Worst-case optimal algorithms for constructing visibility polygons with holes
SCG '86 Proceedings of the second annual symposium on Computational geometry
Art gallery theorems and algorithms
Art gallery theorems and algorithms
An output-sensitive algorithm for computing visibility
SIAM Journal on Computing
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
The Robot Localization Problem
SIAM Journal on Computing
Lectures on Discrete Geometry
Efficient visibility queries in simple polygons
Computational Geometry: Theory and Applications
Efficient computation of query point visibility in polygons with holes
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Visibility testing and counting
FAW-AAIM'11 Proceedings of the 5th joint international frontiers in algorithmics, and 7th international conference on Algorithmic aspects in information and management
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Space/query-time tradeoff for computing the visibility polygon
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
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In this paper we consider query versions of visibility testing and visibility counting. Let S be a set of n disjoint line segments in ℜ2 and let s be an element of S. Visibility testing is to preprocess S so that we can quickly determine if s is visible from a query point q. Visibility counting involves preprocessing S so that one can quickly estimate the number of segments in S visible from a query point q. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, VS(s), of ℜ2 that is weakly visible from a segment s can be represented as the union of a set, CS(s), of O(n2) triangles, even though the complexity of VS(s) can be Ω(n4). We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of S visible from a point p to the number of triangles in ∪s∈S CS(s) that contain p.