Entropy-Preserving Cuttings and Space-Efficient Planar Point Location

Point location is the problem of preprocessing a planar polygonal subdivision S into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. Given the probabilities pz that the query point lies within each cell z S, a natural question is how to design such a structure so as to minimize the expected-case query time. The entropy H of the probability distribution is the dominant term in the lower bound on the expected-case search time. Clearly the number of edges n of the subdivision is a lower bound on the space required. There is no known approach that simultaneously achieves the goals of H + o(H) query time and O(n) space. In this paper we introduce entropy-preserving cuttings and show how to use them to achieve query time H+o(H), using only O(n log # n) space