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

Ondřej Draganov, Herbert Edelsbrunner, Sophie Rosenmeier, Morteza Saghafian

TL;DR

The paper analyzes the planar EMST constant $c_c$ via a Poisson point process and the Delaunay mosaic, establishing a new lower bound $0.6289 \le c_c$ by comparing EMST length to sums of radii of critical radius-function edges and triangles, aided by the pairing of critical events. It then connects these geometric insights to chromatic persistent homology for randomly colored point sets, defining the lunar EMST and showing its cost equals twice the $1$-norm of the degree-1 chromatic persistence diagram, yielding constants dependent on $c_c$ and $c_L c_L$. An eleven-constant framework for the expected $1$-norms of the six diagrams across two colors is developed, with rigorous bounds and relations derived from short exact sequences. Computational experiments on unit square and torus topologies corroborate the analytic bounds, suggesting $c_c \approx 0.646$–$0.648$ and $c_L c_L \approx 0.350$–$0.352$, thereby refining the understanding of EMST length and its topological-data-analysis connections.

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.

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

TL;DR

The paper analyzes the planar EMST constant via a Poisson point process and the Delaunay mosaic, establishing a new lower bound by comparing EMST length to sums of radii of critical radius-function edges and triangles, aided by the pairing of critical events. It then connects these geometric insights to chromatic persistent homology for randomly colored point sets, defining the lunar EMST and showing its cost equals twice the -norm of the degree-1 chromatic persistence diagram, yielding constants dependent on and . An eleven-constant framework for the expected -norms of the six diagrams across two colors is developed, with rigorous bounds and relations derived from short exact sequences. Computational experiments on unit square and torus topologies corroborate the analytic bounds, suggesting – and –, thereby refining the understanding of EMST length and its topological-data-analysis connections.

Abstract

Let be the constant such that the expected length of the Euclidean minimum spanning tree of random points in the unit square is in the limit, when goes to infinity. We improve the prior best lower bound of by Avram and Bertsimas to . The proof is a by-product of studying the persistent homology of randomly -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 -norm of each chromatic persistence diagram is a constant times in the limit, and we determine the constant in terms of and another constant, , which arises for a novel type of Euclidean minimum spanning tree of -colored point sets.

Paper Structure

This paper contains 19 sections, 6 theorems, 27 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Poisson--Delaunay Mosaics
  3. Critical Edges and Triangles
  4. The Poisson Point Process Near the Boundary
  5. The Uniform Distribution Near the Boundary
  6. Euclidean Minimum Spanning Trees
  7. Expected Length
  8. Improved Lower Bound for Asymptotic Constant
  9. Points with Two Colors
  10. Persistent Homology
  11. Chromatic Persistent Homology
  12. The Lunar EMST
  13. Expected 1-norms
  14. The Asymptotic Constants
  15. Comparisons
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $A$ be a stationary Poisson point process with intensity $n > 0$ in ${\mathbb R}{\hbox{${\mathbb R}$}}^2$, and ${\rm Del}{({A})}{\hbox{${\rm Del}{({A})}$}}$ its Delaunay mosaic. Then for every $r_0 > 0$, we have in which $x = n \pi r_0^2$ and $\gamma(k,x) = \int_{t=0}^x t^{k-1} e^{-t} \,{\hbox{$\,$}}{\rm d} t$ is the incomplete gamma function.

Figures (3)

  • Figure 1: A finite set of points in ${\mathbb R}{\hbox{${\mathbb R}$}}^2$, the Delaunay mosaic (all edges), the Gabriel graph (dashed and solid edges), and the Euclidean minimum spanning tree (solid edges). Except for the dotted ones, all edges are critical, the solid ones give death, and the dashed ones give birth.
  • Figure 2: Left panel: the dependence of six constants on $c{\hbox{$c$}}$, which lies somewhere between $0.6289$ and $0.7072$. Within this vertical strip, the affine maps are totally ordered. Right panel: the zero-sets of the two constants that depend on $c{\hbox{$c$}}$ as well as $c_L{\hbox{$c_L$}}$. The non-negativity of these constants and the known bounds on $c{\hbox{$c$}}$ imply the restriction of the possible pairs $(c{\hbox{$c$}}, c_L{\hbox{$c_L$}})$ to the blue quadrangular region.
  • Figure 3: The chromatic persistence diagrams organized in a snake-like fashion, (c)-(b)-(a)-(d)-(e)-(f)-(i)-(h)-(g)-(j)-(k), with relations $\textsc{(c)=(b)}$, $\textsc{(a)=(b)+(d)}$, $\textsc{(e)=(d)+(f)}$, $\textsc{(i)=(f)+(h)}$, $\textsc{(g)=(h)+(j)}$, $\textsc{(k)=(j)}$ implied by short exact sequences. Each plot shows the best-fit curve for the average $1$-norms with the number of points increasing from left to right (blue for unit square and orange for torus). As illustrated by the colored strips along the fitting curves, the standard deviation is consistently small. In six of the eleven cases, we calculate the implied estimate of $c{\hbox{$c$}}$, and in three the implied estimate of $c_L{\hbox{$c_L$}}$.

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 5.1
  • proof
  • ...and 2 more