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：考虑一个点集划分成两个部分，以及原始点集和两个部分的最小生成树长度。若$w(P)$表示$P$的最小生成树长度，我们证明任意满足$n \geq 12$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$1$，且$1$是具备此性质的最大数值。进一步地，我们提出了一种能在$O(1)$时间内计算该二分划分的极快算法，以及一种在$O(n \log{n})$时间内计算对应比值的算法。在某些设定下，比值大于$1$是可以预期的，有时甚至是可以保证的。例如，若$P$是均匀分布在$[0,1]^2$中的$n$个随机点（$n \to \infty$），则对任意$\eps>0$，最大化划分中的上述比值至少为$\sqrt2 -\eps$且概率趋近于$1$。另一个例子，若$P$是$n$个点，其展布不超过$\alpha \sqrt{n}$（其中$\alpha>0$为常数），则最大化划分中的前述比值为$1 + \Omega(\alpha^{-2})$。我们所有的结果和技术均可推广到更高维空间。