Location of a Point in a Planar Subdivision and Its Applications

Given a subdivision of the plane induced by a planar graph with n vertices, in this paper we consider the problem of identifying which region of the subdivision contains a given test point. We present a search algorithm, called point-location algorithm, which operates on a suitably preprocessed data structure. The search runs in time at most 0((log n)2), while the preprocessing task runs in time at most 0(n log n) and requires 0(n) storage. The methods are quite general, since an arbitrary subdivision can be transformed in time at most 0(n log n) into one to which the preprocessing procedure is applicable. This solution of the point-location problem yields interesting and efficient solutions of other geometric problems, such as spatial convex inclusion and inclusion in an arbitrary polygon