Planar Visibility: Testing and Counting
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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 ^2 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, V_S(s), of ^2 that is weakly visible from a segment s can be represented as the union of a set, C_S(s), of O(n^2) triangles, even though the complexity of V_S(s) can be Ω(n^4). 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 C_S(s) that contain p.
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