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
All indices are computed modulo the size of the corresponding vertex set: m + 1 in this case.
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This article belongs to the Topical Collection: Special Issue on International Workshop on Combinatorial Algorithms (IWOCA 2019)
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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