Minimum spanning paths and Hausdorff distance in finite ultrametric spaces

TL;DR

This paper shows that minimum spanning paths can always represent finite ultrametric spaces uniquely and simply, providing new tools for their analysis and an explicit formula for Hausdorff distance.

Contribution

It introduces minimum spanning paths as canonical structures for finite ultrametric spaces and derives an explicit formula for Hausdorff distance in this context.

Findings

Minimum spanning paths uniquely define finite ultrametric spaces.

Explicit formula for Hausdorff distance in finite ultrametric spaces.

Minimum spanning paths can be used to compute Hausdorff distance.

Abstract

It is shown that a minimum weight spanning tree of a finite ultrametric space can be always found in the form of path. As a canonical representing tree such path uniquely defines the whole space and, moreover, it has much more simple structure. Thus, minimum spanning paths are a convenient tool for studying finite ultrametric spaces. To demonstrate this we use them for characterization of some known classes of ultrametric spaces. The explicit formula for Hausdorff distance in finite ultrametric spaces is also found. Moreover, the possibility of using minimum spanning paths for finding this distance is shown.