论文标题
最大距离问题和最小的跨越树木
The Maximum Distance Problem and Minimal Spanning Trees
论文作者
论文摘要
给定一个紧凑型$ e \ subset \ mathbb {r}^n $和$ s> 0 $，最大距离问题寻求$ \ mathbb {r}^n $的紧凑和连接子集的最小hausdorff的$ s $ neighborhood heighborhood offers $ e $ e $ e $。对于$ e \ subset \ mathbb {r}^2 $，我们证明，将覆盖$ e $的Radius $ s $中心的最小跨越树中的最小跨越的树木最小化解决了最大距离问题。 The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $Γ$ in $\mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $Γ_\ast$, where $ \ MATHCAL {H}^1（γ_\ ast）$任意接近$ \ Mathcal {H}^1（γ）$。 我们还提供了一个开源包，用于使用最小跨越树来计算最大距离问题的计算探索，请访问https://github.com/mtdaydream/mdp_mst。
Given a compact $E \subset \mathbb{R}^n$ and $s > 0$, the maximum distance problem seeks a compact and connected subset of $\mathbb{R}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $E\subset \mathbb{R}^2$, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius $s$, which cover $E$, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $Γ$ in $\mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $Γ_\ast$, where $\mathcal{H}^1(Γ_\ast)$ is arbitrarily close to $\mathcal{H}^1(Γ)$. We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at https://github.com/mtdaydream/MDP_MST.
