## Abstract

In this paper we investigate the upper bounds on the numbers of transitions of minimum and maximum spanning trees (MinST and MaxST for short) for linearly moving points. Here, a transition means a change on the combinatorial structure of the spanning trees. Suppose that we are given a set of n points in d-dimensional space, S={p_{ 1}, p_{ 2}, ... p_{ n} }, and that all points move along different straight lines at different but fixed speeds, i.e., the position of p_{ i} is a linear function of a real parameter t. We investigate the numbers of transitions of MinST and MaxST when t increases from-∞ to +∞. We assume that the dimension d is a fixed constant. Since there are O(n^{ 2}) distances among n points, there are naively O(n^{ 4}) transitions of MinST and MaxST. We improve these trivial upper bounds for L_{ 1} and L_{ ∞} distance metrics. Let k_{ p} (n) (resp.[Figure not available: see fulltext.]) be the number of maximum possible transitions of MinST (resp. MaxST) in L_{ p} metric for n linearly moving points. We give the following results in this paper: κ_{1}(n)=O(n^{ 5/2} α(n)), κ_{ ∞}(n)=O(n^{ 5/2} α(n)),[Figure not available: see fulltext.], and[Figure not available: see fulltext.] where α(n) is the inverse Ackermann's function. We also investigate two restricted cases, i.e., the c-oriented case in which there are only c distinct velocity vectors for moving n points, and the case in which only k points move.

Original language | English |
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Pages (from-to) | 161-176 |

Number of pages | 16 |

Journal | Discrete & Computational Geometry |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1995 Dec |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics