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: ....

[Article contains additional citation context not shown here]

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 ....

[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.

....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.

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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.

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

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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.

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

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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.

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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