Linear-Time Algorithms for Linear Programming in $R^3 $ and Related Problems

Megiddo, Nimrod

doi:10.1137/0212052

Cited by 752 publications

(353 citation statements)

References 18 publications

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“…The algorithm is asymptotically optimal, and therefore improves the current best result on trees, given by Bhattacharya et al [6]. The strategy of our algorithm is prune-and-search, which is widely applied in solving distance-related problems [10,14]. The rest of this paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Fortunately, due to the strict quasiconvexity of ψ 1 and the piecewise linearity of E L and E R , one can apply the strategy of prune-and-search [10,14] to obtain an optimal solution in linear time. The quasiconvexity of a function f implies that a local minimum of f is the global minimum of f , and the idea of the prune-and-search algorithm is to search the local minimum over an interval [λ 1 , λ 2 ], which is guaranteed to contain the solution.…”

Section: A Linear Time Algorithmmentioning

confidence: 99%

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An Optimal Algorithm for the Weighted Backup 2-Center Problem on a Tree

Wang

2015

Algorithmica

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Abstract. In this paper, we are concerned with the weighted backup 2-center problem on a tree. The backup 2-center problem is a kind of center facility location problem, in which one is asked to deploy two facilities, with a given probability to fail, in a network. Given that the two facilities do not fail simultaneously, the goal is to find two locations, possibly on edges, that minimize the expected value of the maximum distance over all vertices to their closest functioning facility. In the weighted setting, each vertex in the network is associated with a nonnegative weight, and the distance from vertex u to v is weighted by the weight of u. With the strategy of prune-and-search, we propose a linear time algorithm, which is asymptotically optimal, to solve the weighted backup 2-center problem on a tree.

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Section: Introductionmentioning

confidence: 99%

Section: A Linear Time Algorithmmentioning

confidence: 99%

An Optimal Algorithm for the Weighted Backup 2-Center Problem on a Tree

Wang

2015

Algorithmica

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“…For weighted data, for arbitrary dags the fastest known approach for computing either is to first determine the transitive closure. Given the transitive closure, Prefix can easily be determined in Θ(n 2 ) time since pre(v) only involves predecessors of v. The combination of predecessors and successors used in Basic makes it more complicated, though it too can be determined in Θ(n 2 ) time by using an approach described in [24].…”

Section: ∞ Isotonic Regressionmentioning

confidence: 99%

Isotonic Regression for Multiple Independent Variables

Stout

2013

Algorithmica

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This paper gives algorithms for determining isotonic regressions for weighted data at a set of points P in multidimensional space with the standard componentwise ordering. The approach is based on an orderpreserving embedding of P into a slightly larger directed acyclic graph (dag) G, where the transitive closure of the ordering on P is represented by paths of length 2 in G. Algorithms are given which, compared to previous results, improve the time by a factor ofΘ(|P |) for the L 1 , L 2 , and L ∞ metrics. A yet faster algorithm is given for L 1 isotonic regression with unweighted data. L ∞ isotonic regression is not unique, and algorithms are given for finding L ∞ regressions with desirable properties such as minimizing the number of large regression errors.

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“…Standard combinatorial algorithms for linear programming, including algorithms for linear programming in small dimensions [8,11,18,19,29,30] as well as the simplex method (cf., [7]), can generate a basic infeasible subsystem given an infeasible linear program. The details of finding a basic infeasible subsystem using linear programming are discussed further in Appendix A.…”

Section: An Algorithm For Points In R Dmentioning

confidence: 99%

Output-sensitive algorithms for Tukey depth and related problems

Bremner

Chen

Iacono³

et al. 2008

Stat Comput

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The Tukey depth (Tukey 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.

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Linear-Time Algorithms for Linear Programming in $R^3 $ and Related Problems

Cited by 752 publications

References 18 publications

An Optimal Algorithm for the Weighted Backup 2-Center Problem on a Tree

An Optimal Algorithm for the Weighted Backup 2-Center Problem on a Tree

Isotonic Regression for Multiple Independent Variables

Output-sensitive algorithms for Tukey depth and related problems

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