平面中欧几里得最小生成树的月球推广及其期望代价
Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs
AI总结：
本文针对带s+1种颜色的平面随机点，推广欧几里得最小生成树为月球EMST，证明其期望代价相关常数存在，该代价随n趋于无穷时为常数乘√n，精确值仍未知。
AI中文摘要：
受近期色持续同调引入的启发，我们将平面内n个点的欧几里得最小生成树（EMST）推广为适用于点具有s+1种颜色的月球EMST。将两两不同颜色的点为中心、半径均为r的s+1个圆盘的交集称为“月形”，该推广后的EMST反映了当r从0变化到∞时月形并集的演化过程，其“代价”为树的弧与节点形成时两个半径差值的两倍。若点均匀随机选取于[0,1]^2且颜色随机分配，当n趋于无穷时，期望代价收敛于某个依赖于s的常数乘以√n。本文的主要贡献在于证明了该常数的存在性，不过与经典EMST的情况类似，其精确值仍难以确定。
英文摘要：
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.
