### Abstract

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_{1} and P_{2} 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(n^{2}) 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 ^{2}n) time and a (1 + ε) -approximation algorithm running in O(n+ 1 / ε^{2}· log ^{2}(1 / ε)) time.

Original language | English |
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Number of pages | 23 |

Journal | Discrete and Computational Geometry |

DOIs | |

Publication status | E-pub ahead of print - 1 Jan 2019 |

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### Keywords

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

### Cite this

*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-019-00059-0

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*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-019-00059-0

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

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Minimum perimeter-sum partitions in the plane

AU - Abrahamsen, Mikkel

AU - de Berg, Mark

AU - Buchin, Kevin

AU - Mehr, Mehran

AU - Mehrabi, Ali D.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

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.

KW - Clustering

KW - Computational geometry

KW - Convex hull

KW - Minimum-perimeter partition

UR - http://www.scopus.com/inward/record.url?scp=85061204117&partnerID=8YFLogxK

U2 - 10.1007/s00454-019-00059-0

DO - 10.1007/s00454-019-00059-0

M3 - Article

AN - SCOPUS:85061204117

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -