Chen’s research was supported in part by NSF under Grant CCF-1217906. H. Wang’s research was supported in part by NSF under Grant CCF-1317143.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen

  2. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Haitao Wang

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  1. Danny Z. Chen
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  2. Haitao Wang
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Correspondence to Haitao Wang.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP 2012).

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Chen, D.Z., Wang, H. Computing the Visibility Polygon of an Island in a Polygonal Domain. Algorithmica 77, 40–64 (2017). https://doi.org/10.1007/s00453-015-0058-y

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