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A simple entropybased algorithm for planar point location
 In Proceedings of the Twelfth Annual ACMSIAM Symposium on Discrete Algorithms
, 2001
"... Abstract Given a planar polygonal subdivision S, point location involves preprocessing this subdivisioninto a data structure so that given any query point q, the cell of the subdivision containing qcan be determined efficiently. Suppose that for each cell z in the subdivision, the probability pz tha ..."
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Abstract Given a planar polygonal subdivision S, point location involves preprocessing this subdivisioninto a data structure so that given any query point q, the cell of the subdivision containing qcan be determined efficiently. Suppose that for each cell z in the subdivision, the probability pz that a query point lies within this cell is also given. The goal is to design the data structureto minimize the average search time. This problem has been considered before, but existing
Nearly Optimal ExpectedCase Planar Point Location
"... 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 ..."
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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 onedimensional keys. In the onedimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expectedcase 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.
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost 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 threedimensional 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. Higherdimensional 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 longstanding limitation, answering, for example, a question of Willard (SODA’92).
An experimental study of point location in general planar arrangements
 In ALENEX/ANALCO
, 2006
"... We study the performance in practice of various pointlocation algorithms implemented in Cgal, including a newly devised Landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line ” strategy and a trapezoidaldecomposition based search structure. The current ..."
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We study the performance in practice of various pointlocation algorithms implemented in Cgal, including a newly devised Landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line ” strategy and a trapezoidaldecomposition based search structure. The current implementation addresses general arrangements of arbitrary planar curves, including arrangements of nonlinear segments (e.g., conic arcs) and allows for degenerate input (for example, more than two curves intersecting in a single point, or overlapping curves). All calculations use exact number types and thus result in the correct point location. In our Landmarks algorithm (a.k.a. Jump & Walk), special points, “landmarks”, are chosen in a preprocessing stage, their place in the arrangement is found, and they are inserted into a datastructure that enables efficient nearestneighbor search. Given a query point, the nearest landmark is located and then the algorithm “walks ” from the landmark to the query point. We report on extensive experiments with arrangements composed of line segments or conic arcs. The results indicate that the Landmarks approach is the most efficient when the overall cost of a query is taken into account, combining both preprocessing and query time. The simplicity of the algorithm enables an almost straightforward implementation and rather easy maintenance. The generic programming implementation allows versatility both in the selected type of landmarks, and in the choice of the nearestneighbor search structure. The end result is a highly effective pointlocation algorithm for most practical purposes. ∗ Work reported in this paper has been supported in part by the IST Programme of the EU as a Sharedcorst RTD
DISTRIBUTIONSENSITIVE POINT LOCATION IN CONVEX SUBDIVISIONS
"... 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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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.
ENTROPY, TRIANGULATION, AND POINT LOCATION IN PLANAR SUBDIVISIONS
, 2009
"... A data structure is presented for point location in connected 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. More specifically, an algorithm is presented that preprocesses a connecte ..."
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A data structure is presented for point location in connected 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. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of pointline comparisons performed by this data structure, when the queries are distributed according to D, is ˜ H + O ( ˜ H2/3 + 1) where ˜ H = ˜ H(G, D) is a lower bound on the expected number of pointline comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has nearminimum entropy.
Computational Geometry through the Information Lens
, 2007
"... revisits classic problems in computational geometry from the modern algorithmic ..."
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revisits classic problems in computational geometry from the modern algorithmic
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"... 1 Introduction Distributionsensitive data structures have running times that can be expressed as a function of some distributional measure of the sequence of operations performed on the structure. Thus such structures can exploit sequences of operations that exhibit some desirable behavior. Because ..."
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1 Introduction Distributionsensitive data structures have running times that can be expressed as a function of some distributional measure of the sequence of operations performed on the structure. Thus such structures can exploit sequences of operations that exhibit some desirable behavior. Because realworld access sequences are rarely uniformly random, if our structures are optimized to perform better on the types of distributions likely to found in a given application, we can obtain running times that are faster than standard (distributioninsensitive) data structures. Distributionsensitive dictionaries. Distributionsensitive structures are wellstudied for the dictionary problem: maintain a collection of keyvalue pairs subject to queries for the value associated with a given key. For this problem, two major types of distributions have been studied: proximity and working sets.