Efficient Expected-Case Algorithms for Planar Point Location

Planar point location is among the most fundamental search problems in computational geometry. Although this problem has been heavily studied from the perspective of worst-case query time, there has been surprisingly little theoretical work on expected-case query time. We are given an n-vertex planar polygonal subdivision S satisfying some weak assumptions (satisfied, for example, by all convex subdivisions). We are to preprocess this into a data structure so that queries can be answered efficiently. We assume that the two coordinates of each query point are generated independently by a probability distribution also satisfying some weak assumptions (satisfied, for example, by the uniform distribution). In the decision tree model of computation, it is well-known from information theory that a lower bound on the expected number of comparisons is entropy(S). We provide two data structures, one of size O(n) that can answer queries in 2 entropy(S) + O(1) expected number of comparisons, and another of size O(n) that can answer queries in (4 + O(1/ √ log n)) entropy(S)+O(1) expected number of comparisons. These structures can be built in O(n) and O(n log n) time respectively. Our results are based on a recent result due to Arya and Fu, which bounds the entropy of overlaid subdivisions.