Expected Length of the Euclidean Minimum Spanning Tree and 1-norms of Chromatic Persistence Diagrams in the Plane

2025-10-27

AI Summary

This paper investigates the asymptotic constant (c) governing the expected length of the Euclidean minimum spanning tree (EMST) of (n) independent, uniformly random points in the unit square, as well as the asymptotic behavior of the 1-norm of the colored persistent diagram for a corresponding bichromatic point set. Methodologically, it introduces a novel filtration based on sublevel sets of a bichromatic distance function, thereby establishing a theoretical link between EMST length and the colored persistent 1-norm; it further integrates stochastic geometry, persistent homology, and probabilistic analysis via inclusion relations among monochromatic and bichromatic sublevel sets. Key contributions include: (i) improving the lower bound on the EMST expected length from (0.6008) to (0.6289); and (ii) determining, for the first time, the asymptotic constant for the colored persistent 1-norm—thereby revealing a deep asymptotic equivalence between these two quantities under random bichromatic sampling.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Data transparency and provenance

Abstract

Let $c$ be the constant such that the expected length of the Euclidean minimum spanning tree of $n$ random points in the unit square is $c sqrt{n}$ in the limit, when $n$ goes to infinity. We improve the prior best lower bound of $0.6008 leq c$ by Avram and Bertsimas to $0.6289 leq c$. The proof is a by-product of studying the persistent homology of randomly $2$-colored point sets. Specifically, we consider the filtration induced by the inclusions of the two mono-chromatic sublevel sets of the Euclidean distance function into the bi-chromatic sublevel set of that function. Assigning colors randomly, and with equal probability, we show that the expected $1$-norm of each chromatic persistence diagram is a constant times $sqrt{n}$ in the limit, and we determine the constant in terms of $c$ and another constant, $c_L$, which arises for a novel type of Euclidean minimum spanning tree of $2$-colored point sets.

Problem

• Improving lower bound for Euclidean minimum spanning tree length constant
• Studying persistent homology of randomly colored point sets
• Calculating expected norms of chromatic persistence diagrams

Innovation

• Improved lower bound for Euclidean MST length constant
• Used persistent homology of randomly colored point sets
• Analyzed chromatic persistence diagrams 1-norm asymptotics