TY - JOUR

T1 - On minimum spanning tree-like metric spaces

AU - Hayamizu, Momoko

AU - Fukumizu, Kenji

N1 - Funding Information:
This work was supported by Japan Science and Technology Agency (JST) PRESTO ‘Collaborative Mathematics for Real World Issues’ Grant Number JPMJPR16EB and by JSPS KAKENHI Grant Number 25120012 and 26280009. The authors thank the anonymous reviewer for helpful suggestions. Thanks are extended to Katsutoshi Shinohara for his constructive criticism on an earlier version of the manuscript and Yoshimasa Uematsu for his insights concerning the concept of median.
Publisher Copyright:
© 2017 The Authors

PY - 2017/7/31

Y1 - 2017/7/31

N2 - 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.

AB - 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.

KW - Median graph

KW - Minimum spanning tree

KW - Tree-like metric space

UR - http://www.scopus.com/inward/record.url?scp=85018268133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018268133&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2017.04.001

DO - 10.1016/j.dam.2017.04.001

M3 - Article

AN - SCOPUS:85018268133

VL - 226

SP - 51

EP - 57

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -