Results 1  10
of
114
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 ..."
Abstract

Cited by 302 (22 self)
 Add to MetaCart
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...
Approximating extent measure of points
 Journal of ACM
"... We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our tec ..."
Abstract

Cited by 117 (28 self)
 Add to MetaCart
(Show Context)
We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our technique include�approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms. 1
Coordination And Geometric Optimization Via Distributed Dynamical Systems
, 2003
"... This paper discusses dynamical systems for diskcovering and spherepacking problems. We present facility location functions from geometric optimization and characterize their differentiable properties. We design and analyze a collection of distributed control laws that are related to nonsmooth grad ..."
Abstract

Cited by 98 (26 self)
 Add to MetaCart
This paper discusses dynamical systems for diskcovering and spherepacking problems. We present facility location functions from geometric optimization and characterize their differentiable properties. We design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems. The resulting dynamical systems promise to be of use in coordination problems for networked robots; in this setting the distributed control laws correspond to local interactions between the robots. The technical approach relies on concepts from computational geometry, nonsmooth analysis, and the dynamical system approach to algorithms.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
Abstract

Cited by 81 (17 self)
 Add to MetaCart
(Show Context)
The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Exact and Approximation Algorithms for Clustering
, 1997
"... In this paper we present a n O(k1�1=d) time algorithm for solving the kcenter problem in R d, under L1 and L2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete kcenter problem, as well. We also describe a simple (1 +)approximation algorithm for the kcenter pr ..."
Abstract

Cited by 79 (6 self)
 Add to MetaCart
In this paper we present a n O(k1�1=d) time algorithm for solving the kcenter problem in R d, under L1 and L2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete kcenter problem, as well. We also describe a simple (1 +)approximation algorithm for the kcenter problem, with running time O(n log k) + (k = ) O(k1�1=d). Finally, we present a n O(k1�1=d) time algorithm for solving the Lcapacitated kcenter problem, provided that L = (n=k 1�1=d) or L = O(1). We conclude with a simple approximation algorithm for the Lcapacitated kcenter problem.
Geometric Applications of a Randomized Optimization Technique
, 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 ..."
Abstract

Cited by 54 (11 self)
 Add to MetaCart
(Show Context)
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.
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
Abstract

Cited by 43 (3 self)
 Add to MetaCart
(Show Context)
Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
Distributed control of robotic networks: a mathematical approach to motion coordination algorithms
, 2009
"... (i) You are allowed to freely download, share, print, or photocopy this document. (ii) You are not allowed to modify, sell, or claim authorship of any part of this document. (iii) We thank you for any feedback information, including errors, suggestions, evaluations, and teaching or research uses. 2 ..."
Abstract

Cited by 41 (1 self)
 Add to MetaCart
(i) You are allowed to freely download, share, print, or photocopy this document. (ii) You are not allowed to modify, sell, or claim authorship of any part of this document. (iii) We thank you for any feedback information, including errors, suggestions, evaluations, and teaching or research uses. 2 “Distributed Control of Robotic Networks ” by F. Bullo, J. Cortés and S. Martínez