Simple linear flow decomposition algorithms on trees, circles, and augmented trees

Vaidyanathan, Balachandran

doi:10.1002/net.21470

Cited by 2 publications

(4 citation statements)

References 18 publications

(16 reference statements)

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“…This algorithm, followed by a linear time flow decomposition algorithm (Vaidyanathan [23]), also solves transportation problems, matching problems, and manyto-one matching problems on trees in O(n log n) time. All these problems have not been studied previously.…”

Section: Introductionmentioning

confidence: 99%

“…For the CFPT, our approach is similar and the run‐time of our algorithm is O ( n 2 ) using simple data structures and can be improved to

O (n \log n)

with the use of dynamic trees. Below, we list the contributions of our article: We develop specialized algorithms for convex cost flow problems [with piecewise linear costs and O ( n ) total pieces] on circles and trees that are faster by a factor of n log n or more compared to general purpose algorithms. The algorithm for the CFPC followed by a linear time flow decomposition algorithm (Vaidyanathan ) solves transportation problems and many‐to‐one matching problems on lines and circles in

O (sort (n) + n α (n))

time. This improves on the

O (n \log n)

run‐time for the transportation problem on a line or a circle (Aggarwal et al ) that uses a priority queue (such as a red‐black tree).…”

Section: Introductionmentioning

confidence: 99%

“…Their algorithm runs in

O (n \log n)

time and relies on dynamic trees. The algorithm for the CFPT solves the CFPT in

O (n \log n)

time. This algorithm, followed by a linear time flow decomposition algorithm (Vaidyanathan ), also solves transportation problems, matching problems, and many‐to‐one matching problems on trees in

O (n \log n)

time. All these problems have not been studied previously. We describe new applications of the problems studied.…”

Section: Introductionmentioning

confidence: 99%

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

Orlin

Vaidyanathan

2013

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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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Section: Introductionmentioning

confidence: 99%

“…For the CFPT, our approach is similar and the run‐time of our algorithm is O ( n 2 ) using simple data structures and can be improved to

O (n \log n)

O (sort (n) + n α (n))

time. This improves on the

O (n \log n)

run‐time for the transportation problem on a line or a circle (Aggarwal et al ) that uses a priority queue (such as a red‐black tree).…”

Section: Introductionmentioning

confidence: 99%

“…Their algorithm runs in

O (n \log n)

time and relies on dynamic trees. The algorithm for the CFPT solves the CFPT in

O (n \log n)

time. This algorithm, followed by a linear time flow decomposition algorithm (Vaidyanathan ), also solves transportation problems, matching problems, and many‐to‐one matching problems on trees in

O (n \log n)

time. All these problems have not been studied previously. We describe new applications of the problems studied.…”

Section: Introductionmentioning

confidence: 99%

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

Orlin

Vaidyanathan

2013

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“…Aggarwal et al [1] and Vaidyanathan and Ahuja [19] developed O ( n log n ) algorithms for the transportation problem on a line. A combination of Orlin and Vaidyanathan [14]'s algorithm and Vaidyanathan [18]'s algorithm improves the run‐time for the transportation problem on a line to O (sort ( n ) + n α( n )), where sort( n ) is the time to sort n real numbers and α( n ) is the inverse Ackermann function.…”

Section: Introductionmentioning

confidence: 99%

Faster strongly polynomial algorithms for the unbalanced transportation problem and assignment problem with monge costs

Vaidyanathan

2013

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We consider a transportation problem where the cost matrix satisfies the Monge property. The problem has supply nodes N 1 (|N 1 | = n 1 ), demand nodes N 2 (|N 2 | = n 2 ), supply s i ≥ 0 for i ∈ N 1 , and demand d j ≥ 0 for j ∈ N 2 (n = n 1 + n 2 , m = n 1 n 2 ). When the total supply is equal to the total demand, the problem can be solved in O(n) time using the northwest corner rule (Hoffman [10]). This algorithm and run-time, however, do not extend to the unbalanced case, where the total supply is not equal to the total demand. The fastest strongly polynomial run-time for the unbalanced transportation problem with Monge costs is O(n log n(m + n log n)), and for the unbalanced assignment problem (unit supplies and demands) with Monge costs is O(n(m + n log n)). In this article, we describe a simple algorithm that solves the unbalanced transportation problem with Monge costs in O(mn 1 ) time and the unbalanced assignment problem with Monge costs in O(m) time using elementary data structures. We also develop a faster implementation of the algorithm that utilizes a heap, a self-balancing binary search tree, and dynamic trees, and solves the transportation problem with Monge costs in O(m log n 1 ) time. Our algorithms improve the run-times for: (i) the unbalanced transportation problem with Monge costs by a factor of n log n/ log n 1 or better and (ii) the unbalanced assignment problem with Monge costs by a factor of n or better.

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Simple linear flow decomposition algorithms on trees, circles, and augmented trees

Cited by 2 publications

References 18 publications

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

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

Faster strongly polynomial algorithms for the unbalanced transportation problem and assignment problem with monge costs

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