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(logn,1+logγiΔp2area(Pi))), where
Δp is the shortest Euclidean distance of the query point~p to the
boundary of Pi. 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∈Pi, the quadrilateral or triangle
containing~p has area Ω(Δp2). 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(logn,logΔp21)). The latter solution can be extended to
convex subdivisions in three dimensions