We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for one-dimensional keys. In the one-dimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expected-case search time, and further there exist search trees achieving expected search times of at most H + 2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H + o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.

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 2 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. 1 Introduction Planar point location is an important problem in computational geometry. We are given a polygonal subdivision S consisting of n edges, and the goal is ...

We present the first spatially adaptive data structure that answers approximate nearest neighbor (ANN) queries to points that reside in a geometric space of any constant dimension d. The Lt-norm approximation ratio is O(d 1+1/t), and the running time for a query q is O(d 2 lg δ(p, q)), where p is the result of the preceding query and δ(p, q) is the number of input points in a suitably-sized box containing p and q. Our data structure has O(dn) size and requires O(d 2 n lg n) preprocessing time, where n is the number of points in the data structure. The size of the bounding box for δ depends on d, and our results rely on the Random Access Machine (RAM) model with word size Θ(lg n). 1

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A data structure is presented for point location in convex planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal.

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revisits classic problems in computational geometry from the modern algorithmic

Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linear-space data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unit-cost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA’92). Key words. Computational geometry, word-RAM algorithms, data structures, sorting, searching, convex hulls, Voronoi diagrams, segment intersection AMS subject classifications. 68Q25, 68P05, 68U05 Abbreviated title. Point location in sublogarithmic time

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