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16
Approximation Algorithms for Projective Clustering
 Proceedings of the ACM SIGMOD International Conference on Management of data, Philadelphia
, 2000
"... We consider the following two instances of the projective clustering problem: Given a set S of n points in R d and an integer k ? 0; cover S by k hyperstrips (resp. hypercylinders) so that the maximum width of a hyperstrip (resp., the maximum diameter of a hypercylinder) is minimized. Let w ..."
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Cited by 246 (21 self)
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We consider the following two instances of the projective clustering problem: Given a set S of n points in R d and an integer k ? 0; cover S by k hyperstrips (resp. hypercylinders) so that the maximum width of a hyperstrip (resp., the maximum diameter of a hypercylinder) is minimized. Let w be the smallest value so that S can be covered by k hyperstrips (resp. hypercylinders), each of width (resp. diameter) at most w : In the plane, the two problems are equivalent. It is NPHard to compute k planar strips of width even at most Cw ; for any constant C ? 0 [50]. This paper contains four main results related to projective clustering: (i) For d = 2, we present a randomized algorithm that computes O(k log k) strips of width at most 6w that cover S. Its expected running time is O(nk 2 log 4 n) if k 2 log k n; it also works for larger values of k, but then the expected running time is O(n 2=3 k 8=3 log 4 n). We also propose another algorithm that computes a c...
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 94 (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.
Geometric Applications of a Randomized Optimization Technique
 Discrete Comput. Geom
, 1999
"... We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, ..."
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Cited by 53 (6 self)
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We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal kpoint subsets, matching point sets under translation, computing rectilinear pcenters and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm randmin 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a wellknown fact [27, 44], the expected number of times that step 5 is execut...
Rectilinear and Polygonal pPiercing and pCenter Problems
 In Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axispa ..."
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Cited by 30 (1 self)
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We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axisparallel rectangles or convex coriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions) . We also investigate the planar rectilinear (and polygonal) pcenter problem, in which we are given a set S of n points in the plane, and wish to find p axisparallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a lineartime solution for the rectilinear 3center problem (by showing that this problem can be formulated as an LPtype problem and by exhibiting a relation to Helly numbers). We give O(n log n...
Enclosing K Points in the Smallest Axis Parallel Rectangle
, 1998
"... We consider the following clustering problem. Given a set S of n points in the plane, and given an integer k, n 2 ! k n, we want to find the smallest axis parallel rectangle (smallest perimeter or area) that encloses exactly k points of S. We present an algorithm which runs in time O(n + k(n \Ga ..."
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Cited by 9 (1 self)
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We consider the following clustering problem. Given a set S of n points in the plane, and given an integer k, n 2 ! k n, we want to find the smallest axis parallel rectangle (smallest perimeter or area) that encloses exactly k points of S. We present an algorithm which runs in time O(n + k(n \Gamma k) 2 ) improving previous algorithms which run in time O(k 2 n) and do not perform well for larger k values. We present an algorithm to enclose k of n given points in an axis parallel box in ddimensional space which runs in time O(dn+dk(n \Gamma k) 2(d\Gamma1) ) and occupies O(dn) space. We slightly improve algorithms for other problems whose runtimes depend on k.
Optimal Line Bipartitions of Point Sets
 Int. J. Comput. Geom. and Appls
, 1996
"... Let S be a set of n points in the plane. We study the following problem: Partition S by a line into two subsets S a and S b such that maxff(S a ); f(S b )g is minimal, where f is any monotone function defined over 2 S . We first present a solution to the case where the points in S are the vertices o ..."
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Cited by 5 (3 self)
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Let S be a set of n points in the plane. We study the following problem: Partition S by a line into two subsets S a and S b such that maxff(S a ); f(S b )g is minimal, where f is any monotone function defined over 2 S . We first present a solution to the case where the points in S are the vertices of some convex polygon and apply it to some common cases  f(S 0 ) is the perimeter, area, or width of the convex hull of S 0 ` S  to obtain linear solutions (or O(n log n) solutions if the convex hull of S is not given) to the corresponding problems. This solution is based on an efficient procedure for finding a minimal entry in matrices of some special type, which we believe is of independent interest. For the general case we present a linear space solution which is in some sense output sensitive. It yields solutions to the perimeter and area cases that are never slower and often faster than the best previous solutions.
Covering a Set of Points By Two AxisParallel Boxes
, 1999
"... In this paper we consider the following covering problem. Given a set S of n points in ddimensional space, d 2, find two axisparallel boxes that together cover the set S such that the measure of the largest box is minimized, where the measure is a monotone function of the box. We present a simple ..."
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Cited by 5 (0 self)
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In this paper we consider the following covering problem. Given a set S of n points in ddimensional space, d 2, find two axisparallel boxes that together cover the set S such that the measure of the largest box is minimized, where the measure is a monotone function of the box. We present a simple algorithm for finding boxes in O(n log n + n d\Gamma1 ) time and O(n) space. Key words: Algorithms, Computational geometry, optimization, axisparallel 1 Introduction We consider the following minmax two box problem. Given a set S of n points in ddimensional space, d 2, find two axisparallel boxes b 1 and b 2 that together cover the set S and minimize the maximum of measures (b 1 ) and (b 2 ), where is a monotone function of the box, i.e. b 1 ` b 2 implies (b 1 ) (b 2 ). Examples of the box measure are the volume of the box, the perimeter of the box (in higher dimensions it can be defined as the sum of 1dimensional edges or as the area of the boundary of the box), the length of...
Rectilinear 2Center Problems
 IN PROCEEDINGS OF THE 11TH CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY (CCCGâ€™99
, 1999
"... We present efficient algorithms for two problems of facility location. In both problems we want to optimize the location of two facilities with respect to n given sites. The first problem, the continuous version, has no restrictions for facility locations but in the second one, the discrete version, ..."
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Cited by 5 (0 self)
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We present efficient algorithms for two problems of facility location. In both problems we want to optimize the location of two facilities with respect to n given sites. The first problem, the continuous version, has no restrictions for facility locations but in the second one, the discrete version, facilities are chosen from a specified set of possible locations. We consider the rectilinear metric L1 and arbitrary dimension d and determine the locations that minimize, over all sites, the maximum distance to the closest facility. The algorithms for the continuous and discrete versions take O(n) and O(n log d\Gamma2 n log log n + n log n) running time respectively.
Discrete Rectilinear 2Center Problems
 Comput. Geom. Theory Appl
, 2000
"... Given a set P of n points in the plane, we seek two squares such that their center points belong to P, their union contains P, and the area of the larger square is minimal. We present efficient algorithms for three variants of this problem: In the first the squares are axis parallel, in the second t ..."
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Cited by 4 (0 self)
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Given a set P of n points in the plane, we seek two squares such that their center points belong to P, their union contains P, and the area of the larger square is minimal. We present efficient algorithms for three variants of this problem: In the first the squares are axis parallel, in the second they are free to rotate but must remain parallel to each other, and in the third they are free to rotate independently.
Algorithmic techniques for geometric optimization
 In Computer Science Today: Recent Trends and Developments, Lecture Notes in Computer Science
, 1995
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