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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 303 (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 121 (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.
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 ..."
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Cited by 90 (20 self)
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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...
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 59 (11 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...
Computing envelopes in four dimensions with applications
 SIAM J. Comput
, 1997
"... Abstract. Let F be a collection of ndvariate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n d+ε) for any ..."
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Cited by 42 (18 self)
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Abstract. Let F be a collection of ndvariate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n d+ε) for any ε>0. For d = 3, by combining this algorithm with the pointlocation technique of Preparata and Tamassia, we can compute, in randomized expected time O(n 3+ε), for any ε>0, a data structure of size O(n 3+ε) that, for any query point q, can determine in O(log 2 n) time the function(s) of F that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several problems in computational geometry, including (a) computing the width of a point set in 3space, (b) computing the “biggest stick ” in a simple polygon in the plane, and (c) computing the smallestwidth annulus covering a planar point set. The solutions to these problems run in randomized expected time O(n 17/11+ε), for any ε>0, improving previous solutions that run in time O(n 8/5+ε). We also present data structures for (i) performing nearestneighbor and related queries for fairly general collections of objects in 3space and for collections of moving objects in the plane and (ii) performing rayshooting and related queries among n spheres or more general objects in 3space. Both of these data structures require O(n 3+ε) storage and preprocessing time, for any ε>0, and support polylogarithmictime queries. These structures improve previous solutions to these problems.
Almost tight upper bounds for vertical decompositions in four dimensions
 In Proc. 42nd IEEE Symposium on Foundations of Computer Science
, 2001
"... We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major proble ..."
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Cited by 35 (5 self)
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We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n 2d−4+ε), for any ε> 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems. 1
Approximation Algorithms for kLine Center
, 2002
"... Given a set P of n points in Rd and an integer k> = 1, let w * denote the minimumvalue so that P can be covered by k cylinders of width at most w*. We describe analgorithm that, given P and an "> 0, computes k cylinders of width at most (1 + ")w*that cover P. The running time ..."
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Cited by 34 (5 self)
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Given a set P of n points in Rd and an integer k> = 1, let w * denote the minimumvalue so that P can be covered by k cylinders of width at most w*. We describe analgorithm that, given P and an &quot;> 0, computes k cylinders of width at most (1 + &quot;)w*that cover P. The running time of the algorithm is O(n log n), with the constant ofproportionality depending on k, d, and &quot;. The running times of the fastest algorithmsthat compute w * exactly are of the order of nO(dk). An approximation algorithm withnearlinear dependence on n for k> 1 was only known for the planar 2line centerproblem, i.e., the case k = 2, d = 2.We believe that the techniques used in showing this result are quite useful in themselves. We first show that there exists a small &quot;certificate &quot; Q ` P, whose size doesnot depend on n, such that for any kcylinders that cover Q, an enlargement of thesecylinders by a factor of (1 + &quot;) covers P. We only establish the existence of a small certificate and our proof does not give us an efficient way of constructing one. We then observe that a wellknown scheme based on sampling and iterated reweighting gives usan efficient algorithm for solving the problem. Only the existence of a small certificate is used to establish the correctness of the algorithm. This technique is quite generaland can be used in other contexts as well.
FixedDimensional Linear Programming Queries Made Easy
 Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We derive two results from Clarkson's randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a random ..."
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Cited by 33 (9 self)
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We derive two results from Clarkson's randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a randomized data structure with O(n) space and O(n 1\Gamma1=bd=2c 2 O(log n) ) query time for linear programming on n halfspaces (d ? 3). The second result is a simpler proof of the following: a sequence of q linear programming queries on n halfspaces can be answered in O(n log q) time, if q n ff d for a certain constant ff d ? 0. Unlike previous methods, our algorithms do not require parametric searching. 1 Introduction One of the major discoveries in computational geometry is that fixeddimensional linear programming can be solved in linear time [Meg84]. It was observed that the introduction of randomization leads to considerably simpler solutions [Sei91, Cla95]. The goal of this paper is...
Penetration Depth of Two Convex Polytopes in 3D
 Nordic J. Computing
, 2000
"... with m and n facets, respectively. The penetration depth of A and B, denoted as (A; B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes (A; B) in O(m + m ) expected time, for any constant " & ..."
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Cited by 26 (2 self)
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with m and n facets, respectively. The penetration depth of A and B, denoted as (A; B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes (A; B) in O(m + m ) expected time, for any constant " > 0. It also computes a vector t such that ktk = (A; B) and int(A + t) \ B = ;. We show that if the Minkowski sum B ( A) has K facets, then the expected running time of our algorithm is O K , for any " > 0.
Approximation and Exact Algorithms for MinimumWidth Annuli and Shells
 Discrete Comput. Geom
, 1999
"... Let S be a set of n points in R d . The "roundness" of S can be measured by computing the width ! = ! (S) of the thinnest spherical shell (or annulus in R 2 ) that contains S. This paper contains three main results related to computing ! : (i) For d = 2, we can compute in O(n ..."
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Cited by 23 (14 self)
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Let S be a set of n points in R d . The "roundness" of S can be measured by computing the width ! = ! (S) of the thinnest spherical shell (or annulus in R 2 ) that contains S. This paper contains three main results related to computing ! : (i) For d = 2, we can compute in O(n log n) time an annulus containing S whose width is at most 2! (S). We extend this algorithm, so that for any given parameter " ? 0, an annulus containing S whose width is at most (1 + ")! , is computed in time O(n log n + n=" 2 ). (ii) For d 3, given a parameter " ? 0, we can compute a shell containing S of width at most (1+ ")! either in time O \Gamma n " d log( \Delta ! " ) \Delta or in time O \Gamma n " d\Gamma2 \Gamma log n + 1 " \Delta log \Gamma \Delta ! " \Delta\Delta . Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by NSF grants EIA9870724, and CCR9732787, by an NYI award, and by a grant from ...