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44
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 117 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
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 57 (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.
An ExpanderBased Approach to Geometric Optimization
 IN PROC. 9TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1993
"... We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach ..."
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Cited by 42 (15 self)
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We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach
A NearLinear Algorithm for the Planar 2Center Problem
 Discrete & Computational Geometry
, 1996
"... We present an O(n log n)time algorithm for computing the 2center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous O(n log n)time algorithm of [10]. ..."
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Cited by 42 (6 self)
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We present an O(n log n)time algorithm for computing the 2center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous O(n log n)time algorithm of [10].
Dynamic Planar Convex Hull Operations in NearLogarithmic Amortized Time
 JOURNAL OF THE ACM
, 1999
"... We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangentfinding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum siz ..."
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Cited by 41 (6 self)
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We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangentfinding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of P and " is any fixed positive constant. For some advanced queries such as bridgefinding, both our bounds increase to O(log 3=2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log 2 n) time per update. 1 Introduction Although the algorithmic study of convex hulls is as old as computational geometry itself, the basic problem of optimally maintaining the planar convex hull under insertions and deletions of points [30, 34] remains unsolved and has been regarded by some as one of the foremost open problems in the area [14, 26]. Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often us...
Geometry helps in bottleneck matching and related problems
 Algorithmica
, 2001
"... This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (onetoone)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of th ..."
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Cited by 36 (5 self)
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This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (onetoone)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match respectively. Bottleneck matchinga matching that minimizesmax( Match)is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblanceare proposed. The running time of all the algorithms involving planar objects is roughly O(n1.5). For instance, if the objects are points in the plane, the running time of the exactalgorithm is O(n1.5 log n). A semidynamic datastructure for answering containmentproblems for a set of congruent disks in the plane is developed. This data structure may be of independent interest.Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are pointsetsin the plane, an O(n5 log n) time algorithm for determining whether for some translatedcopy the resemblance gets below a given ae is presented, thus improving the previousresult of Alt, Mehlhorn, Wagener and Welzl by a factor of almost n. This result is usedto compute the smallest such ae in time O(n5 log2 n), and an efficient approximationscheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or thefair matching problem) is to find Match*U, a matching that minimizes max(Match)min ( Match). A minimum deviation matching Match*D is a matching that minimizes(1 /n)\Sigma (Match) min(Match). Algorithms for computing Match*U and Match*D inroughly O(n10/3) time are presented. These algorithms are more efficient than theprevious
A Randomized Approximation Scheme for Metric MAXCUT
"... Metric MAXCUT is the problem of dividing a set of points in metric space into two parts so as to maximize the sum of the distances between points belonging to distinct parts. We show that metric MAXCUT has a polynomial time randomized approximation scheme. ..."
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Cited by 31 (5 self)
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Metric MAXCUT is the problem of dividing a set of points in metric space into two parts so as to maximize the sum of the distances between points belonging to distinct parts. We show that metric MAXCUT has a polynomial time randomized approximation scheme.
Geometric Clusterings
, 1990
"... A kclustering of a given set of points in the plane is a partition of the points into k subsets ("clusters"). For any fixed k, we can find a kclustering which minimizes any monotone function of the diameters or the radii of the clusters in polynomial time. The algorithm is based on the f ..."
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Cited by 29 (1 self)
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A kclustering of a given set of points in the plane is a partition of the points into k subsets ("clusters"). For any fixed k, we can find a kclustering which minimizes any monotone function of the diameters or the radii of the clusters in polynomial time. The algorithm is based on the fact that any two clusters in an optimal solution can be separated by a line.
More Planar TwoCenter Algorithms
 Comput. Geom. Theory Appl
, 1997
"... This paper considers the planar Euclidean twocenter problem: given a planar npoint set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(n log 2 n log 2 log n), improving the previous O(n log 9 n) bound ..."
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Cited by 27 (1 self)
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This paper considers the planar Euclidean twocenter problem: given a planar npoint set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(n log 2 n log 2 log n), improving the previous O(n log 9 n) bound of Sharir and almost matching the randomized O(n log 2 n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O(n log n) time with high probability. Keywords: twocenter, randomization, parametric search 1 Introduction Consider the following "facility location" problem: given a set S of n "demand" points in IR d and a number p, find a set T of p "supply"points in IR d minimizing max s2S min t2T d(s; t), where d(s; t) denotes the Euclidean distance between s and t. Geometrically, the problem is equivalent to finding p congruent disks of the smallest radius covering S and is referred to as the (Euclidean) pcenter problem. Th...
On Some Geometric Selection and Optimization Problems Via Sorted Matrices
, 1998
"... In this paper we apply the selection and optimization technique of Frederickson and Johnson to a number of geometric selection and optimization problems, some of which have previously been solved by parametric search, and provide efficient and simple algorithms. Our technique improves the solutions ..."
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Cited by 20 (2 self)
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In this paper we apply the selection and optimization technique of Frederickson and Johnson to a number of geometric selection and optimization problems, some of which have previously been solved by parametric search, and provide efficient and simple algorithms. Our technique improves the solutions obtained by parametric search by a log n factor. For example, we apply the technique to the twoline center problem, where we want to find two strips that cover a given set S of n points in the plane, so as to minimize the width of the largest of the two strips. Key words: computational geometry, algorithm, selection, optimization, twoline center. A version of this paper appeared in Fourth Workshop on Algorthms and Data Structures, S.G. Akl, F. Dehne, J. Sack and N. Santoro, editors, Lecture Notes in Computer Science 955, SpringerVerlag, pp. 2635. Work by K. Kedem has been supported by a grant from the U.S.Israeli Binational Science Foundation, and by a grant from the Israel Science ...