Distance-Sensitive Planar Point Location

Let S be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γi that the query point lies in a polygon Pi of S. We show how to preprocess S such that the query time for a point p ∈ Pi depends on γi and, in addition, on the distance from p to the boundary of Pi—the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time O ( min ( log n, 1 + log area(Pi) γi∆p )) , where ∆p is the shortest Euclidean distance of the query point p to the boundary of Pi. Our structure uses O(n) space and O(n log n) preprocessing time. It is based on a decomposition of the regions of S into convex quadrilaterals and triangles with the following property: for any point p ∈ Pi, the quadrilateral or triangle containing p has area Ω(∆p). For the special case where S is a subdivision of the unit square and γi = area(Pi), we present a simpler solution that achieves a query time of O ( min ( log n, log 1 ∆p )) . The latter solution can be extended to convex subdivisions in three