H. Hershberger and S. Suri. Finding tailored partitions. Journal of Algorithms, 12:431--463, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For example, Hershberger [147] described an O(n = log log n) time algorithm for partitioning a given set S of n points into two subsets so that the sum of their diameters is minimized. If we want to minimize the maximum of the two diameters, the running time can be improved to O(n log n) [149]. Glozman et al. 131] have studied problems of covering S by several different kinds of shapes. Maass [195] showed that the problem of covering S with the minimum number of unit width annuli is NP Hard even for d = 1 (a unit width annulus in 1 dimension is a union of two unit length intervals) ....

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms, 12 (1991), 431--463.

....problems. In the discrete problems the centers of the shapes are constrained to lie on points of P , whereas in the non discrete problems the centers are not constrained. Hence, the discrete problems are somewhat more di#cult than the non discrete ones. For our knowledge only Hershberger and Suri [11] and later Agarwal et al. 3] who improved the results of [11] worked on discrete covering problems (namely, on the discrete two center problem) Thus our survey below describes mainly non discrete covering problems, with the exception of [3] The two center problem (two covering discs) was ....

....constrained to lie on points of P , whereas in the non discrete problems the centers are not constrained. Hence, the discrete problems are somewhat more di#cult than the non discrete ones. For our knowledge only Hershberger and Suri [11] and later Agarwal et al. 3] who improved the results of [11], worked on discrete covering problems (namely, on the discrete two center problem) Thus our survey below describes mainly non discrete covering problems, with the exception of [3] The two center problem (two covering discs) was solved in time O(n log by Sharir [21] and recently in O(n ....

J. Hershberger and S. Suri, "Finding tailored partitions", Journal of Algorithms, 12 (1991), 431--463.

....maximum of the diameters of the sets S 1 and S 2 . Asano, Bhattacharya, Keil and Yao ( 1] improved the bound on the time complexity of this problem, obtaining an optimal O(n log n) algorithm. Monma and Suri ( 13] gave an O(n for minimizing the sum of diameters. Recently, Hershberger and Suri ([8]) have considered the problem in which the measure of size (S i ) is (a) the diameter, b) the area, perimeter, or diagonal of the smallest enclosing axes parallel rectangle, or (c) the side length of the smallest enclosing axes parallel square. They provide O(n log n) time algorithms to find ....

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431--463, 1991.

....r is determined either by a pair of points of P , in which case r is half the distance between these points, or by a triple of points of P , in which case r is the radius of the circumcircle of the triangle spanned by these points. We use the recent algorithm of Hershberger [23] see also [6, 24]) whose running time is O(n ) for the corresponding decision problem: Given a value r 0, determine whether r r, r r, or r = r. We begin our algorithm with a preliminary stage in which we perform a binary search for r among the half distances determined by pairs of ....

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms 12 (1991), 431-- 463.

....Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA 1 details) We refer to this problem as the 2DC (2 disk cover) problem. The best previous solution of the 2DC problem runs in O(n ) time [6] see also [7]) Our strategy is to assume that such a pair of disks exist, call them D 1 , D 2 , and to conduct a search for their centers. Let c i denote the center of D i , and let C i denote the circle bounding D i , for i = 1; 2. We may assume, with no loss of generality, that jc 1 c 2 j is as small as ....

....we wish to determine whether the intersection K(P ) p2P B r (p) is nonempty, where B r (p) is the closed disk of radius r centered at p. This condition is equivalent to the condition that P can be covered by a disk of radius r. Such a procedure is also used in the preceding algorithms of [6, 7]. We give here a slightly inferior implementation of this procedure. This is done because it is easier to describe, and, more importantly, it is easier to parallelize, which is required by the parametric searching technique. To keep track of K(P ) as P is being updated, we maintain separately the ....

[Article contains additional citation context not shown here]

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms 12 (1991), 431-- 463.

....point p, find some pointinP farther than D from p, or report that no such point exists# (2) delete a given point from P . As noted by Aggarwal et al. 2] these operations can be performed in O(log n) amortized time and linear space using the circular hull data structure of Hershberger and Suri [25]. We start the labeling process by marking each unmatched pointaseven (it has a zero length path to an unmatched point) We build the data structure above, letting P consist of all unmarked points (initially, that is simply the matched points) We then process each marked point in turn, ....

J. Hershberger and S. Suri. Finding tailored partitions. In 5th ACM Symp. Comput. Geom., pages 255--265, 1989.

.... ( 1] 11] there are also lots of geometric variants most of them concerning points distributed in the Euclidean plane [3] many of them, as far as dealing with arbitrary many covering components being NP hard [6,9] On the other hand, there are also certain partition or clustering problems [4,5,8], which could be related to partition variants of the problem at hand. Whereas there are tiling problems [10,13] which in some sense are dual resp. complementary. The paper is organized as follows. After having presented some notation and a precise de nition of the problem, in the remainder of ....

J. Hershberger, S. Suri, Finding Tailored Partitions, Journal of Algorithms 12 (1991) 431-463.

....Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA 1 details) We refer to this problem as the 2DC (2 disk cover) problem. The best previous solution of the 2DC problem runs in O(n 2 ) time [6] see also [7]) Our strategy is to assume that such a pair of disks exist, call them D 1 , D 2 , and to conduct a search for their centers. Let c i denote the center of D i , and let C i denote the circle bounding D i , for i = 1; 2. We may assume, with no loss of generality, that jc 1 c 2 j is as small as ....

....we wish to determine whether the intersection K(P ) T p2P B r (p) is nonempty, where B r (p) is the closed disk of radius r centered at p. This condition is equivalent to the condition that P can be covered by a disk of radius r. Such a procedure is also used in the preceding algorithms of [6, 7]. We give here a slightly inferior implementation of this procedure. This is done because it is easier to describe, and, more importantly, it is easier to parallelize, which is required by the parametric searching technique. To keep track of K(P ) as P is being updated, we maintain separately the ....

[Article contains additional citation context not shown here]

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms 12 (1991), 431-- 463.

....S into k subsets S 1 , S 2 , S k , such that f( S 1 ) S 2 ) S k ) Here, and k are part of the inputs and f is some function of the measures of the subsets. The choice of and f depends on actual applications. In most cases, these problems are NP complete for arbitrary k [17, 21, 22]. So, most of the research has been concentrated on the bipartition problem (i.e. when k = 2) The problem of minimizing the maximum diameter for a bipartition was first considered by Avis [2] Later, an optimal O(n log n) time algorithm was given by Asano et al. [3] In [27] an O(n 2 ) time ....

....problem (i.e. when k = 2) The problem of minimizing the maximum diameter for a bipartition was first considered by Avis [2] Later, an optimal O(n log n) time algorithm was given by Asano et al. [3] In [27] an O(n 2 ) time algorithm was given for 3 minimizing the sum of the two diameters. In [21], these problems were considered from a different point of view. Instead of imposing a single constraint on the maximum or sum of the parameter, they considered the problem of finding a bipartition where each partition satisfies some constraint. Such a formulation is useful for a finer cluster ....

[Article contains additional citation context not shown here]

J. Hershberger and S. Suri, "Finding tailored partitions", J. Algorithms, 12, (1991), pp. 431-463. 38

....the collection of subsets of S. f is monotone if S 1 S 2 ) f(S 1 ) f(S 2 ) for S 1 ; S 2 S. The problem of efficiently finding an optimal bipartition in respect to some criterion is the simplest form of the more general k partition problem, and it has been studied extensively, see e.g. [1, 2, 3, 6, 7, 8, 9, 10, 11, 13]. The problem considered here was studied by Mitchell and Wynters [12] who present solutions for two instances of the problem, in which f(S 0 ) is either the perimeter or area of the convex hull of S 0 . Their solutions require O(n 3 ) time and O(n) space. They also present an O(n 2 ) time ....

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431--463, 1991.

....For example, Hershberger [125] described an O(n 2 = log log n) time algorithm for partitioning a given set S of n points into two subsets so that the sum of their diameters is minimized. If we want to minimize the maximum of the two diameters, the running time can be improved to O(n log n) [127]. Glozman et al. 111] have studied problems of covering S by several different kinds of shapes. Maass [167] showed that the problem of covering S with the minimum number of unit width annuli is NP Hard even for d = 1 (a unit width annulus in 1 dimension is a union of two unit length intervals) ....

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms, 12 (1991), 431--463.

.... 3 log 5 n) by Agarwal et al. 1] the two line center problem, solved in time O(n 2 log 2 n) by Jaromczyk et al. 16] see also [18, 12] the two square center problem, where the squares are with mutually parallel sides solved in time O(n 2 ) by Jaromczyk et al. 14] Hershberger and Suri [13] and Glozman et al. 12] considered the problem of covering the set S by two axis parallel rectangles such that the size of the larger rectangle is minimized. They present an O(n log n) algorithm for this problem. In [22, 19, 20] several algorithms are presented that deal with a number of squares ....

....space (d 2) find two axis parallel rectangles that cover all the points of S and are centered at the points of C and size of the larger rectangle is minimized. Let us call the solution of this problem minimal rectangular cover. Here we consider the size as a perimeter. Hershberger and Suri [13], Glozman et al. 12] and Bespamyatnikh and Segal [4] consider a similar problem, but without constraining the centers of the rectangles to be in C. They present an algorithm which runs in time O(n log n) Our algorithm runs in time O(mn log m log n) 3.1 The decision algorithm Assume we are ....

[Article contains additional citation context not shown here]

Hershberger J., Suri S.: Finding Tailored Partitions. J. Algorithms 12 (1991) 431-- 463

....over the collection of subsets of S. f is monotone if S 1 S 2 ) f(S 1 ) f(S 2 ) for S 1 ; S 2 S. The problem of eOEciently nding an optimal bipartition in respect to some criterion is the simplest form of the more general k partition problem, and it has been studied extensively, see e.g. [1, 2, 3, 6, 7, 8, 9, 10, 11, 13]. The problem considered here was studied by Mitchell and Wynters [12] who present solutions for two instances of the problem, in which f(S 0 ) is either the perimeter or area of the convex hull of S 0 . Their solutions require O(n 3 ) time and O(n) space. They also present an O(n 2 ) time ....

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431463, 1991.

....or variance, is applied to each cluster individually. Then individual measures are combined into an overall criterion using a function such as the sum or the maximum. For the case of fixed k, there are polynomial time algorithms for minimizing various combinations of cluster diameters [CRW91, HS91b] and for minimizing the sum of variances [BH89, IKI94] but other individual cluster criteria, such as the sum of pairwise distances, give open problems. For example, Euclidean max cut, equivalent to asking for the two clusters that minimize the sum of all pairwise intracluster distances, is ....

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431--463, 1991.

.... the minimum enclosed circle [9] On other hand it belongs to the class of partition problems where we are interested in partitioning a Preprint submitted to Elsevier Preprint 29 March 1999 set of given points into two subsets in order to optimize some function of the sizes of two subsets [1,4,7,8]. This problem is also closely related to the rectilinear p center problem (and in particular to the 2 center problem) The p center problem is defined as follows. Let R be the set of compact convex regions with nonempty interior in the plane, where every region r 2 R is assigned a scaling point ....

....sizes, then we have the weighted rectilinear p center problem, and if R is a set of arbitrary axisparallel rectangles (and the scaling points are also arbitrary) then we face the general rectilinear p center problem. Results for the problems defined above, for p = 2 and d = 2, were obtained by [3,4,11,12]. Hershberger and Suri [4] solve the following clustering problem: Given a planar set of points S, a rectangular measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 [S 2 satisfying (S i ) i for i = 1; 2) They present an algorithm which solves this ....

[Article contains additional citation context not shown here]

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431-- 463, 1991.

.... (Euclidean and even bidimensional, i.e. m = 2) versions of representative clustering problems [4, 22, 27, 28, 30, 39] Work concentrated on suitable approximation algorithms for them [8, 41] Others concentrated on special cases where polynomial algorithms can be found (for example, the case p = 2 [23]) More recently, theoretical results have concentrated on polynomial approximation schemes for the representative based clustering approaches [1, and references ] 3 TWGD restricted to CH disjoint partitions is NPhard TWGD is very different from representative approaches like p median and ....

J. Hershberger and A. Suri. Finding tailored partitions. J. of Algorithms, 12:431--463, 1991.

....the two covering disks can be any pair of points in the plane. This latter problem has been studied extensively, where the best algorithm, due to Sharir [14] and slightly improved by Eppstein [6] runs in randomized expected O(n log 2 n) time. The discrete 2 center problem has been studied in [7], where a near quadratic algorithm is proposed (such an algorithm is briefly described later in this introduction) Before discussing it further, we note that the discrete 1 center problem, seeking the smallest disk Pankaj Agarwal and Micha Sharir have been supported by a joint grant from the ....

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms 12 (1991), 431--463.

....solved in worst case O(n) time for any fixed dimension d [5, 10, 19] simple randomized O(n) time methods are also known [6, 24] The next easiest case, the 2 center problem in two dimensions, is the subject of several recent papers in the computational geometry literature. Hershberger and Suri [14] considered the weaker problem of deciding whether S can be covered by two disks of radius r for a given r. They showed that this decision problem can be solved in O(n 2 log n) time (a small improvement was subsequently noted by Hershberger [13] Agarwal and Sharir [1] used this result in ....

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431--463, 1991.

.... and even bidimensional, i.e. m = 2) versions of representative clustering problems [4, 22, 27, 28, 30, 39] Work concentrated on suitable approximation algorithms for them [8, 41] Others concentrated in special cases where polynomial algorithms can be found (for example, the case p = 2 [23]) More recently, theoretical results have concentrated on polynomial approximation schemes for the representative based clustering approaches [1, and references] 3 TWGD restricted to CH disjoint partitions is NPhard It is not hard to see that TWGD is very different from representative ....

J. Hershberger and A. Suri. Finding tailored partitions. J. of Algorithms, 12:431--463, 1991.

.... unsolved and has been regarded by some as one of the foremost open problems in the area [13, 25] Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often used as subroutines for tackling more difficult geometric problems (see the references [5, 11, 12, 15, 16, 19, 20, 21, 23, 24, 26, 27, 34] for a mere sampling) Among the earliest proposed methods for dynamic hull maintenance in the plane was one by Overmars and van Leeuwen [31] and dated back to 1981. The worst case update time is O(log 2 n) where n is the maximum number of points. Since this data structure actually stores the ....

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431--463, 1991.

....of the two covering disks can be any pair of points in the plane. This latter problem has been studied extensively, where the best algorithm, due to Sharir [14] and slightly improved by Eppstein [6] runs in randomized expected O(n log 2 n) time. The discrete 2 center problem has been studied in [7], where a near quadratic algorithm is proposed (such an algorithm is briefly described later in this introduction) Before discussing it further, we note that the discrete 1 center problem, seeking the smallest disk centered at a point of P and containing P , is much easier to solve, in time O(n ....

J. Hershberger and S. Suri, Finding tailored partitions, J. Algorithms 12 (1991), 431--463.

....problem as given by a set X of points in some metric space and seeking the best partition of X into two clusters. To measure the quality of the partition, some criterion is applied to each cluster individually. Some criterions for which polynomialtime algorithms exist are the diameter [19, 11] and the variance [1, 12] but there is no polynomial time algorithm known for minimizing the sum of pairwise distances, which is equivalent to maximizing the sum of distances between points in different clusters, i.e. to metric MAX CUT. Thus Bern raises the question of designing an efficient ....

H. Hershberger and S. Suri. Finding tailored partitions. Journal of Algorithms, 12:431--463, 1991.

.... is the covering problem such as the classical problem of finding the minimum enclosed circle [9] On other hand it belongs to partition problems where we are interested in partitioning a set of points into two subsets in such a way to optimize some function of the sizes of two subsets [1,4,7,8]. Preprint submitted to Elsevier Preprint 5 November This problem is also closely related to the rectilinear p center problems (and in particular to the 2 center problem) The p center problem is defined as follows. Let R be the set of compact convex regions with nonempty interior in the plane, ....

....sizes, then we have the weighted rectilinear p center problem, and if R is a set of arbitrary axisparallel rectangles (and the scaling points are also arbitrary) then we face the general rectilinear p center problem. Results for the above defined problems, for p = 2 and d = 2, were obtained by [4,3,12,11]. Hershberger and Suri [4] solve the following clustering problem: Given a planar set of points S, a rectangular measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 [S 2 satisfying (S i ) i for i = 1; 2) They present an algorithm which solves this ....

[Article contains additional citation context not shown here]

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, pages 431--463, 1991.

No context found.

H. Hershberger and S. Suri. Finding tailored partitions. Journal of Algorithms, 12:431--463, 1991.

No context found.

J. Hershberger and S. Suri. Finding tailored partitions. J. Algorithms, 12:431-463, 1991.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC