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19
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.
Deformable spanners and applications
 In Proc. of the 20th ACM Symposium on Computational Geometry (SoCG’04
, 2004
"... For a set S of points in R d,ansspanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)spanner with O(n/ε d) edges, where ε is a spe ..."
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Cited by 35 (5 self)
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For a set S of points in R d,ansspanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), wellseparated pair decomposition, and approximate kcenters. 1
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...
Algorithms for Proximity Problems in Higher Dimensions
 Comput. Geom. Theory Appl
, 1996
"... We present algorithms for five interdistance enumeration problems that take as input a set S of n points in IR d (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that ..."
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Cited by 23 (2 self)
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We present algorithms for five interdistance enumeration problems that take as input a set S of n points in IR d (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that takes as additional input a distance # and outputs all k pairs of points in S separated by a distance of # or less; an O(n log n + k log k) time and O(n+k) space algorithm that enumerates in nondecreasing order the k closest pairs of points in S; an O(n log n + k) time algorithm for the same problem without any order restrictions; an O(nk log n) time and O(n) space algorithm that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S; and an O(n log n + kn) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the ...
Fast Greedy Triangulation Algorithms
 Proc. 10th Ann. Symp. Computational Geometry
, 1994
"... this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n log n) and space O(n) for points uniformly distributed over any convex shape. A variant of this algorithm should also be fast for many other distributions. We first describe a surprisingly ..."
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Cited by 16 (2 self)
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this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n log n) and space O(n) for points uniformly distributed over any convex shape. A variant of this algorithm should also be fast for many other distributions. We first describe a surprisingly simple method for testing the compatibility of a candidate edge with edges in a partially constructed greedy triangulation. The new edge is tentatively added to the embedding of the partial GT and at most four constant time tests are done involving edges lying clockwise and counterclockwise from the candidate edge at each vertex. Even though there can be O(n) edges adjacent to one of the endpoints, we are able to show that if we can determine where in angular order the new edge falls among a subset of at most 10 of those edges then we can perform the compatibility test and if necessary update the triangulation. Our method therefore provides a \Theta(1) time edge test that requires only \Theta(1) time to update the structure, \Theta(n) time for initialization, and \Theta(n) space. This compares favorably with the previous method of Gilbert [10], which requires \Theta(log n) time for an edge test, \Theta(n log n) time for an update, \Theta(n
Triangulations Intersect Nicely
, 1996
"... We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulatio ..."
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Cited by 13 (3 self)
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We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimumweight triangulation which can be computed in polynomial time by matching and networkflow techniques. We exhibit an easytorecognize class of point sets for which the minimumweight triangulation coincides with the greedy triangulation.
Robust DistanceBased Clustering with Applications to Spatial Data Mining
, 1999
"... In this paper, we present a method for clustering georeferenced data suitable for applications in spatial data mining, based on the medoid method. The medoid method is related to kMeans, with the restriction that cluster representatives be chosen from among the data elements. Although the medoid m ..."
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Cited by 12 (2 self)
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In this paper, we present a method for clustering georeferenced data suitable for applications in spatial data mining, based on the medoid method. The medoid method is related to kMeans, with the restriction that cluster representatives be chosen from among the data elements. Although the medoid method in general produces clusters of high quality, especially in the presence of noise, it is often criticized for the\Omega\Gamma n 2 ) time that it requires. Our method incorporates both proximity and density information to achieve highquality clusters in subquadratic time; it does not require that the user specify the number of clusters in advance. The time bound is achieved by means of a fast approximation to the medoid objective function, using Delaunay triangulations to store proximity information.
A Simple Linear Time Greedy Triangulation Algorithm for Uniformly Distributed Points
 REPORT IIG408, INSTITUTES FOR INFORMATION PROCESSING, TECHNISCHE UNIVERSIT AT GRAZ
, 1995
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