Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs

2026-08-27

AI Summary

该研究解决了多色点集的欧几里得最小生成树问题，通过引入月牙状结构，并证明了随机情况下其期望成本收敛于与点数平方根成比例的常数。

Abstract

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a \emph{lune}, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its \emph{cost} is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

Problem

• Euclidean Minimum Spanning Tree
• lunar EMST
• expected cost

Innovation

• lunar EMST
• lune
• expected cost
• random coloring
• convergence