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 γ i that the query point lies in a polygon P i of S . We show how to preprocess S such that the query time for a point~p∈P i depends on~γ i and, in addition, on the distance from p to the boundary of~P i ---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(logn,1+logarea(P i )γ i Δ 2 p )) , where Δ p is the shortest Euclidean distance of the query point~p to the boundary of P i . Our structure uses O(n) space and O(nlogn) 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∈P i , the quadrilateral or triangle containing~p has area Ω(Δ 2 p ) . For the special case where S is a subdivision of the unit square and γ i =area(P i ) , we present a simpler solution that achieves a query time of O(min(logn,log1Δ 2 p )) . The latter solution can be extended to convex subdivisions in three dimensions.