A faster algorithm for the limited-capacity many-to-many point matching in one dimension

Rajabi-Alni, Fatemeh; Bagheri, Alireza

doi:10.48550/arxiv.1904.03015

Cited by 2 publications

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“…In this section, we present an O(n 2 ) algorithm for finding a minimum-cost OMMD between two sets S and T lying on the real line. Our algorithm is based on the algorithm of Rajabi-Alni and Bagheri [11]. We begin with some useful lemmas.…”

Section: Our Algorithm For Ommd Problemmentioning

confidence: 99%

“…Also, assume we have computed C(b i , h) for all 1 ≤ h ≤ k − 1, and now we want to compute C(b i , k). In fact, we use the idea of [11], that is we first insert the point b 1 ∈ A w+1 and determine that whether the point b 1 decreases the cost of the Proof. Note that deg(a h ) > Demand(a h ) implies that a h decreases the cost of the OMMD.…”

Section: Our Algorithm For Ommd Problemmentioning

confidence: 99%

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A Fast and Efficient algorithm for Many-To-Many Matching of Points with Demands in One Dimension

Rajabi-Alni¹,

Bagheri²,

Minaei-Bidgoli³

2019

Preprint

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Given two point sets S and T , we first study the many-to-many matching with demands (MMD) problem, where each point of one set must be matched to at least a given number of the points of the other set. We propose an O n 2 time algorithm for computing a one dimensional MMD (OMMD) of minimum cost, where |S| + |T | = n. In an OMMD problem, the input point sets S and T lie on the real line and the cost of matching a point to another point is equal to the distance between the two points. We also study a generalized version of the MMD problem, the many-to-many matching with demands and capacities (MMDC) problem, that in which each point has a limited capacity in addition to a demand. We give an O(n 2 ) time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.

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Section: Our Algorithm For Ommd Problemmentioning

confidence: 99%

Section: Our Algorithm For Ommd Problemmentioning

confidence: 99%

A Fast and Efficient algorithm for Many-To-Many Matching of Points with Demands in One Dimension

Rajabi-Alni¹,

Bagheri²,

Minaei-Bidgoli³

2019

Preprint

Self Cite

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“…Furthermore, the authors state the task of generalizing their work to this version in the plane as an important open problem, especially since geometry has helped in designing efficient algorithms for the computation of the minimum-weight perfect matching for points in the plane (see, e.g., [17,18]). Several variants of the 1-dimensional many-to-many matching problem are considered in [13,14,15].…”

Section: Introductionmentioning

confidence: 99%

Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

Bandyapadhyay¹,

Maheshwari²,

Smid³

2021

Preprint

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Given two sets S and T of points in the plane, of total size n, a many-to-many matching between S and T is a set of pairs (p, q) such that p ∈ S, q ∈ T and for each r ∈ S ∪ T , r appears in at least one such pair. The cost of a pair (p, q) is the (Euclidean) distance between p and q. In the minimum-cost many-to-many matching problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in O(n 3 ) time. In a more restricted setting where all the points are on a line, the problem can be solved in O(n log n) time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an O(n 2 • poly(log n)) time exact algorithm and an O(n 3/2 • poly(log n)) time (1 + ϵ)-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].

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A faster algorithm for the limited-capacity many-to-many point matching in one dimension

Cited by 2 publications

References 0 publications

A Fast and Efficient algorithm for Many-To-Many Matching of Points with Demands in One Dimension

A Fast and Efficient algorithm for Many-To-Many Matching of Points with Demands in One Dimension

Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

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