Two trees are better than one
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909057925775360 |
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| author | Dumitrescu, Adrian Pach, János Tóth, Géza |
| author_facet | Dumitrescu, Adrian Pach, János Tóth, Géza |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09916 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Two trees are better than one Dumitrescu, Adrian Pach, János Tóth, Géza Computational Geometry Combinatorics 52Cxx, 05Cxx 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. |
| title | Two trees are better than one |
| topic | Computational Geometry Combinatorics 52Cxx, 05Cxx |
| url | https://arxiv.org/abs/2312.09916 |