## Abstract

We give an optimal bound on the number of transitions of the minimum weight base of an integer valued parametric polymatroid. This generalizes and unifies Tamal Dey's O(k^{1/3}n) upper bound on the number of k-sets (and the complexity of the k-level of a straight-line arrangement), David Eppstein's lower bound on the number of transitions of the minimum weight base of a parametric matroid, and also the Θ(kn) bound on the complexity of the at-most-k level (the union of i-levels for i = 1, 2,..., k) of a straight-line arrangement. As applications, we improve Welzl's upper bound on the sum of the complexities of multiple levels, and apply this bound to the number of different equal-sized-bucketings of a planar point set with parallel partition lines. We also consider an application to a special parametric transportation problem.

Original language | English |
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Pages (from-to) | 429-443 |

Number of pages | 15 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 12 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2002 Oct 1 |

## Keywords

- Combinatorial geometry
- K-level
- Matroid

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics