[Paper Review] Calculation of Minimum Spanning Tree Edges Lengths using Gromov--Hausdorff Distance

Alexey Avgustinovich Tuzhilin|arXiv (Cornell University)|May 5, 2016

Topological and Geometric Data Analysis5 references6 citations

TL;DR

This paper establishes a novel connection between minimum spanning tree (MST) edge lengths in finite metric spaces and Gromov–Hausdorff distances to simplices of large diameter. By leveraging a minimax formula analogous to eigenvalue computation, it derives exact formulas for MST length, Steiner minimal tree length, and minimal filling length through optimization of Gromov–Hausdorff distances to scaled simplices.

ABSTRACT

In the present paper we show how one can calculate the lengths of edges of a minimum spanning tree constructed for a finite metric space, in terms of the Gromov-Hausdorff distances from this space to simplices of sufficiently large diameter. Here by simplices we mean finite metric spaces all of whose nonzero distances are the same. As an application, we reduce the problems of finding a Steiner minimal tree length or a minimal filling length to maximization of the total distance to some finite number of simplices considered as points of the Gromov-Hausdorff space.

Motivation & Objective

• To establish a mathematical relationship between the edge lengths of a minimum spanning tree (MST) in a finite metric space and the Gromov–Hausdorff distance to simplices.
• To provide a new computational framework for MST, Steiner minimal tree (SMT), and minimal filling (MF) lengths using Gromov–Hausdorff distances.
• To demonstrate that these geometric optimization problems can be reduced to maximizing total distances to a finite set of simplex points in the Gromov–Hausdorff space.
• To formalize a minimax principle for MST edge lengths analogous to spectral theory for self-adjoint operators.
• To extend the framework to compute SMT and MF lengths by optimizing over ambient or super-structures containing the input point set.

Proposed method

• Uses the Gromov–Hausdorff distance $ d_{GH}(X, riangle_{k+1}) $ between a finite metric space $ X $ and a simplex $ riangle_{k+1} $ of diameter 1.
• Applies the distortion formula for correspondences: $ d_{GH}(X,Y) = \frac{1}{2} \inf_R \operatorname{dis}(R) $, where $ R $ ranges over correspondences between $ X $ and $ \triangle_{k+1} $.
• Characterizes optimal correspondences $ R \in \mathcal{R}_{\text{opt}}(\triangle_{k+1}, X) $ that induce partitions $ \{X_1, \dots, X_{k+1}\} $ of $ X $, leading to $ \operatorname{dis}(R) = 1 - \alpha(\{X_i\}) $.
• Derives $ 2d_{GH}(\triangle_{k+1}, X) = 1 - \sigma_k $, where $ \sigma_k $ is the maximum inter-cluster distance over all $ (k+1) $-partitions of $ X $.
• Scales the simplex via $ \lambda \geq 2\operatorname{diam}X $, yielding $ \sigma_k = \lambda - 2d_{GH}(X, \lambda\triangle_{k+1}) $.
• Sums over $ k $ to obtain the MST length: $ \operatorname{mst}(X) = \sum_{k=1}^{n-1} \left[ \lambda - 2d_{GH}(X, \lambda\triangle_{k+1}) \right] $, valid for $ \lambda \geq 2\operatorname{diam}X $.

Experimental results

Research questions

• RQ1 Can the lengths of edges in a minimum spanning tree of a finite metric space be expressed in terms of Gromov–Hausdorff distances to simplices?
• RQ2 Is there a minimax formula for MST edge lengths analogous to eigenvalue computation in spectral theory?
• RQ3 Can the length of a Steiner minimal tree be computed via optimization of Gromov–Hausdorff distances to scaled simplices?
• RQ4 Can the minimal filling length of a finite metric space be reduced to a maximization problem over Gromov–Hausdorff distances?
• RQ5 What is the role of the diameter scaling parameter $ \lambda $ in stabilizing the Gromov–Hausdorff distance-based computation of geometric invariants?

Key findings

• The length of the minimum spanning tree on a finite metric space $ X $ is given by $ \operatorname{mst}(X) = \lambda(n-1) - 2\sum_{k=1}^{n-1} d_{GH}(X, \lambda\triangle_{k+1}) $, valid for any $ \lambda \geq 2\operatorname{diam}X $.
• For each $ k $, the $ k $-th largest MST edge length corresponds to $ \lambda - 2d_{GH}(X, \lambda\triangle_{k+1}) $, with $ \sigma_k = \lambda - 2d_{GH}(X, \lambda\triangle_{k+1}) $ being the $ k $-th largest inter-partition distance.
• The Steiner minimal tree length satisfies $ \operatorname{smt}(M,X) = \inf \left\{ \sum_{k=1}^\infty \left[ \lambda - 2d_{GH}(V, \lambda\triangle_{k+1}) \right] : V \in \mathcal{M}(M,X,d) \right\} $, where $ d > \operatorname{smt}(M,X) $ and $ \lambda \geq 2d $.
• The minimal filling length of a finite metric space $ M $ is given by $ \operatorname{mf}(M) = \inf \left\{ \sum_{k=1}^\infty \left[ \lambda - 2d_{GH}(V, \lambda\triangle_{k+1}) \right] : V \in \mathcal{M}(M,d) \right\} $, for $ d > \operatorname{mf}(M) $ and $ \lambda \geq 2d $.
• For $ m > n $, $ d_{GH}(X, \lambda\triangle_m) = \lambda/2 $, so $ \lambda - 2d_{GH}(X, \lambda\triangle_m) = 0 $, ensuring convergence of the infinite sum in the MST formula.
• The method reduces the computation of complex geometric invariants (MST, SMT, MF) to optimization over Gromov–Hausdorff distances to a finite set of simplex configurations.