## Abstract

This paper presents a new method of partition, named π-splitting, of a point set in d-dimensional space. Given a point G in a d-dimensional simplex T, T(G;i) is the subsimplex spanned by G and the ith facet of T. Let S be a set of n points in T, and let π be a sequence of nonnegative integers π_{1}, ..., n_{d+1} satisfying σ_{i=1}^{d+1}π_{1}=n The π-splitter of (T, S) is a point G in T such that T(G;i) contains at least π_{i} points of S in its closure for every i=1, 2, ..., d + 1. The associated dissection is the re-splitting. The existence of a π-splitting is shown for any (T, S) and π, and two efficient algorithms for finding such a splitting are given. One runs in O(d^{2}n log n + d^{3}n) time, and the other runs in O(n) time if the dimension d can be considered as a constant. Applications of re-splitting to mesh generation, polygonal-tour generation, and a combinatorial assignment problem are given.

Original language | English |
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Pages (from-to) | 649-668 |

Number of pages | 20 |

Journal | Algorithmica |

Volume | 9 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1993 Jun 1 |

Externally published | Yes |

## Keywords

- Assignment problem
- Computational geometry
- Partition of point sets

## ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics