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13
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 . . . . . . . . . . . . . . . . . . ...
Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
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Cited by 56 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
Proximity Problems on Moving Points
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair o ..."
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Cited by 50 (15 self)
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A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably e#cient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree. The method for maintaining the closest pair of points can be extended to the maintenance of closest pair of other distance functions which allows us to maintain the closest pair of a set of moving objects with similar sizes and of a set of points on a smooth manifold.
Dynamic Euclidean Minimum Spanning Trees and Extrema of Binary Functions
, 1995
"... We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n 1/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n # ) per update. Our algorithm uses a novel ..."
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Cited by 40 (4 self)
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We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n 1/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n # ) per update. Our algorithm uses a novel construction, the ordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. 1 Introduction A dynamic geometric data structure is one that maintains the solution to some problem, defined on a geometric input such as a point set, as the input undergoes update operations such as insertions or deletions of single points. Dynamic algorithms have been studied for many geometric optimization problems, including closest pairs [7, 23, 25, 26], diameter [7, 26], width [4], convex hulls [15, 22], linear ...
An Optimal Algorithm for Closest Pair Maintenance
 Discrete Comput. Geom
, 1995
"... Given a set S of n points in kdimensional space, and an L t metric, the dynamic closest pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric L t , we give a data structure of size O(n) ..."
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Cited by 35 (0 self)
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Given a set S of n points in kdimensional space, and an L t metric, the dynamic closest pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric L t , we give a data structure of size O(n) that maintains a closest pair of S in O(logn) time per insertion and deletion. The running time of algorithm is optimal up to constant factor because \Omega\Gammaaus n) is a lower bound, in algebraic decisiontree model of computation, on the time complexity of any algorithm that maintains the closest pair (for k = 1). The algorithm is based on the fairsplit tree. The constant factor in the update time is exponential in the dimension. We modify the fairsplit tree to reduce it. 1 Introduction The dynamic closest pair problem is one of the very wellstudied proximity problem in computational geometry [6, 1720, 22, 2426, 2831]. We are given a set S of n points in kdimensional space...
Efficient Construction of a Bounded Degree Spanner with Low Weight
 IN PROC. 2ND ANNU. EUROPEAN SYMPOS. ALGORITHMS (ESA
, 1994
"... Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner for S is a graph having the points of S as its vertices such that for any pair p; q of points there is a path between them of length at most t times the Euclidean distance between p and q. An efficient implementatio ..."
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Cited by 26 (3 self)
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Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner for S is a graph having the points of S as its vertices such that for any pair p; q of points there is a path between them of length at most t times the Euclidean distance between p and q. An efficient implementation of a greedy algorithm is given that constructs a tspanner having bounded degree such that the total length of all its edges is bounded by O(log n) times the length of a minimum spanning tree for S. The algorithm has running time O(n log d n). Applying recent results of Das, Narasimhan and Salowe to this tspanner gives an O(n log d n) time algorithm for constructing a tspanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree for S. Previously, no o(n 2 ) time algorithms were known for constructing a tspanner of bounded degree. In the final part of the paper, an application to the problem of distance enumeration is...
An Optimal Algorithm for the onLine Closest Pair Problem
 Algorithmica
, 1994
"... We give an algorithm that computes the closest pair in a set of n points in k dimensional space online, in O(n log n) time. The algorithm only uses algebraic functions and, therefore, is optimal. The algorithm maintains a hierarchical subdivision of kspace into hyperrectangles, which is stored i ..."
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Cited by 19 (3 self)
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We give an algorithm that computes the closest pair in a set of n points in k dimensional space online, in O(n log n) time. The algorithm only uses algebraic functions and, therefore, is optimal. The algorithm maintains a hierarchical subdivision of kspace into hyperrectangles, which is stored in a binary tree. Centroids are used to maintain a balanced decomposition of this tree. 1 Introduction The closest pair problem is one of the classical problems in computational geometry. In this problem, we have to compute the closest pairor its distancein a set of n points in kdimensional space. Distances are measured in an arbitrary, but fixed, L t metric. Let p = (p 1 ; : : : ; p k ) and q = (q 1 ; : : : ; q k ) be two points in kdimensional space. Then the L t distance d t (p; q) between p and q is defined by d t (p; q) := / k X i=1 jp i \Gamma q i j t !1=t ; if 1 t ! 1, and for t = 1, it is defined by d1 (p; q) := max 1ik jp i \Gamma q i j: We observe, as many o...
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 ...
Kinetic and Dynamic Data Structures for Closest Pairs and All Nearest Neighbors
, 2008
"... We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic twodimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or del ..."
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Cited by 3 (2 self)
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We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic twodimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses O(n log n) space, and processes O(n 2 βs+2(n) log n) critical events, each in O(log 2 n) time. Here s is the maximum number of times where the distances between any two specific pairs of points can become equal, βs(q) = λs(q)/q, and λs(q) is the maximum length of DavenportSchinzel sequences of order s on q symbols. The dynamic version of the problem incurs a slight degradation in performance: If m ≥ n insertions and deletions are performed, the structure still uses O(n log n) space, and processes O(mnβs+2(n) log³ n) events, each in O(log³ n) time. Our kinetic data structure for all nearest neighbors uses O(n log² n) space, and processes O(n 2 β 2 s+2 (n) log3 n) critical events. The expected time to process all events is O(n 2 β 2 s+2 (n) log4 n), though processing a single event may take �(n) expected time in the worst case. If m ≥ n insertions and deletions are performed, then the expected number of events is O(mnβ 2 s+2 (n) log3 n) and processing them all takes O(mnβ 2 s+2 (n) log4 n). An insertion or deletion takes O(n) expected time.
9 1995 SpringerVerlag New York Inc. Dynamic HalfSpace Range Reporting and Its Applications 1
"... Abstract. We consider the halfspace rangereporting problem: Given a set S of n points in ~a, preprocess it into a data structure, so that, given a query halfspace 7, all k points of S n? can be reported efficiently. We extend previously known static solutions to dynamic ones, supporting insertion ..."
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Abstract. We consider the halfspace rangereporting problem: Given a set S of n points in ~a, preprocess it into a data structure, so that, given a query halfspace 7, all k points of S n? can be reported efficiently. We extend previously known static solutions to dynamic ones, supporting insertions and deletions of points of S. For a given parameter m, n < m < n Ld/2j and an arbitrarily small positive constant e, we achieve O(m 1 +~) space and preprocessing time, O((n/m La/zJ) tog n + k) query time, and O(rn 1 +~/n) amortized update time (d> 3). We present, among others, the following applications: an O(n 1 +~)time algorithm for computing convex layers in R 3, and an output sensitive algorithm for computing a level in an arrangements of planes in R 3, whose time complexity is O((b + n)'n~), where b is the size of the level. Key Words.