 
Summary: A Simple EntropyBased Algorithm for Planar Point Location
Sunil Arya
Theocharis Malamatos
David M. Mount§
February 2, 2007
Abstract
Given a planar polygonal subdivision S, point location involves preprocessing this subdivision
into a data structure so that given any query point q, the cell of the subdivision containing q
can be determined efficiently. Suppose that for each cell z in the subdivision, the probability
pz that a query point lies within this cell is also given. The goal is to design the data structure
to minimize the average search time. This problem has been considered before, but existing
data structures are all quite complicated. It has long been known that the entropy H of the
probability distribution is the dominant term in the lower bound on the averagecase search
time. In this paper, we show that a very simple modification of a wellknown randomized
incremental algorithm can be applied to produce a data structure of expected linear size that
can answer pointlocation queries in O(H) average time. We also present empirical evidence for
the practical efficiency of this approach.
1 Introduction
The planar pointlocation problem is one of the most fundamental query problems in computational
geometry. The problem is to preprocess a planar polygonal subdivision S consisting of n edges into
