Minimum perimeter-sum partitions in the plane

Mikkel Abrahamsen; Mark de Berg; Kevin Buchin; Mehran Mehr; Ali D. Mehrabi

doi:10.1007/s00454-019-00059-0

Minimum perimeter-sum partitions in the plane

Mikkel Abrahamsen (Corresponding author), Mark de Berg, Kevin Buchin, Mehran Mehr, Ali D. Mehrabi

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Abstract

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P₁ and P₂ such that the sum of the perimeters of CH(P1) and CH(P2) is minimized, where CH(Pi) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n²) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(nlog ²n) time and a (1 + ε) -approximation algorithm running in O(n+ 1 / ε²· log ²(1 / ε)) time.

Original language	English
Pages (from-to)	483-505
Number of pages	23
Journal	Discrete and Computational Geometry
Volume	63
Issue number	2
Early online date	1 Jan 2019
DOIs	https://doi.org/10.1007/s00454-019-00059-0
Publication status	Published - 1 Mar 2020

Keywords

Clustering
Computational geometry
Convex hull
Minimum-perimeter partition

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10.1007/s00454-019-00059-0

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AU - Abrahamsen, Mikkel

AU - de Berg, Mark

AU - Buchin, Kevin

AU - Mehr, Mehran

AU - Mehrabi, Ali D.

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AB - Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P1 and P2 such that the sum of the perimeters of CH(P1) and CH(P2) is minimized, where CH(Pi) denotes the convex hull of Pi. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(nlog 2n) time and a (1 + ε) -approximation algorithm running in O(n+ 1 / ε2· log 2(1 / ε)) time.

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Minimum perimeter-sum partitions in the plane

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