P.K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Computing Surveys, 30(4):412--458, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of parametric search, the number of calls to the decision process can be reduced by a log factor, thus improving the running time to O(PT p T p T s T s logP) We will explain his idea and comment on it in Section 4. One of the drawbacks of parametric search mentioned by Agarwal and Sharir [1] is that it requires the design of an efficient parallel algorithm for the generic version of the decision problem, which is not always easy. However, it is instructive to point out that the generic algorithm does not necessarily have to solve the same problem as the concrete version; all that is ....

....is generally assumed that the overhead involved (i.e. the hidden constants in the O notation) are too large to be of practical use. Partly because of the presumed difficulties with parametric search, alternatives have been proposed for many specific problems; see for instance Agarwal and Sharir [1] and Chan [7] However, parametric search is currently still the most general method. We would like to counter balance the advice of avoiding parametric search in real world applications by making some observations that considerably simplify sorting based parametric search (Sections 3 and 4) and ....

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P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Comput. Surv., 30:412--458, 1998.

....favorable properties of the latter, as analyzed in Section 2. We thus adopt Intersection Over Maximum throughout the sequel. Previous work. The ShareCam problem bears some resemblance to other geometric optimization problems, in particular to variants of the p center and the p median problems [AS98] However, no question substantially similar to the ShareCam problem has, to our knowledge, been previously studied. The ShareCam problem was introduced by Song, Van der Stappen, and Goldberg [SvdSG02] who have also outlined preliminary solutions. They discretize the range of zoom values that ....

P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Comput. Surv., 30:412--458, 1998.

....give an n O(k 1 Gamma1=d ) time exact algorithm and a polynomial time approximation scheme with running time O(n log k) k=ffl) O(k 1 Gamma1=d ) for the k center problem in R d with L p distances, for all p, using dynamic programming. See also the survey of Agarwal and Sharir [2] for previous and related work. A different idea is advocated by Drineas, Frieze, Kannan, Vempala, and Vinay [18] They give a 2 approximation for k clustering (fixed k) for the case of squared Euclidean distances, using methods from linear algebra (specifically, singular value decomposition, see ....

P.K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Computing Surveys, 30(4):412--458, 1998.

.... we will use this decision algorithm in combination with the well known parametric search technique [7, 18] In what follows, we assume that the reader is familiar with this technique; for instance, see earlier papers on the 2 center problem [1, 12, 22] or the survey by Agarwal and Sharir [2]. In order to apply parametric search, we need an efficient parallel version of the decision algorithm; we do not have to parallelize preprocessing steps that do not depend on the parameter r, and we can use Valiant s comparison model of computation [23] Unfortunately, the algorithm in Theorem ....

P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. Tech. Report CS-1996-19, Dept. of Computer Science, Duke Univ., Durham, 1996.

....algorithm, so that comparisons can be batched. Running time typically increases by logarithmic factors, even when an improvement by Cole [25] is applicable. As many researchers have commented, the resulting algorithms tend to be complicated and impractical; see the survey by Agarwal and Sharir [3]. In contrast, our randomized reductions use the decision algorithms purely as black boxes, avoid the extra logarithmic factors in the running time, and are easier to implement. A number of alternatives to parametric search have been proposed in the geometry literature [3] First, if the search ....

....by Agarwal and Sharir [3] In contrast, our randomized reductions use the decision algorithms purely as black boxes, avoid the extra logarithmic factors in the running time, and are easier to implement. A number of alternatives to parametric search have been proposed in the geometry literature [3]. First, if the search space has linear size, then an ordinary binary search is sufficient. For many rectilinear problems, the search space forms a matrix with sorted rows columns, and one can use Frederickson and Johnson s selection algorithm [37] to carry out the binary search [20, 39, 66] that ....

P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. Technical Report CS1996 -19, Department of Computer Science, Duke University, Durham, NC, 1996.

....this parallel algorithm sequentially. Running time typically increases by logarithmic factors, even when an improvement by Cole [20] is applicable. Furthermore, as many researchers have commented, the resulting algorithms tend to be complicated and impractical; see the survey by Agarwal and Sharir [3]. In contrast, our randomized reductions use the decision algorithms as black boxes only, avoid the extra logarithmic factors in the running time, and are easier to implement. A number of alternatives to parametric search have been proposed in the geometry literature [3] First, if the search ....

....by Agarwal and Sharir [3] In contrast, our randomized reductions use the decision algorithms as black boxes only, avoid the extra logarithmic factors in the running time, and are easier to implement. A number of alternatives to parametric search have been proposed in the geometry literature [3]. First, if the search space has linear size, then an ordinary binary search is sufficient. For many rectilinear problems, the search space forms a matrix with sorted rows columns, and one can use Frederickson and Johnson s selection algorithm [30] to carry out the binary search [32] In other ....

P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. Technical Report CS-199619, Department of Computer Science, Duke University, Durham, NC, 1996.

....there are at most L points in each cluster. The capacitated k center problem is to compute a k clustering of the smallest size so that each cluster has at most L points. Previous results. There is a vast literature on clustering problems, see, for example, the books [3, 11, 18] the survey paper [1], and the references there in. Even the simplest clustering problems are known to be NP Hard, including the Euclidean k center problem in the plane [13, 24] In fact, it is NP Hard to approximate the two dimensional k center problem within a factor of 2 even under the L1 metric [12] The greedy ....

....This algorithm requires O(kn) distance computations. The running time was improved by Feder and Greene [12] to O(n log k) for any L p metric. Several efficient algorithms have been developed for Euclidean and rectilinear k center problems when k is small; see the survey by Agarwal and Sharir [1] for a summary of such results. Gonzalez [15] presented an n O(l) time algorithm for the k center problem in R under the L1 metric, when the points lie in a horizontal strip of height l; it can be extended to the L 2 metric as well. Hwang et al. 17] gave an n time algorithm for the ....

P. K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, Tech. Rep. CS-1996-19, Dept. Computer Science, Duke University, 1996.

....in geometric optimization, as it would be an impossible task. Additional applications of randomization in geometric optimization include volume estimation of convex bodies [69] learning geometric concepts, and shape matching. Interested readers can find more material on geometric optimization in [10, 31, 90] and on randomized geometric algorithms in various chapters of this book as well as in [157, 139, 51] ....

P. K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, ACM Comput. Surv., 30 (1998), 412--458.

....in each cluster. Given a capacity L, the L capacitated k center problem is to compute a k clustering of the smallest size so that each cluster has at most L points. Previous results. There is a vast literature on clustering problems, see, for example, the books [3, 10, 17] the survey paper [1], and the references therein. Even the simplest clustering problems are known to be NPHard, including the Euclidean k center problem in the plane [12, 23] In fact, it is NP Hard to approximate the 2 dimensional k center problem within a factor smaller than 2 even under the L1 metric [11] The ....

....This algorithm requires O(kn) distance computations. The running time was improved by Feder and Greene [11] to O(n log k) for any L p metric. Several efficient algorithms have been developed for Euclidean and rectilinear k center problems when k is small. See the survey by Agarwal and Sharir [1] for a summary of such results. The algorithm described by Gonzalez [14] can solve the planar k center problem under the L1 metric in n O(l) time, when the points lie in a horizontal strip of height l; it can be extended to higher dimensions. 1 Another commonly used definition of the size of ....

P. K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, ACM Comput. Surveys, (to appear).

....in geometric optimization have been attacked by techniques that reduce the problem to constructing and searching in various substructures of surface arrangements. Hence, the area of geometric optimization is a natural extension, and a good application area, of the study of arrangements. See [24] for a recent survey on geometric optimization. One of the basic techniques for geometric optimization is the parametric searching technique, originally proposed by Megiddo [264] This technique reduces the optimization problem to a decision problem, where one needs to compare the optimal value to ....

....of the decision procedure with the (unknown) optimum value as an input parameter. In most applications, careful implementation of this technique leads to a solution of the optimization problem whose running time is larger than that of the decision algorithm only by a polylogarithmic factor. See [24] for a more detailed survey of parametric searching and its applications. Several alternatives to parametric searching have been developed during the past decade. They use randomization [25, 95, 251] expander graphs [227] and searching in monotone matrices [175] Like parametric searching, all ....

P. K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, ACM Comput. Surv., to appear.

....there are at most L points in each cluster. The capacitated k center problem is to compute a k clustering of the smallest size so that each cluster has at most L points. Previous results. There is a vast literature on clustering problems, see, for example, the books [3, 11, 18] the survey paper [1], and the references there in. Even the simplest clustering problems are known to be NP Hard, including the Euclidean k center problem in the plane [13, 24] In fact, it is NP Hard to approximate the two dimensional k center problem within a factor of 2 even under the L1 metric [12] The greedy ....

....This algorithm requires O(kn) distance computations. The running time was improved by Feder and Greene [12] to O(n log k) for any L p metric. Several efficient algorithms have been developed for Euclidean and rectilinear k center problems when k is small; see the survey by Agarwal and Sharir [1] for a summary of such results. Gonzalez [15] presented an n O(l) time algorithm for the k center problem in R d under the L1 metric, when the points lie in a horizontal strip of height l; it can be extended to the L 2 metric as well. Hwang et al. 17] gave an n O( p k) time algorithm ....

P. K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, Tech. Rep. CS-1996-19, Dept. Computer Science, Duke University, 1996.

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P.K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Computing Surveys, 30(4):412--458, 1998.

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P. K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, ACM Comput. Surveys, (to appear).

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P.J. Agarwal and M. Sharir (1998). Efficient Algorithms for Geometric Optimization. ACM Computing Surveys 30(4), 412-458.

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