Optimal Point Location in a Monotone Subdivision

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 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 Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594–606] for locating a point in 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.