On minimum spanning tree-like metric spaces

Momoko Hayamizu, Kenji Fukumizu

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)51-57
Number of pages7
JournalDiscrete Applied Mathematics
Volume226
DOIs
Publication statusPublished - 2017 Jul 31
Externally publishedYes

Keywords

  • Median graph
  • Minimum spanning tree
  • Tree-like metric space

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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