Efficient Many-To-Many Matching of Points with Demands in One Dimension

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Bibliographic Details
Main Authors:	Rajabi-Alni, Fatemeh, Minaei-Bidgoli, Behrouz
Format:	Preprint
Published:	2019
Subjects:	Computational Geometry
Online Access:	Acceder al recurso
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author	Rajabi-Alni, Fatemeh Minaei-Bidgoli, Behrouz
author_facet	Rajabi-Alni, Fatemeh Minaei-Bidgoli, Behrouz
contents	Given two point sets $S$ and $T$, the minimum-cost many-to-many matching with demands (MMD) problem is the problem of finding a minimum-cost many-to-many matching between $S$ and $T$ such that each point of $S$ (respectively $T$) is matched to at least a given number of the points of $T$ (respectively $S$). We propose the first $O\left(n^2\right)$-time algorithm for computing a one dimensional MMD (OMMD) of minimum cost between $S$ and $T$, where $\left\|S\right\|+\left\|T\right\|=n$. 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 equals the Euclidean 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 the first $O(n^2)$-time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.
format	Preprint
id	arxiv_https___arxiv_org_abs_1904_05184
institution	arXiv
publishDate	2019
record_format	arxiv
spellingShingle	Efficient Many-To-Many Matching of Points with Demands in One Dimension Rajabi-Alni, Fatemeh Minaei-Bidgoli, Behrouz Computational Geometry Given two point sets $S$ and $T$, the minimum-cost many-to-many matching with demands (MMD) problem is the problem of finding a minimum-cost many-to-many matching between $S$ and $T$ such that each point of $S$ (respectively $T$) is matched to at least a given number of the points of $T$ (respectively $S$). We propose the first $O\left(n^2\right)$-time algorithm for computing a one dimensional MMD (OMMD) of minimum cost between $S$ and $T$, where $\left\|S\right\|+\left\|T\right\|=n$. 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 equals the Euclidean 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 the first $O(n^2)$-time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.
title	Efficient Many-To-Many Matching of Points with Demands in One Dimension
topic	Computational Geometry
url	https://arxiv.org/abs/1904.05184

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