On minimum spanning tree-like metric spaces

Momoko Hayamizu*, Kenji Fukumizu

*この研究の対応する著者

研究成果: Article査読

3 被引用数 (Scopus)

抄録

We attempt to shed new light on the notion of ‘tree-like’ metric spaces by focusing on an approach that does not use the four-point condition. Our key question is: Given metric space M on n points, when does a fully labelled positive-weighted tree T exist on the same n vertices that precisely realises M using its shortest path metric? We prove that if a spanning tree representation, T, of M exists, then it is isomorphic to the unique minimum spanning tree in the weighted complete graph associated with M, and we introduce a fourth-point condition that is necessary and sufficient to ensure the existence of T whenever each distance in M is unique. In other words, a finite median graph, in which each geodesic distance is distinct, is simply a tree. Provided that the tie-breaking assumption holds, the fourth-point condition serves as a criterion for measuring the goodness-of-fit of the minimum spanning tree to M, i.e., the spanning tree-likeness of M. It is also possible to evaluate the spanning path-likeness of M. These quantities can be measured in O(n4) and O(n3) time, respectively.

本文言語English
ページ(範囲)51-57
ページ数7
ジャーナルDiscrete Applied Mathematics
226
DOI
出版ステータスPublished - 2017 7月 31
外部発表はい

ASJC Scopus subject areas

  • 離散数学と組合せ数学
  • 応用数学

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