Two trees are better than one
Adrian Dumitrescu, János Pach, Géza Tóth
TL;DR
This paper studies how partitioning a planar point set into two colors affects the total length of Euclidean minimum spanning trees. It defines the MST-ratio $\gamma(P)=\max_{P=R\cup B}\frac{w(R)+w(B)}{w(P)}$ and proves that for any $n\ge 12$ points, $\gamma(P)>1$, with tight lower and upper bounds, and presents an $O(1)$-time method to obtain a maximizing bipartition and an $O(n\log n)$ method to compute the ratio. It strengthens understanding by proving stronger results for random point sets (where $\gamma(P) \ge \sqrt{2}-\varepsilon$ w.p.1 as $n\to\infty$) and for dense point sets (where $\gamma(P) \ge 1+\Omega(\alpha^{-2})$ when $\Delta(P)\le \alpha\sqrt{n}$). The work extends to higher dimensions via the disjoint-balls property and raises open questions about tighter gaps, computation, and constants. Overall, the paper combines geometric decompositions, MST sensitivity lemmas, and efficient algorithms to illuminate when partitioning can strictly increase total MST length.
Abstract
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 $α\sqrt{n}$, for some constant $α>0$, then the aforementioned ratio in a maximizing partition is $1 + Ω(α^{-2})$. All our results and techniques are extendable to higher dimensions.