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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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Cited by 20 (5 self)
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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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Cited by 17 (5 self)
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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.
EntropyPreserving Cuttings and SpaceEfficient Planar Point Location
 In Proceedings of the Twelfth Annual ACMSIAM Symposium on Discrete Algorithms
, 2001
"... 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 ho ..."
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Cited by 14 (4 self)
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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 expectedcase query time. The entropy H of the probability distribution is the dominant term in the lower bound on the expectedcase 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 entropypreserving 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 ...
Proximate point searching
 In Proceedings of the 14th Canadian Conference on Computational Geometry (CCCG
, 2002
"... In the 2D point searching problem, the goal is to preprocess n points P = {p1,..., pn} in the plane so that, for an online sequence of query points q1,..., qm, it can quickly determined which (if any) of the elements of P are equal to each query point qi. This problem can be solved in O(log n) time ..."
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Cited by 11 (5 self)
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In the 2D point searching problem, the goal is to preprocess n points P = {p1,..., pn} in the plane so that, for an online sequence of query points q1,..., qm, it can quickly determined which (if any) of the elements of P are equal to each query point qi. This problem can be solved in O(log n) time by mapping the problem to one dimension. We present a data structure that is optimized for answering queries quickly when they are geometrically close to the previous successful query. Specifically, our data structure executes queries in time O(log d(qi−1, qi)), where d is some distance function between two points, and uses O(n log n) space. Our structure works with a variety of distance functions. In contrast, it is proved that, for some of the most intuitive distance functions d, it is impossible to obtain an O(log d(qi−1, qi)) runtime, or any bound that is o(log n).
Optimal Planar Point Location
 IN PROCEEDINGS OF THE TWELFTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2001
"... Given a fixed distribution of point location queries among the regions of a triangulation of the plane, a data structure is presented that achieves, within constant multiplicative factors, the entropy bound on the expected point location query time. ..."
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Cited by 10 (2 self)
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Given a fixed distribution of point location queries among the regions of a triangulation of the plane, a data structure is presented that achieves, within constant multiplicative factors, the entropy bound on the expected point location query time.
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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Cited by 4 (3 self)
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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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Cited by 1 (1 self)
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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.
Nearly Optimal ExpectedCase Planar Point Location
, 2000
"... 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 ..."
Abstract
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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.
A Static Optimality Transformation with Applications to Planar Point Location
, 2012
"... Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is finetuned for the distribution. All these methods suffer from the requirement that the query distr ..."
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Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is finetuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2d analogue of the jump from Knuth’s optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information. 1