M. J. Katz and F. Nielsen, On piercing sets of objects, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 113--121.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for computing a rectilinear 4 center, using the matrix searching technique by Frederickson and Johnson; they have shown that this algorithm is optimal in the worst case. Recently Chan [54] developed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center. See [177, 257] for additional related results. 7.2 Euclidean p line center and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem of determining whether w = 0 ....

M. J. Katz and F. Nielsen, On piercing sets of objects, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 113--121.

....2.3, Chan [36] proposed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center, which is optimal in the worstcase. Chan also presented an O(n log n) expected time algorithm for computing the smallest square that contains k of a given set of n points in the plane. See [114, 172] for additional related results. 6.2 Euclidean p line center Let D be a set of n points in R d and be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem ....

M. J. Katz and F. Nielsen, On piercing sets of objects, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 113-121.

....algorithm for computing a rectilinear 4 center (and have shown that this algorithm is worst case optimal) and an O(n log 5 n) time algorithm for computing a rectilinear 5 center. The algorithms for the 4 center and 5 center employ the Frederickson Johnson matrix searching technique. See [152, 219] for additional related results. 7.2 Euclidean p line center Let D be a set of n points in R d and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem of ....

M. J. Katz and F. Nielsen, On piercing sets of objects, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 113--121.

....2.3, Chan [38] proposed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center, which is optimal in the worstcase. Chan also presented an O(n log n) expected time algorithm for computing the smallest square that contains k of a given set of n points in the plane. See [120, 182] for additional related results. 6.2 Euclidean p line center Let D be a set of n points in R d and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir [151] showed that the ....

M. J. Katz and F. Nielsen, On piercing sets of objects, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 113--121.

....for which we supply an O(c 2 n log n) solution, and show that it is worst case optimal when c is fixed. 2 piercing of a set of homothetic triangles, and the triangular 2 center problem, can both be solved in linear time. The same holds for 4 oriented trapezoids. Independently, Katz and Nielsen [21] have recently obtained several results on the piercing problems studied in this paper, and also results concerning 2 piercing of boxes, simplices, and c oriented polytopes in higher dimensions. They have not addressed the related center problems. Methods. In order to obtain the linear bound for ....

....It is easily verified that this condition holds for the above problems. Remark. One can apply Lemma 3.2(c) using a machinery similar to that above, to obtain a linear time randomized algorithm for the general rectilinear 2 center problem in any dimension d. See also a related piercing result in [21]. 3.2 Rectilinear 4 and 5 center problems Let S be a set of n points in the plane. We want to find the smallest , so that S can be covered by the union of four (or five) axis parallel squares of side length . We solve this problem by providing a general scheme for transforming a p piercing ....

M. Katz and F. Nielsen, On piercing sets of objects, These Proceedings.

....to finding whether the intersection of rectangles empty or not. In Sharir and Welzl [7] 2 and 3 piercing problems in the plane are solved in linear time, while they reach only O(n log 3 n) bound for the 4 piercing problem and O(n log 4 n) bound for the 5 piercing problem. Katz and Nielsen [2] present a linear time algorithm for d dimensional boxes (d 2) when p = 2. In this paper we present a new technique which allows to obtain simple linear time algorithms for p = 1; 2; 3, and obtain an O(n log n) time solution for p = 4; 5, thus improving the previous results of [7] We improve ....

M. Katz, F. Nielsen, "On piercing sets of objects", In Proc. 12th ACM Symp. on Computational Geometry, 1996.

....for computing a rectilinear 4 center, using the matrix searching technique by Frederickson and Johnson; they have shown that this algorithm is optimal in the worst case. Recently Chan [56] developed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center. See [178, 259] for additional related results. 7.2 Euclidean p line center Let D be a set of n points in R d and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem ....

M. J. Katz and F. Nielsen, On piercing sets of objects, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 113--121.

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M.J. Katz and F. Nielsen, "On piercing sets of objects", Proc. 12th ACM Symp. on Computational Geometry , 113--121, 1996.

....The decision algorithm The decision problem is stated as follows: Given a set P of n points, are there two constrained axis parallel squares, each of a given area A, whose union covers P . We present an O(n log n) algorithm for solving the decision problem. We adopt the notation of [20] see also [12, 17]) Denote by R the set of axisparallel squares of area A centered at the points of P . R is p pierceable if there exists a set X of p points which intersects each of the squares in R. The set X is called a piercing set for R. Notice that X is a piercing set for R if and only if the union of the ....

M.J. Katz and F. Nielsen, "On piercing sets of objects", Proc. 12th ACM Symp. on Computational Geometry, 113--121, 1996.

....two axis aligned constrained squares, each of a given area A, whose union covers P . We present an O(n log n) algorithm for solving the decision problem. Applying the sorted matrices technique [10] we obtain an O(n log 2 n) time optimization algorithm. We adopt the notation of [22] see also [14, 19]) Denote by R the set of axis aligned squares of area A centered at the points of P . R is called p pierceable if there exists a set of p points which intersects every member in R. These points are called piercing points and the union of the axisaligned squares of area A centered at these points ....

M.J. Katz and F. Nielsen, "On piercing sets of objects", Proc. 12th ACM Symp. on Computational Geometry , 113--121, 1996.

....Can we compute a maximal cell in a better running time This would improve the running time of the greedy algorithm. Another aspect of this problem that is currently being investigated is to give eOEcient algorithms to detect whether a set of objects is k pierceable or not for small values of k [KN95, KN96] Acknowledgements. I would like to thank Jean Daniel Boissonnat, Herv# Br#n nimann and Mariette Yvinec for helpful discussions. RR n Sigma2854 28 F. Nielsen ....

M. J. Katz and F. Nielsen. On piercing sets of objects. Research Report # para#tre, INRIA, BP93, 06902 Sophia-Antipolis, France, 1995.

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