We consider partitions of a point set into two parts, and the lengths of the minimum spanning trees of the original set and of the two parts. If $w(P)$ denotes the length of a minimum spanning tree of $P$, we show that every set $P$ of $n \geq 12$ points admits a bipartition $P= R \cup B$ for which the ratio $\frac{w(R)+w(B)}{w(P)}$ is strictly larger than $1$; and that $1$ is the largest number with this property. Furthermore, we provide a very fast algorithm that computes such a bipartition in $O(1)$ time and one that computes the corresponding ratio in $O(n \log{n})$ time. In certain settings, a ratio larger than $1$ can be expected and sometimes guaranteed. For example, if $P$ is a set of $n$ random points uniformly distributed in $[0,1]^2$ ($n \to \infty$), then for any $\eps>0$, the above ratio in a maximizing partition is at least $\sqrt2 -\eps$ with probability tending to $1$. As another example, if $P$ is a set of $n$ points with spread at most $\alpha \sqrt{n}$, for some constant $\alpha>0$, then the aforementioned ratio in a maximizing partition is $1 + \Omega(\alpha^{-2})$. All our results and techniques are extendable to higher dimensions.

翻译：我们考虑将一个点集划分为两部分，并分析原始点集及其两部分的最小生成树长度。设$w(P)$表示点集$P$的最小生成树长度，我们证明：对于任意满足$n \geq 12$的点集$P$，存在一个二分划分$P= R \cup B$，使得比值$\frac{w(R)+w(B)}{w(P)}$严格大于$1$；并且$1$是具有这一性质的最大数值。此外，我们提出了一种能在$O(1)$时间内计算此类二分划分的极快算法，以及一种能在$O(n \log{n})$时间内计算对应比值的算法。在特定场景下，该比值可能大于$1$，且有时能得到保证。例如，若$P$是由$[0,1]^2$上均匀分布的$n$个随机点构成的集合（$n \to \infty$），则对于任意$\eps>0$，最大化划分中的上述比值以趋于$1$的概率至少为$\sqrt2 -\eps$。另一个例子是：若$P$是$n$个点构成的集合，其展度至多为$\alpha \sqrt{n}$（其中$\alpha>0$为常数），则最大化划分中的上述比值为$1 + \Omega(\alpha^{-2})$。我们所有的结果和方法均可推广至更高维空间。