Minimum perimeter-sum partitions in the plane

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

Research output: Contribution to journalArticleAcademicpeer-review

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 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.

Original languageEnglish
Number of pages23
JournalDiscrete and Computational Geometry
DOIs
Publication statusE-pub ahead of print - 1 Jan 2019

Fingerprint

Perimeter
Partition
Approximation algorithms
Exact Algorithms
Pi
Convex Hull
Approximation Algorithms
Partitioning
Denote
Subset

Keywords

  • Clustering
  • Computational geometry
  • Convex hull
  • Minimum-perimeter partition

Cite this

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title = "Minimum perimeter-sum partitions in the plane",
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 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. / Abrahamsen, Mikkel (Corresponding author); de Berg, Mark; Buchin, Kevin; Mehr, Mehran; Mehrabi, Ali D.

In: Discrete and Computational Geometry, 01.01.2019.

Research output: Contribution to journalArticleAcademicpeer-review

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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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