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

TL;DR

This paper improves the lower bound for the expected length constant of Euclidean minimum spanning trees in the plane and explores the expected 1-norms of chromatic persistence diagrams in the context of random 2-colored point sets.

Contribution

It provides a tighter lower bound for the Euclidean MST length constant and introduces a novel analysis of chromatic persistence diagrams related to random 2-colored point sets.

Findings

Lower bound for c improved to 0.6289

Expected 1-norm of chromatic persistence diagrams scales as constant times √n

Introduces a new type of Euclidean minimum spanning tree for 2-colored points

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.