H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for applications of the random sampling technique in computational geometry. In this section, we describe a randomized algorithm for linear programming by Clarkson [71] based on random sampling, which is actually quite general and can be applied to any geometric set cover and related problems [6, 51]. Other randomized algorithms for linear programming, which run in expected linear time for any fixed dimension, are proposed by Dyer and Frieze [104] Seidel [248] and Matousek et al. 207] Let H be the set of constraints. We assign a weight (h) 2 Z to each constraint; initially (h) 1 for ....

....Computing k , the minimum number of balls of radius r that cover D, is also NP Complete [123] A greedy algorithm can construct k log n balls of radius r that cover D. Hochbaum and Maass gave a polynomial time algorithm to compute a cover of size (1 )k , for any 0 [152] see also [51, 118, 134]. No constant factor approximation algorithm is known for the capacitated covering problem, with unit radius disks, that is, the problem of partitioning a given point set S in the plane into the minimum number of clusters, each of which consists of at most c points and can be covered by a disk of ....

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H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), 263--279.

....O(n log n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an O(n log n) time algorithm with an improved approximation ratio of 3, where n is the number of 1 and 2 hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n log n) time. 1 Introduction Wireless ad hoc networks can be flexibly and quickly deployed for many applications such as automated battlefield, search and rescue, and disaster relief. Unlike wired networks or cellular networks, no wired backbone ....

....Cover problem disk centers can be chosen arbitrarily in the plane, the algorithms for this problem do not apply to the Minimum Forwarding Set problem where disks must be centered at 1 hop neighbors only. The Minimum Forwarding Set problem is a special case of the NP Hard Disk Cover problem [2], which asks for a minimum size subset of a given set of disks covering a given set of points. The complexity of Minimum Forwarding Set problems is not known. A constant ratio approximation algorithm for Disk Cover, and therefore also for Minimum Forwarding Set, was given by Bronnimann and ....

[Article contains additional citation context not shown here]

H. Bronnimann and M.T. Goodrich, Almost Optimal Set Covers in Finite VC-Dimension. Proc. 10th ACM Symp. on Computational Geometry (SCG), 1994, 293--302.

....metric in Example 1 [7, 10] that it is NP Complete to compute a partitioning with a value less than two times the optimum to several new metrics. A preliminary version of the results in this paper appeared as part of [26] Our algorithms are based on the framework of Bronnimann and Goodrich in [5], and exploit an interesting relationship between partitioning problems and the construction of small nets. In particular, the partitioning problem for the MAX SUM metric can be reduced to a Set Cover problem with small VC dimension. It is shown in [5] that such Set Cover problems can be ....

....framework of Bronnimann and Goodrich in [5] and exploit an interesting relationship between partitioning problems and the construction of small nets. In particular, the partitioning problem for the MAX SUM metric can be reduced to a Set Cover problem with small VC dimension. It is shown in [5] that such Set Cover problems can be efficiently approximated using nets and an elegant analysis provided by Clarkson in [8] We show how to translate and expand these ideas to obtain very simple and fast algorithms for the above general classes of metrics. 4 Related Work Applications: ....

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H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete and Computational Geometry, 14, 463--479, 1995.

....on heuristic searches, we address the approximability of the camera placement problem. It is well known (and easy to see) that this problem can be cast as a hitting set problem. While the approximability of generic instances of the hitting set problem is well understood, Bronnimann and Goodrich[3] presented improved approximation algorithms for the problem in the case that the input instances have bounded Vapnik Chervonenkis (VC) dimension. In this paper we explore the VC dimension of set systems associated with the camera placement problem described above. We show a constant bound for ....

....of all camera locations that can see p. The hitting set problem assumes a finite set X and we have to implicitly deal with this issue when we attempt to pose MFGP as such a problem. As stated earlier the general hitting set problem cannot be approximated to better than a log factor. However [3] shows that if the VC dimension can be bounded by d and the optimal hitting set has size c, then we can produce an O(d log cd) approximation. Thus we need to examine the VC dimension of hitting set instances that can be produced from MFGP. We are able to determine the VC dimension both in 2D and ....

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H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. GEOMETRY: Discrete & Computational Geometry, 14, 1995.

....the Art Gallery and the related problems with our connected sensor coverage problem is that the Art Gallery and related problems require an optimal placement of observers, while our problem deals with an optimal selection of already placed sensors. From that perspective, the geometric variations [16, 17, 5] of the classic set cover problem are more related to our problem. However, none of the geometric set cover variations addressed in the literature deal with the notion of connectivity. For the disk cover problem [5] there is a polynomial algorithm that delivers a constant factor approximation, ....

....of already placed sensors. From that perspective, the geometric variations [16, 17, 5] of the classic set cover problem are more related to our problem. However, none of the geometric set cover variations addressed in the literature deal with the notion of connectivity. For the disk cover problem [5], there is a polynomial algorithm that delivers a constant factor approximation, however, the approximation algorithm does not generalize to other geometric regions (not even rectangles) and due to involved computation required, the straightforward distributed implementation would incur a very ....

H. Bonnimann and M. Goodrich. Almost optimal set covers in finite vc-dimension. Discrete Computational Geometry, 14, 1995.

....intersection alone are unlikely to give a o(log n) approximation factor for this problem. Further, our result shows that constant VC dimension alone does not help in getting a o(log n) approximation for the Set Covering problem. This is to be contrasted with the result of Bronnimann and Goodrich[BG94] which shows that if the VC dimension is a constant and an O( ffl ) sized (weighted) ffl net can be constructed in polynomial time, then a constant factor approximation can be obtained. Finally, for the problem of covering points with lines, we observe that the NP Hardness proof of Megiddo and ....

H. Bronnimann, M. Goodrich. Almost Optimal Set Covers in Finite VCDimension. Discrete Comput. Geom., 14, 1995, pp. 263--279.

....to solving the convex set covering problem. This problem is discussed by Clarkson [Clarkson, 1993] who describes a O#cn log n# time randomized algorithm for finding covering sets of cardinality within O#log c# of the optimal set covering c. More recent results by Bronniman and Goodrich [Bronnimann and Goodrich, 1994] on the dual problem of finding minimal hitting sets improves on these bounds. They demonstrate an O#n log n# algorithm that finds a hitting set of size O#1# from the optimal set size. They employ work by Matousek [Matousek, 1990] using # nets. 3 RISC Robotics RISC robotics [Canny and ....

....Clarkson [Clarkson, 1993] describes a randomized algorithm for computing the three dimensional convex point set cover from an initial set of n points to within O#log c# of the optimal cover of c points. His algorithm has a running time of O#cn log n#. 23 Recent work by Bronniman and Goodrich [Bronnimann and Goodrich, 1994] improve on both the running time and approximation to the optimal convex set covering. Their deterministic algorithm solves the equivalent problem of finding a minimal hitting set, where a hitting set is a subset H # X such that H has a non empty intersection with every set R in a collection of ....

Bronnimann, H. and Goodrich, M. (1994). Almost optimal set covers in finite vc-dimension. In Proc. 10th ACM Symp. on Computational Geometry (SCG), pages 293--302.

....two polyhedra L and U with U L, and we want to find a separating polytope S with U S L with small number of facets. If we ask for the smallest possible number of facets, this problem is NP complete [DJ90] Several approximation algorithms for this problem have been presented, deterministic [DJ90, BG94], and randomized [Cla93] Consider U as the intersection of a set of half planes. In the above mentioned algorithms, S is found by eliminating facets (halfplanes) from U , while making sure that the new polytope is still contained in L. Applying a suitable duality transformation, we have again two ....

....the approximation factor for the cover. This fact does not use the geometry of the situation at all, and one wonders whether the lower bound on s could be reduced in this specific instance where the points of U are in (non strictly) convex position. In fact 6 the deterministic algorithm in [BG94] which uses nets reduces the approximation factor to O(1) for the three dimensional version of a closely related problem, see section 5. Finally, to obtain a bound on the running time, we need to bound the number of iterations of the loop in find cover: Lemma 3.3 (Clarkson) If j 2c the number ....

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H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite vc-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

....an O(n log n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an O(n 2 ) time algorithm with an improved approximation ratio of 3, where n is the number of 1 and 2 hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n 3 log n) time. 1. INTRODUCTION Wireless ad hoc networks can be exibly and quickly deployed for many applications such as automated battle eld, search and rescue, and disaster relief. Unlike wired networks or cellular networks, no wired backbone ....

....Cover problem disk centers can be chosen arbitrarily in the plane, the algorithms for this problem do not apply to the Minimum Forwarding Set problem where disks must be centered at 1 hop neighbors only. The Minimum Forwarding Set problem is a special case of the NP Hard Disk Cover problem [2], which asks for a minimum size subset of a given set of disks covering a given set of points. The complexity of Minimum Forwarding Set problems is not known. A constant ratio approximation algorithm for Disk Cover, and therefore also for Minimum Forwarding Set, was given by Bronnimann and ....

[Article contains additional citation context not shown here]

H. Bronnimann and M.T. Goodrich, Almost Optimal Set Covers in Finite VC-Dimension. Proc. 10th ACM Symp. on Computational Geometry (SCG), 1994, 293-302.

.... the greedy algorithm for finding a set cover [11] to obtain, in polynomial time, a piercing set whose size is larger than the optimal size by a factor of (1 log l) where l n is the depth of the arrangement of C (the maximum number of objects containing a common point) Bronnimann and Goodrich [10] (see also Clarkson [12] presented a polynomial time algorithm for computing a set cover in which the approximation factor depends both on the optimal cover size a and on the VC dimension of the Dynamic Data Structures October 5, 1999 Dynamic Data Structures for Fat Objects 4 underlying set ....

H. Bronnimann and M.T. Goodrich, Almost optimal set covers in finite VCDimension, Discrete and Computational Geometry 14 (1995), 263--279.

....for applications of the random sampling technique in computational geometry. In this section, we describe a randomized algorithm for linear programming by Clarkson [62] based on random sampling, which is actually quite general and can be applied to any geometric set cover and related problems [6, 45]. Other randomized algorithms for linear programming, which run in expected linear time for any fixed dimension, are proposed by Dyer and Frieze [90] Seidel [211] Kalai [149] and Matousek et al. 178] Let H be the set of constraints. We assign a weight (h) 2 Z to each constraint; initially ....

....points. Computing k , the minimum number of balls of radius r that cover D, is also NP complete [104] A greedy algorithm can construct k log n balls of radius r that cover D. Hochbaum and Maass gave a polynomial time algorithm to compute a cover of size (1 )k , for any 0 [130] see also [45, 101, 114]. No constant factor approximation algorithm is known for the capacitated covering problem, with unit radius disks, that is, the problem of partitioning a given point set S in the plane into the minimum number of clusters, each of which consists of at most c points and can be covered by a disk of ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), 263--279.

....polytopes. This has recently been improved by Clarkson in [Cla93] he proposes a randomized algorithm for computing an approximation of size O(k o log k o ) in expected time O(k o n 1 ffi ) for any ffi 0 (the constant of proportionality depends on ffi , and tends to 1 as ffi tends to 0) In [BG94] 3 Bronnimann and Goodrich observed that a variant of Clarkson s algorithm yields a polynomial time deterministic algorithm that computes an approximation of size O(k 0 ) Working with polyhedral terrains, AS94] present a polynomial time algorithm that computes an ffl approximation of size O(k ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. In Proceedings Tenth ACM Symposium on Computational Geometry, pages 293--302, 1994. 15

....progress on the problems studied and on other related problems. We conclude by mentioning the following open problems: ffl Can one compute all k bichromatic intersections in the setup of Section B for the case of segments, in O(n log n k) time ffl Can one use the hitting set technique of [BG95] see also [PA95] and Section A.2) to get a better approximation for the problem of eliminating cycles of rods in space ffl Can one speed up the algorithms presented in this paper by using Chazelle s hierarchical cuttings [Cha93] instead of using the technique of [HP99b] Acknowledgments The ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VCdimension. Discrete Comput. Geom., 14:263--279, 1995. 11

.... linear programming [Meg84] In the case of P and Q being two convex polyhedra, this problem is efficiently solved in [DK85] The problem of finding a polygon with minimum number of vertices, separating two given sets was studied in numerous papers: EP88] ABO 89] DJ90] Mou92] MS92] BG94] The interest in circular separability was fuelled by applications in pattern recognition and image processing, KA84] Fis86] Notice that for two finite sets of points, following the idea of Lay [Lay71] an instance of a spherical separability problem in E d may be transformed into a linear ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite vc-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994.

....system Sigma is finite; see [100] for the definition of VC dimension. Clarkson [48] modified the ITERATIVE lp algorithm for computing a convex polytope of small complexity that lies between two nested convex polytopes, by reducing it to a geometric set cover problem. Later Bronnimann and Goodrich [35] showed that Clarkson s algorithm works for any instance of set cover. For simplicity, we describe the algorithm for computing a hitting set. It performs a binary search on the size of the hitting set. At each stage, given an integer k, it either returns a hitting set of size O(k log k) or it ....

....) time algorithm for computing a nested polytope with O(kOPT log n) vertices. Clarkson [48] showed that the randomized technique described in Section 5 can compute a nested polytope with O(kOPT log kOPT ) vertices in O(n log c n) expected time, for some constant c 0. Bronnimann and Goodrich [35] extended Clarkson s algorithm to obtain a polynomial time deterministic algorithm that constructs a nested polytope with O(kOPT ) vertices. A widely studied special case of surface simplification, motivated by applications in geographic information systems and scientific computing, is when P is ....

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), 263--279.

....and that compute solutions that are guaranteed to be within a small constant factor of optimal. The algorithms provide an interesting application of the framework for 13 approximating Set Cover for set systems with bounded Vapnik Chervonenkis (VC) dimension described by Bronnimann and Goodrich [4] (see also the discussion in Section 6.3. 6.1 NP Hardness Results For the special case of the MAX SUM ID metric, Charikar, Chekuri, Feder, and Motwani [6] have shown that it is NP hard to approximate the minimum heft of any p Theta p partitioning to within a factor of less than 2. This result ....

....each x 2 X , and define w(Y ) P y2Y w(y) for any subset Y of X . Definition 1. Given a weight function w, we say that a p Theta p partitioning is ff good if every tile r i;j of H satisfies w(R i;j ) ff Delta w(X) We remark that our ff good partitionings correspond to the ffl nets used in [4] and originally introduced in [12] which have found many applications in computational geometry. 14 6.3 Upper Bounds for MAX Metrics We now present approximation results for the p Theta p partitioning problem where the cumulative metric is the MAX metric, i.e. the heft of a partition is the ....

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Bronnimann and Goodrich. Almost optimal set covers in finite VC-dimension. In Proceedings of the 10th Annual Symposium on Computational Geometry, 1994.

....from planar 3 SAT. Whether the special case of polyhedron approximation is NP complete seems to be unknown. Mitchell and Suri [MS92] gave an O(logn) approximation algorithm (which works in general dimension) In this section we describe a recent improvementdue to Bronnimann and Goodrich [BG94] that achieves a constant factor approximation in IR 3 . Both algorithms reduce the separation problem to the hitting set problem [GJ79] given a collection S = S 1 , S 2 , S n of sets of points (elements of a ground set) one seeks a minimum cardinality set of points S such ....

H. Bronnimann and M.T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th ACM Symp. Computational Geometry, pages 293--302, 1994.

....case that the points lie in d for some fixed d, but with a running time that s doubly exponential in 1=ffl and d. We remark that in the special case for achieving a constant factor approximation when the points lie in 2 , an approximation algorithm for disk cover of Bronnimann and Goodrich [3] can be used in place of the disk cover algorithm of [7] as a subroutine, to reduce the running time for class cover in this case to O(n 5 log n) First we show the class cover problem is NP Complete, and that approximating the class cover problem within a factor better than O(log n) is not ....

....and Goodrich, who present a constant factor approximation of the 2 dimensional disk cover algorithm that runs in O(n 3 log n) time. Substituting this in for the Hochbaum Maass algorithm above gives us an O(1) approximation to the class cover problem with a running time of O(n 5 log n) see [3]) 4 Applications to Classification Given a solution S to the class cover problem, it can be incorporated into a simple classifier as follows: declare a new unclassified point to be in the positive class if it lies within distance max v2B d(v; S) from some blue center, and in the negative class ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite vc-dimension. Discrete Comput. Geom., 14:463--479, 1995.

.... presented a simple randomized algorithm with the same approximate ratio [Cl93] and most recently Bronnimann and Goodrich obtained a constant factor approximate solution for this problem using a beautiful combination of set covers in finite VC dimension and randomized natural selection algorithms [BG95]. However, their algorithm is very slow (O(n 4 log n) time) when the optimal solution has size n ffi since it uses the algorithm of Matousek et al. as a subroutine which spends O(n 3 ) time to compute ffl nets in a set system [MSW90] We note that in 2D the algorithms of [II86; II88; ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. Disc. Comput. Geom., 14:463--479, 1995.

....take exponential time, and it would clearly be impractical. As with curves, certain problems in optimal surface approximation are well understood, while others are not. It is known that L# optimal polygonal approximation of convex surfaces is NP hard (requires exponential time, in practice) [19, 9]. This implies, of course, that L# optimal approximation of height fields and more general surfaces (in the space of all triangulations) is also NP hard, since they are a superset of convex surfaces. We do not know if there are polynomial time algorithms for optimal surface simplification using ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annual ACM Symp. on Computational Geometry, pages 293--302, 1994.

....seminal papers on this topic. It has been shown that computing the minimal facet approximation within a certain error bound is NP hard for convex polytopes [9] Thus algorithms for approximation of convex objects focus mainly on fast heuristics that produce approximations close to the optimal [3, 7, 28]. Polyhedral Terrains: Simplification of polyhedral terrains has been an active area of research for almost two decades because of its considerable importance to the GIS (Geographical Information System) community. It is impossible to do full justice to such a vast area in a mere section. We ....

Bronnimann, H. and Goodrich, M., "Almost optimal set covers in finite VC-dimension", Proceedings Tenth ACM Symposium on Computational Geometry, 1994, 293-302.

....n# for convex polytopes. This has recently been improved by Clarkson in [3] he proposes a randomized algorithm for computing an approximation of size O#k o log k o # in expected time O#k o n 1 # # for any ##0 (the constant of proportionality depends on #, and tends to 1 as # tends to 0) In [2] Bronnimann and Goodrich observed that a variant of Clarkson s algorithm yields a polynomial time deterministic algorithm that computes an approximation of size O#k 0 #. Working with polyhedral terrains, 1] present a polynomial time algorithm that computes an # approximation of size O#k o log k o ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. In Proceedings Tenth ACM Symposium on Computational Geometry, pages 293--302, 1994.

....q dimensional hypercylinders. More precisely, if S can be covered by k hyper cylinders of radius r; then the greedy algorithm covers S by O(k log n) hyper cylinders of radius r in time n O(d) The approximation factor can be improved to O(k log k) using the technique by Bronimann and Goodrich [14]. For example, this approach computes a cover of S R 2 by O(k log k) strips of a given width r in time O(n 3 k log k) assuming that S can be covered by k strips of width r each. No algorithm with roughly nk running time is known for this problem. Our results. In this paper we first consider ....

....ranges. The problem of computing a strip cover Sigma Pi of S of width 2w is equivalent to finding a hitting set for the set system X . We construct the hitting set by a method based on Clarkson s algorithm for polytope approximation [19] which was later extended by Bronimann and Goodrich [14]. Since we will be using the framework of Clarkson s algorithm, we sketch its main ideas. The algorithm works in phases. It assigns weights to each element in E: Initially, wt(e) 1 for all e 2 E: In each phase, it chooses a random subset M E of size O(k log k) so that each element of E is ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), 263--279.

.... There are few geometric situations where the existence of (1=r) nets of size O(r) has been established (improving the general O(r log r) bound) most notably for halfspaces in IR 3 [MSW90] This has a nice algorithmic application in a polytope approximation problem; see Bronniman and Goodrich [BG95] The VC dimension and nets are frequently used also in other fields, in particular in statistics (this is where they come from, after all) and in learning theory. The applicability of approximation to approximate ham sandwich cuts is noted in Lo et al. LMS94] a similar trick can be used ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. Discr. & Comput. Geom., 14:463--484, 1995.

.... (NP hard) 22, 21] but some polynomial time algorithms are known for computing a nearly optimal (i.e. nearly minimum facet) approximating surface, guaranteed to be within a factor O(log n) of optimal (see [5, 16, 76, 79] or within a constant factor of optimal, if the surface is convex (see [10]) Unfortunately, the polynomial time bounds for these theoretically good approaches is rather high (at least cubic) In contrast, from the practical point of view, most of the previous computer graphics and geography research in the area is based on heuristics for generating triangulations that ....

....than just on DEM arrays. Conceptually, there are no changes needed to the algorithm. A somewhat less trivial modification will be to generalize the algorithm to approximate arbitrary (non terrain) polyhedral surfaces and to find approximations to a minimumfacet separating surface (as done in [10, 16, 79], in the convex case) Another straightforward extension of our method allows one to use it to build hierarchical representations of terrain. For example, we can simply start with an extremely crude terrain approximation (e.g. just two triangles) and then adjust ffl to be smaller and smaller, ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite vc-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

....K o is the size of an optimal approximation. Extending their work, Clarkson [7] has proposed an O(K o n 1 ffi ) expected time randomized algorithm for computing an approximation of size O(K o log K o ) where ffi can be an arbitrarily small positive number. Recently Bronnimann and Goodrich [5] have refined Clarkson s algorithm and have given a polynomial time algorithm for computing an approximation of size O(K 0 ) In this paper, we study the approximation problem for surfaces that correspond to graphs of bivariate functions. We show that it is NP Hard to decide if a surface can be ....

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Proc. 10th ACM Symp. on Computational Geometry, 1994, pp. 293--302.

....a factor O(log n) of optimal for the case of separating a convex polyhedron from a (possibly nonconvex) polyhedron. Clarkson [4] has provided an algorithm that gets within a factor O(log k ) of optimal, where k is the complexity of an optimal separator. Most recently, Bronnimann and Goodrich [3] have improved the results to obtain a constant factor approximation algorithm for the separation problem for nested convex polytopes in three dimensions. There have been several papers on the surface approximation problem in the graphics community (e.g. see [20] however, these algorithms do ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

....0 , and we want to find a separating polytope S 0 with U 0 S 0 L 0 with small number of facets. If we ask for the smallest possible number of facets, this problem is NP complete [DJ90] Several approximation algorithms for this problem have been presented both deterministic [DJ90, BG94] and randomized [Cla93] Consider U 0 as the intersection of a set of half planes. In the above mentioned algorithms, S 0 is found by eliminating facets (half planes) from U 0 , while making sure that the new polytope is still contained in L 0 . We assume that all polytopes involved ....

....approximation algorithm for the MinCover P problem (or its dual) using a greedy heuristic, with an approximation factor of d log n. It runs in time O(n d 1 ) There is also a deterministic analogue of Clarkson s algorithm which solves this problem and which provides a O(d log c) approximation [BG94] in time O i (n bd=2c c d n log d (dc) c log(n=c) j . It is easy to verify that these hitting set techniques also apply to our MinCover B problem as well as more general versions. It is unclear however whether these techniques can be applied to yield approximation algorithms for ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

....q dimensional hypercylinders. More precisely, if S can be covered by k hyper cylinders of radius r; then the greedy algorithm covers S by O(k log n) hyper cylinders of radius r in time n O(d) The approximation factor can be improved to O(k log k) using the technique by Bronimann and Goodrich [7]. For example, when q = 1; d = 2; this approach computes a cover of S by O(k log k) strips of given width in time roughly n 3 k log k: Our Results. In this paper we consider the simplest case of q = 1; d = 2: Thus, given a set S of n points and a positive integer k; we want to cover S by k ....

.... S by O(k log k) strips of width at most 6w in expected time O(nk 2 log 4 n) Our algorithm also works for larger values of k; but then the expected running time is O(n 2=3 k 8=3 log 4 n) The algorithm is based on the randomized algorithms by Clarkson [9] and Bronimann and Goodrich [7] for computing a hitting set. The main contribution of our paper is obtaining an algorithm whose running time is near linear as a function of n: We also propose an approximation algorithm for the 2 line center problem in the plane. The algorithm finds a cover by two strips of width at most 3w ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), 263--279.

....for applications of the random sampling technique in computational geometry. In this section, we describe a randomized algorithm for linear programming by Clarkson [72] based on random sampling, which is actually quite general and can be applied to any geometric set cover and related problems [6, 52]. Other randomized algorithms for linear programming, which run in expected linear time for any fixed dimension, are proposed by Dyer and Frieze [105] Seidel [250] and Matousek et al. 208] Clarkson s algorithm proceeds as follows. Let H be the set of constraints. We assign a weight (h) 2 Z to ....

....Computing k , the minimum number of balls of radius r that cover D, is also NP Complete [124] A greedy algorithm can construct k log n balls of radius r that cover D. Hochbaum and Maass gave a polynomial time algorithm to compute a cover of size (1 )k , for any 0 [153] see also [52, 119, 135]. No constant factor approximation algorithm is known for the capacitated covering problem, with unit radius disks, that is, the problem of partitioning a given point set S in the plane into the minimum number of clusters, each of which consists of at most c points and can be covered by a disk of ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), 263--279.

.... has recently been improved by Clarkson in [6] he proposes a randomized algorithm for computing an approximation of size O(k o log k o ) in expected time O(k o n 1 ffi ) for any ffi 0 (the constant of proportionality depends on ffi, and tends to 1 as ffi tends to 0) Bronnimann and Goodrich [3] observed that a variant of Clarkson s algorithm yields a polynomial time deterministic algorithm that computes an approximation of size O(k 0 ) Working with polyhedral terrains, Agarwal and Suri [1] present a polynomial time algorithm that computes an ffl approximation of size O(k o log k o ) ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. In Proceedings Tenth ACM Symposium on Computational Geometry, pages 293--302, 1994.

....polytopes. This has recently been improved by Clarkson in [3] he proposes a randomized algorithm for computing an approximation of size O(k o log k o ) in expected time O(k o n 1 ffi ) for any ffi 0 (the constant of proportionality depends on ffi , and tends to 1 as ffi tends to 0) In [2] Bronnimann and Goodrich observed that a variant of Clarkson s algorithm yields a polynomial time deterministic algorithm that computes an approximation of size O(k 0 ) Working with polyhedral terrains, 1] present a polynomial time algorithm that computes an ffl approximation of size O(k o log k ....

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. In Proceedings Tenth ACM Symposium on Computational Geometry, pages 293--302, 1994.

.... O(n 3 ) time algorithm that computes a convex polytope Q with O(c log n) vertices such that (1 Gamma )P Q (1 )P ; c is the size of the smallest polytope that lies between (1 Gamma )P and (1 )P [31] These bounds were subsequently improved by Clarkson [7] and Bronnimann and Goodrich [5]; the latter gave an algorithm, with O(nc(c log n) log(n=c) running time, to compute a polytope Q with O(c) vertices. It is an open question whether an approximate convex polytope with the minimum number of vertices can be computed in polynomial time. None of the algorithms for convex surfaces ....

....on ffi. As we will see below, the algorithm produces an approximation of size roughly O(c log c) in time O(n 2 ffi ) for some cases. We will be combining the ideas from the algorithms of Agarwal and Suri [3] and Clarkson [7] along with some new ideas to obtain the faster algorithm. As in [5, 7] we will formulate the problem as a twodimensional hitting set problem and use a variant of Clarkson s randomized algorithm to compute a small hitting set. In order to expedite the running time, we do not construct the underlying set system explicitly. Instead, we run the Clarkson s algorithm on ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete and Comput. Geom., 15 (1995), 463--479.

.... problem arises in a dual form in the context of separating two nested polyhedra [MS92, Cla93] These problems can in turn be cast as Hitting Set problems [MS92] There is also a deterministic analogue of Clarkson s algorithm which solves this problem and which provides a O(d log c) approximation [BG94] in time O i n bd=2c c d n log d (dc) j c log(n=c) It is easy to verify that these hitting set techniques also apply to our MinCover B problem as well as more general versions. It is unclear however whether these techniques can be applied to yield approximation algorithms for ....

....for k = 2; d 2 and c 43; or for k = 1:501, d 2 and c 6. See [Tei95] for details. Finally, to obtain a bound on the running time, we need to bound the number of iterations of the loop in find cover. Again this lemma is a slight variation of Clarkson s and a similar lemma also appears in [BG94]. Lemma 3.3 The number of successful iterations of the loop in find cover before a cover is found is bounded above by 2k 2k Gamma3 c lg(n=c) which is 4c lg(n=c) for k = 2, 1501c lg(n=c) for k = 1:501. Proof. See [Tei95] for details. This implies together with Lemma 3.2 that the expected number ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite vc-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

....0 , and we want to find a separating polytope S 0 with U 0 S 0 L 0 with small number of facets. If we ask for the smallest possible number of facets, this problem is NP complete [DJ90] Several approximation algorithms for this problem have been presented both deterministic [DJ90, BG94] and randomized [Cla93] Consider U 0 as the intersection of a set of half planes. In the above mentioned algorithms, S 0 is found by eliminating facets (half planes) from U 0 , while making sure that the new polytope is still contained in L. We assume that all polytopes involved contain ....

....approximation algorithm for the MinCover P problem (or its dual) using a greedy heuristic, with an approximation factor of d log n. It runs in time O(n d 1 ) There is also a deterministic analogue of Clarkson s algorithm which solves this problem and which provides a O(d log c) approximation [BG94] in time O i n bd=2c c d n log d (dc) j c log(n=c) It is easy to verify that these hitting set techniques also apply to our MinCover B problem as well as more general versions. It is unclear however whether these techniques can be applied to yield approximation algorithms for ....

[Article contains additional citation context not shown here]

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

....is equivalent to solving the convex set covering problem. This problem is discussed by Clarkson [5] who describes a O(cn log O(1) n) time randomized algorithm for finding covering sets of cardinality within O(log c) of the optimal set covering c. More recent results by Bronniman and Goodrich [3] on the dual problem of finding minimal hitting sets improves on these bounds. They demonstrate an O(n log 2 n) algorithm that finds a hitting set of size O(1) from the optimal set size. They employ work by Matousek [15] using ffl nets. E. Paulos and J. Canny 3 Defining Optimality When ....

....Recently, Clarkson [5] describes a randomized algorithm for computing the three dimensional convex point set cover from an initial set of n points to within O(log c) of the optimal cover of c points. His algorithm has a running time of O(cn log O(1) n) Recent work by Bronniman and Goodrich [3] improve on both the running time and approximation to the optimal convex set covering. Their deterministic algorithm solves the equivalent problem of finding a minimal hitting set, where a hitting set is a subset H X such that H has a non empty intersection with every set R in a collection of ....

H. Bronnimann and M.T. Goodrich. Almost optimal set covers in finite vc-dimension. In Proc. 10th ACM Symp. on Computational Geometry (SCG), pages 293--302, 1994.

.... hard (NP hard) 5, 4] but some polynomial time algorithms are known for computing a nearly optimal (i.e. nearly minimum facet) approximating surface, guaranteed to be within a factor O(log n) of optimal (see [1, 3, 15, 17] or within a constant factor of optimal, if the surface is convex (see [2]) Unfortunately, the polynomial time bounds for these theoretically good approaches is rather high (at least cubic) In contrast, from the practical point of view, most of the previous computer graphics and geography research in the area is based on heuristics for generating triangulations that ....

....than just on DEM arrays. Conceptually, there are no changes needed to the algorithm. A somewhat less trivial modification will be to generalize the algorithm to approximate arbitrary (nonterrain) polyhedral surfaces and to find approximations to a minimum facet separating surface (as done in [2, 3, 17], in the convex case) Another straightforward extension of our method allows one to use it to build hierarchical representations of terrain. For example, we can simply start with an extremely crude terrain approximation (e.g. just two triangles) and then adjust ffl to be smaller and smaller, ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

.... p jRj=jXj) Rg, the VC dimension of (X; R) is defined as the maximum size of a subset A of X such that RjA = 2 A (e.g. see [49] A related and simpler notion, however, is based upon the shatter function, R (m) fjRj A j: A X; jAj = mg: In particular, we say that (X; R) has VC exponent [8, 13] bounded by e if R (m) is O(m e ) For example, if (X; R) is the hyperplane set system, where X is a set of n hyperplanes in IR d and R is the set of all combinatorially distinct ways of intersecting hyperplanes with simplices, then (X; R) has VC exponent bounded by d(d 1) Interestingly, ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete and Computational Geometry, 14, 1995, 463--479.

....to produce such representations of arbitrary 3 connected planar graphs in linear time. Incidentally, another well known instance of a geometric optimization problem for polyhedra, which we did not consider is polyhedral separability, which has been the subject of extensive research, e.g. see [1, 9, 15, 17, 26, 28, 27, 31]. For example, a heavily studied instance of this problem is that one given two concentric polyhedra, and one wishes to find a separating polyhedra with minimum faces nested between the two. The nested polyhedral separability problem that we consider was first raised by Victor Klee [31] which ....

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

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H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

No context found.

H. Bronnimann and M. Goodrich, "Almost optimal set covers in finite VC-dimension," Discrete & Computational Geometry, vol. 14, no. 4, pp. 463--479, 1995.

No context found.

H. Bronnimann and M.T. Goodrich. Almost optimal set covers in finite VCdimension. Discrete & Computational Geometry, 14(4):463--479, 1995.

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H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VCdimension. Discrete Comput. Geom., 14:263--279, 1995.

No context found.

H. Bronnimann and M. Goodrich, "Almost optimal set covers in finite VC-dimension," Discrete & Computational Geometry, vol. 14, no. 4, pp. 463--479, 1995.

No context found.

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom., 14:263--279, 1995.

No context found.

Bronnimann, H., Goodrich, M.T., 1995, Almost optimal set covers in finite VC-dimension, Discrete and Computational Geometry, 15, pp.463-479.

No context found.

H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete and Computational Geometry, 14, 463--479, 1995.

No context found.

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

No context found.

H. Bronnimann and M. T. Goodrich. Almost optimal set covers in finite VCdimension. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 293--302, 1994.

No context found.

H. Bronnimann and M. T. Goodrich. "Almost optimal set covers in finite VCdimension, " Discrete Comput. Geom., 14 (1995), 263--279.

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