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23
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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Cited by 23 (4 self)
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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).
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.
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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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 ...
Instanceoptimal geometric algorithms
"... ... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of ..."
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Cited by 15 (2 self)
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... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of A on input 〈s1,..., sn〉 is at most a constant factor times the maximum running time of A ′ on 〈s1,..., sn〉, where the maximum is taken over all permutations 〈s1,..., sn 〉 of S. In fact, we can establish a stronger property: for every S and A ′ , the maximum running time of A is at most a constant factor times the average running time of A ′ over all permutations of S. We call algorithms satisfying these properties instanceoptimal in the orderoblivious and randomorder setting. Such instanceoptimal algorithms simultaneously subsume outputsensitive algorithms and distributiondependent averagecase algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instancespecific lower bound, we deviate from traditional Ben–Orstyle proofs and adopt an interesting adversary argument. For 2d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound. For 3d convex hulls, we propose a new algorithm. To demonstrate the potential of the concept, we further obtain instanceoptimal results for a few other standard problems in computational geometry, such as maxima in 2d and 3d, orthogonal line segment intersection in 2d, finding bichromatic L∞close pairs in 2d, offline orthogonal range searching in 2d, offline dominance reporting in 2d and 3d, offline halfspace range reporting 1.
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.
Achieving Spatial Adaptivity while Finding Approximate Nearest Neighbors
"... 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 Ltnorm 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 th ..."
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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 Ltnorm 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 suitablysized 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
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
Point location in disconnected planar subdivisions. arXiv:1001.2763
, 2010
"... Abstract. Let G be a (possibly disconnected) planar subdivision and let D be a probability measure over R2. The current paper shows how to preprocess (G,D) into an O(n) size data structure that can answer planar point location queries over G. The expected query time of this data structure, for a qu ..."
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Abstract. Let G be a (possibly disconnected) planar subdivision and let D be a probability measure over R2. The current paper shows how to preprocess (G,D) into an O(n) size data structure that can answer planar point location queries over G. The expected query time of this data structure, for a query point drawn according to D, is O(H + 1), where H is a lower bound on the expected query time of any linear decision tree for point location in G. This extends the results of Collette et al. (2008, 2009) from connected planar subdivisions to disconnected planar subdivisions. A version of this structure, when combined with existing results on succinct point location, provides a succinct distributionsensitive point location structure. 1