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An \(O(n^2)\) Algorithm for the Limited-Capacity Many-to-Many Point Matching in One Dimension

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Abstract

Given two point sets S and T, in a many-to-many matching between S and T each point in S is assigned to one or more points in T and vice versa. A generalization of the many-to-many matching problem is the limited capacity many-to-many matching problem, where the number of points that can be matched to each point (the capacity of each point) is limited. In this paper, we provide an \(O\left( n^2\right) \) time algorithm for the one dimensional minimum-cost limited capacity many-to-many matching problem, where \(\left| S\right| +\left| T\right| =n\). Our algorithm improves the best previous time complexity of \(O(kn^2)\), that in which k is the largest capacity of the points in \(S \cup T\). In this problem, both S and T lie on the real line and the cost of matching \(s \in S\) to \(t \in T\) is equal to the distance between s and t.

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Correspondence to Fatemeh Rajabi-Alni.

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Rajabi-Alni, F., Bagheri, A. An \(O(n^2)\) Algorithm for the Limited-Capacity Many-to-Many Point Matching in One Dimension. Algorithmica 76, 381–400 (2016). https://doi.org/10.1007/s00453-015-0044-4

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Keywords

  • Many-to-many point matching
  • One dimensional point-matching
  • Limited capacity point matching