Optimal Expected-Case Planar Point Location

Abstract

Point location is the problem of preprocessing a planar polygonal subdivision $S$ of size $n$ into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. We consider this problem from the perspective of expected query time. We are given the probabilities $p_z$ that the query point lies within each cell $z \in S$. The entropy $H$ of the resulting discrete probability distribution is the dominant term in the lower bound on the expected-case query time. We show that it is possible to achieve query time $H + O(\sqrt{H}+1)$ with space $O(n)$, which is optimal up to lower order terms in the query time. We extend this result to subdivisions with convex cells, assuming a uniform query distribution within each cell. In order to achieve space efficiency, we introduce the concept of entropy-preserving cuttings.