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+log area(P_i)/γ_iΔ_p²)), 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_p²). 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,log 1/Δ_p²)). The latter solution can be extended to convex subdivisions in three dimensions.