Efficient Minimum Cost Matching and Transportation Using the Quadrangle Inequality

Aggarwal, Alok; Bar-Noy, Amotz; Khuller, Samir; Kravets, Dina; Schieber, Baruch

doi:10.1006/jagm.1995.1030

Cited by 22 publications

(22 citation statements)

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“…Efficient algorithms for searching for minima in Monge arrays have been developed previously. See, for example, [1,3].…”

Section: The Ideamentioning

confidence: 99%

“…An example of a bipartite graph with a distance function that has the Monge property is the following. Consider a bipartite graph ´ µ with the distance function ¢ Ê satisfying the condition ´Ù Ûµ · ´Ú Ü µ ´Ù Üµ · ´Ú Û µ (1) for all nodes Ù Ú ¾ , and Û Ü ¾ . We note that for any right-matching Å there exists a Monge right-matching Å ¼ having no greater cost.…”

Section: Monge Searching Data Structuresmentioning

confidence: 99%

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Planar graphs, negative weight edges, shortest paths, and near linear time

Fakcharoenphol

Rao

2001

Proceedings 42nd IEEE Symposium on Foundations of Computer Science

193

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In this paper, we present an Ç´Ò ÐÓ ¿ Òµ time algorithm for finding shortest paths in a planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by Lipton, Rose, and Tarjan in 1978 which ran in Ç´Ò ¿ ¾ µ time, and the best polynomial algorithm developed by Henzinger, Klein, Subramanian, and Rao in 1994 which ran in Ç´Ò ¿ µ time.We also present significantly improved algorithms for query and dynamic versions of the shortest path problems. Ô Òµ 1 Their algorithm, however, depended on the

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“…Efficient algorithms for searching for minima in Monge arrays have been developed previously. See, for example, [1,3].…”

Section: The Ideamentioning

confidence: 99%

Section: Monge Searching Data Structuresmentioning

confidence: 99%

Planar graphs, negative weight edges, shortest paths, and near linear time

Fakcharoenphol

Rao

2001

Proceedings 42nd IEEE Symposium on Foundations of Computer Science

193

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“…1. link(R, node i, node j, real x)-The operation assumes that node i is a tree root, and that node i and node j belong to different trees in R. The operation combines the trees containing i and j by adding arc (i, j) with real value x and making j the parent of i. W i , Increment i , and r p(i),i for i ∈ [2, n] can be computed in a straightforward manner in O(n) time.…”

Section: The Convex Cost Flow Problem On a Treementioning

confidence: 97%

“…This improves on the O(n log n) run-time for the transportation problem on a line or a circle (Aggarwal et al [1]) that uses a priority queue (such as a red-black tree). It also almost matches the O(n) run-time for the many-to-one matching problem on a line (Colannino et al [11]) when the numbers sorted are integers less than some polynomial in n (the many-to-one matching problem on a circle that has not been studied previously).…”

Section: Introductionmentioning

confidence: 97%

Fast algorithms for convex cost flow problems on circles, lines, and trees

Orlin

Vaidyanathan

2013

Networks

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We develop efficient algorithms to solve convex cost flow problems where the underlying graph is a circle, a line, or a tree. Each node i has an associated supply/demand b(i ). The cost of sending flow on arc (i , j ) is a piecewise linear convex function f ij defined over R. Let n be the number of nodes and m = O(n) be the total number of pieces of all the convex functions. A flow x is feasible if the imbalances on all nodes are nonnegative. Excess e i (x ) stored on node i has an associated linear cost c i × e i (x ). We show that the problem on a circle can be transformed into an equivalent problem on a line in O(n) time. Thereafter, we develop an algorithm that solves the problem on a line in O(sort(n) + nα(n)) time, where sort(n) is the time to sort n real numbers and α(n) is the inverse Ackermann function. We also prove that when the nodes lie on a tree, the problem can be solved in O(n log n) time using the dynamic tree data structure. We describe applications in areas such as distributed computing, lot-sizing, computational biology, computational music, and transportation.

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“…The algorithm is based on a reduction to the problem of computing the single source shortest path problem in a weighted directed acyclic graph. The construction is inspired by the graph-theoretic approach of Eiter and Mannila, and relies heavily on the following lemma, sometimes called the quadrangle inequality [1]. …”

Section: The New O(n 2 ) Algorithmmentioning

confidence: 99%

An algorithm for computing the restriction scaffold assignment problem in computational biology

Colannino

Toussaint

2005

Information Processing Letters

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Let S and T be two finite sets of points on the real line with |S| + |T | = n and |S| > |T |. The restriction scaffold assignment problem in computational biology assigns each point of S to a point of T such that the sum of all the assignment costs is minimized, with the constraint that every element of T must be assigned at least one element of S. The cost of assigning an element s i of S to an element t j of T is |s i −t j |, i.e., the distance between s i and t j . In 2003 Ben-Dor, Karp, Schwikowski and Shamir [2] published an O(n log n) time algorithm for this problem. Here we provide a counter-example to their algorithm and present a new algorithm that runs in O(n 2 ) time, improving the best previous complexity of O(n 3 ).

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Efficient Minimum Cost Matching and Transportation Using the Quadrangle Inequality

Cited by 22 publications

References 0 publications

Planar graphs, negative weight edges, shortest paths, and near linear time

Planar graphs, negative weight edges, shortest paths, and near linear time

Fast algorithms for convex cost flow problems on circles, lines, and trees

An algorithm for computing the restriction scaffold assignment problem in computational biology

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