[Paper Review] Trimming of finite metric spaces

Vladimir Turaev|arXiv (Cornell University)|Dec 20, 2016

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper introduces the concept of trim metric spaces and proves that every finite metric space is isometric to the leaf space of a finite metric forest with a trim base. The key contribution is a canonical construction of a trim core for any finite pseudometric space, which characterizes the space up to isometry and enables a universal decomposition into metric forests with trim bases.

ABSTRACT

We define a class of trim metric spaces and show that every finite metric space is the leaf space of a metric forest with trim base.

Motivation & Objective

• To define and characterize trim metric spaces, a class of finite metric spaces where every point lies between two others.
• To develop a canonical construction of a trim core for any finite pseudometric space, generalizing the notion of metric reduction.
• To establish that every finite metric space is isometric to the leaf space of a finite metric forest with a trim base.
• To show that the trim core of a finite pseudometric space is the unique minimal object in a canonical factorization, up to isometry.
• To prove that the leaf space and base of any finite metric forest are trim equivalent, generalizing the core construction to forests.

Proposed method

• Define a pseudometric space as trim if every point lies between two other distinct points, using Menger's notion of betweenness.
• Construct the trim core $ c(X) $ of a finite pseudometric space $ X $ as the quotient of $ X $ under an equivalence relation identifying points that are not between others, with a canonical non-expansive surjection $ q_X: X \to c(X) $.
• Prove functoriality: isometries between pseudometric spaces induce isometries between their trim cores.
• Introduce metric forests as collections of rooted trees with a base metric space, defining their leaf space via a path-length pseudometric across and within components.
• Use the concept of 'drift' and iterative trimming $ t $ to reduce metric forests to their trim bases, showing convergence to the base under repeated application.
• Apply recursive reduction and tree contraction techniques to prove that the trim core of a leaf space equals its base when the base is trim.

Experimental results

Research questions

• RQ1 Can every finite metric space be represented as the leaf space of a metric forest with a trim base?
• RQ2 What is the canonical minimal object (trim core) to which any finite pseudometric space can be reduced via non-expansive maps?
• RQ3 How do the leaf space and base of a finite metric forest relate in terms of isometry and trim equivalence?
• RQ4 Under what conditions does the trim core of a finite pseudometric space equal the original space?
• RQ5 What structural properties of metric forests ensure that their leaf space and base are trim equivalent?

Key findings

• Every finite pseudometric space $ X $ admits a canonical surjective non-expansive map $ q_X: X \to c(X) $ to its trim core $ c(X) $, which is unique up to isometry.
• The trim core $ c(X) $ is a point if and only if $ X $ is isometric to the leaf space of a finite tree.
• For any finite metric forest $ \zeta $ with trim base $ B $, the trim core of its leaf space satisfies $ c(\partial\zeta) = B $, up to isometry.
• The leaf space and base of any finite metric forest are trim equivalent, meaning their trim cores are isometric.
• Iterative application of the trimming operation $ t $ on the leaf space of a metric forest with trim base eventually yields the base itself, proving $ t^{N+1}(\partial\zeta) = B $ for some $ N \geq 1 $.
• The trim core construction is functorial: isometries between pseudometric spaces induce isometries between their trim cores, preserving the canonical factorization.