Optimal Expected-Case Planar Point Location

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 pz that the query point lies within each cell z ∈ 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( √ 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. ∗Preliminary results appeared in the papers “Nearly Optimal Expected-Case Planar Point Location” in Proc. 41st Sympos. Foundations of Computer Science, 2000, 208–218 and “Entropy Preserving Cuttings and Space Efficient Planar Point Location” in Proc. 12th ACM-SIAM Sympos. Discrete Algorithms, 2001, 256–261. The work of Arya, Malamatos, and Wong was supported in part by the Research Grants Council, Hong Kong, China (HKUST6229/00E and HKUST6080/01E) and the work of Mount was supported by the National Science Foundation (CCR–0098151 and CCF–0635099). †Department of Computer Science, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: {arya,odan}@cs.ust.hk. ‡Max Planck Institut für Informatik, Saarbrücken, Germany. Part of this research was conducted while the author was at the Hong Kong University of Science and Technology. Email: [email protected]. §Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland. Email: [email protected].