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Closest Pair Queries in Spatial Databases
 In Proceedings of the ACMSIGMOD Conference on Management of Data
, 2000
"... This paper addresses the problem of finding the K closest pairs between two spatial data sets, where each set is stored in a structure belonging in the Rtree family. Five different algorithms (four recursive and one iterative) are presented for solving this problem. The case of 1 closest pair is tr ..."
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Cited by 65 (9 self)
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This paper addresses the problem of finding the K closest pairs between two spatial data sets, where each set is stored in a structure belonging in the Rtree family. Five different algorithms (four recursive and one iterative) are presented for solving this problem. The case of 1 closest pair is treated as a special case. An extensive study, based on experiments performed with synthetic as well as with real point data sets, is presented. A wide range of values for the basic parameters affecting the performance of the algorithms, especially the effect of overlap between the two data sets, is explored. Moreover, an algorithmic as well as an experimental comparison with existing incremental algorithms addressing the same problem is presented. In most settings, the new algorithms proposed clearly outperform the existing ones. 1
ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 65 (14 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
Randomized Data Structures for the Dynamic ClosestPair Problem
, 1993
"... We describe a new randomized data structure, the sparse partition, for solving the dynamic closestpair problem. Using this data structure the closest pair of a set of n points in Ddimensional space, for any fixed D, can be found in constant time. If a frame containing all the points is known in adv ..."
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Cited by 10 (2 self)
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We describe a new randomized data structure, the sparse partition, for solving the dynamic closestpair problem. Using this data structure the closest pair of a set of n points in Ddimensional space, for any fixed D, can be found in constant time. If a frame containing all the points is known in advance, and if the floor function is available at unitcost, then the data structure supports insertions into and deletions from the set in expected O(log n) time and requires expected O(n) space. Here, it is assumed that the updates are chosen by an adversary who does not know the random choices made by the data structure. This method is more efficient than any deterministic algorithm for solving the problem in dimension D ? 1. The data structure can be modified to run in O(log 2 n) expected time per update in the algebraic computation tree model of computation. Even this version is more efficient than the currently best known deterministic algorithm for D ? 2. 1 Introduction We ...
On Enumerating and Selecting Distances
 Int. J. Comput. Geom. Appl
, 1999
"... Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selec ..."
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Cited by 9 (2 self)
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Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n + n 2=3 k 1=3 log 5=3 n). We also describe outputsensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search. Keywords: distance enumeration, distance selection, closest pairs, range counting, randomized algorithms. 1 Introduction Finding the closest pair of an npoint set has a long history in computational geometry (see [34] for a nice survey). In the plane, the problem can be solved in O(n log n) time using the Delaunay triangulation. In an arbitrary fixed dimension d, the first O(n log n) algorithm, based on di...
I/Oefficient wellseparated pair decomposition and its applications
 In Proc. Annual European Symposium on Algorithms
, 2000
"... Abstract. We present an external memory algorithm to compute a wellseparated pair decomposition (WSPD) of a given point set P in £ d in O ¤ sort ¤ N¥¦ ¥ I/Os using O ¤ N § B ¥ blocks of external memory, where N is the number of points in P, and sort ¤ N ¥ denotes the I/O complexity of sorting N ite ..."
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Cited by 8 (1 self)
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Abstract. We present an external memory algorithm to compute a wellseparated pair decomposition (WSPD) of a given point set P in £ d in O ¤ sort ¤ N¥¦ ¥ I/Os using O ¤ N § B ¥ blocks of external memory, where N is the number of points in P, and sort ¤ N ¥ denotes the I/O complexity of sorting N items. (Throughout this paper we assume that the dimension d is fixed). We also show how to dynamically maintain the WSPD in O ¤ log B N ¥ I/O’s per insert or delete operation using O ¤ N § B ¥ blocks of external memory. As applications of the WSPD, we show how to compute a linear size tspanner for P within the same I/O and space bounds and how to solve the Knearest neighbor and Kclosest pair problems in O ¤ sort ¤ KN¥¦¥ and O ¤ sort ¤ N ¨ K¥¦ ¥ I/Os using O ¤ KN § B ¥ and O¤¦ ¤ N ¨ K¥© § B ¥ blocks of external memory, respectively. Using the dynamic WSPD, we show how to dynamically maintain the closest pair of P in O ¤ log B N ¥ I/O’s per insert or delete operation using O ¤ N § B ¥ blocks of external memory. 1
An O(N 3/4 log 2 N) Quantum Algorithm for the 1dimensional ClosestPair Problem
"... Abstract. We present an algorithm for solving the closest pair problem using O(N 3/4 log 2 N) queries. This improves the simple approach using Dürr and Høyer’s minimum algorithm which would result in O(N) queries. The classical deterministic and probabilistic complexity is, respectively, of O(N log ..."
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Abstract. We present an algorithm for solving the closest pair problem using O(N 3/4 log 2 N) queries. This improves the simple approach using Dürr and Høyer’s minimum algorithm which would result in O(N) queries. The classical deterministic and probabilistic complexity is, respectively, of O(N log N) and O(N) queries. We also show a quantum lower bound of Ω(N 2/3) queries for the problem, and we discuss some open issues. 1.
Multimodal Partial Estimates Fusion
"... Fusing partial estimates is a critical and common problem in many computer vision tasks such as partbased detection and tracking. It generally becomes complicated and intractable when there are a large number of multimodal partial estimates, and thus it is desirable to find an effective and scalabl ..."
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Fusing partial estimates is a critical and common problem in many computer vision tasks such as partbased detection and tracking. It generally becomes complicated and intractable when there are a large number of multimodal partial estimates, and thus it is desirable to find an effective and scalable fusion method to integrate these partial estimates. This paper presents a novel and effective approach to fusing multimodal partial estimates in a principled way. In this new approach, fusion is related to a computational geometry problem of finding the minimumvolume orthotope, and an effective and scalable branch and bound search algorithm is designed to obtain the global optimal solution. Experiments on tracking articulated objects and occluded objects show the effectiveness of the proposed approach. 1.
and
, 1993
"... The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest p ..."
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The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest pair: Assume that a multiset of n points in the ddimensional Euclidean space is given, where d � 1 is a fixed integer. Each point is represented as a dtuple of integers in the range 0,..., U � 14 Ž or of arbitrary real numbers.. Find a closest pair, i.e., a pair of points whose distance is minimal over all such pairs.
Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing ✩
"... Motivated by the desire to cope with data imprecision [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in th ..."
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Motivated by the desire to cope with data imprecision [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(nα(n) log ∗ n) expected time. If we further assume that the points are “oblivious ” with respect to the data structure, the running time improves to O(nα(n)). The same result holds when L is a set of line segments (in general position). We present several extensions, including a tradeoff between space and query time and an outputsensitive algorithm. We also study the “dual problem ” where we show how to efficiently compute the ( ≤ k)level of n lines in the plane, each of which is incident to a distinct point (given in advance). We complement our results by Ω(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their xorder), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the “standard ” convex hull and sorting problems, in which the two problems require Θ(n log n) steps in the worst case (under the algebraic computation tree model).