Minimum perimeter-sum partitions in the plane

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

doi:10.4230/LIPIcs.SoCG.2017.4

Minimum perimeter-sum partitions in the plane

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

Department of Computer Science

3 Citations (Scopus)

39 Downloads (Pure)

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(P_i) 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(n log⁴ n) time and a (1 + ϵ)-approximation algorithm running in O(n + 1/ϵ² · log⁴ (1/ϵ)) time.

Original language	English
Title of host publication	33rd International Symposium on Computational Geometry (SoCG 2017)
Editors	Boris Aronov, Matthew J. Katz
Number of pages	15
Publisher	Schloss Dagstuhl - Leibniz-Zentrum für Informatik
Publication date	2017
Article number	4
ISBN (Electronic)	978-3-95977-038-5
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2017.4
Publication status	Published - 2017
Event	33rd International Symposium on Computational Geometry - Brisbane, Australia Duration: 4 Jul 2017 → 7 Jul 2017 Conference number: 33

Conference

Conference	33rd International Symposium on Computational Geometry
Number	33
Country/Territory	Australia
City	Brisbane
Period	04/07/2017 → 07/07/2017

Series	Leibniz International Proceedings in Informatics
Volume	77
ISSN	1868-8969

Keywords

Clustering
Computational geometry
Convex hull
Minimum-perimeter partition

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Abrahamsen_2017_Minimum_perimeter-sumFinal published version, 652 KBLicence: CC BY

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Minimum perimeter-sum partitions in the plane. / Abrahamsen, Mikkel; de Berg, Mark; Buchin, Kevin et al.
33rd International Symposium on Computational Geometry (SoCG 2017). ed. / Boris Aronov; Matthew J. Katz. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. 4 (Leibniz International Proceedings in Informatics, Vol. 77).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Abrahamsen, M, de Berg, M, Buchin, K, Mehr, M & Mehrabi, AD 2017, Minimum perimeter-sum partitions in the plane. in B Aronov & MJ Katz (eds), 33rd International Symposium on Computational Geometry (SoCG 2017)., 4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Leibniz International Proceedings in Informatics, vol. 77, 33rd International Symposium on Computational Geometry, Brisbane, Queensland, Australia, 04/07/2017. https://doi.org/10.4230/LIPIcs.SoCG.2017.4

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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(n log4 n) time and a (1 + ϵ)-approximation algorithm running in O(n + 1/ϵ2 · log4 (1/ϵ)) time.

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Minimum perimeter-sum partitions in the plane

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