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Optimal point location in a monotone subdivision. (English) Zbl 0602.68102
Point location, often known in graphics as ”hit detection”, is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let $${\mathcal S}$$ denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of m edges. In this paper we exhibit a substantial refinement of the technique of D. T. Lee and F. P. Preparata [SIAM J. Comput. 6, 594-606 (1977; Zbl 0357.68034)] for locating a point in $${\mathcal S}$$ based on separating chains. The new data structure, called a layered dag, can be built in O(m) time, uses O(m) storage, and makes possible point location in O(log m) time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight- line segments.

##### MSC:
 68U99 Computing methodologies and applications 68R10 Graph theory (including graph drawing) in computer science 68Q25 Analysis of algorithms and problem complexity
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