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

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.