SIAM Journal on Discrete Mathematics, Volume 35, Issue 3, Page 1592-1614, January 2021.
Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter $c \geq 1$, we call $P$ a $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$ for every triple $(i,j,k)$, $1 \leq i<j<k \leq n$. The stretch factor is a global property: it measures how close $P$ is to a straight line, and it involves all the vertices of $P$; being a $c$-chain, on the other hand, is a fingerprint property: it only depends on subsets of $O(1)$ vertices of the chain. We investigate how the $c$-chain property influences the stretch factor in the plane: (i) we show that for every $\varepsilon > 0$, there is a noncrossing $c$-chain that has stretch factor $\Omega(n^{1/2-\varepsilon})$ for sufficiently large constant $c=c(\varepsilon)$; (ii) on the other hand, the stretch factor of a $c$-chain $P$ is $O(n^{1/2})$ for every constant $c\geq 1$, regardless of whether $P$ is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain $P$ in $\mathbb{R}^2$ with $n$ vertices, the minimum $c\geq 1$ for which $P$ is a $c$-chain in $O(n^{2.5}\ {\rm polylog}\ n)$ expected time and $O(n\log n)$ space. These results generalize to $\mathbb{R}^d$. For every dimension $d\geq 2$ and every $\varepsilon>0$, we construct a noncrossing $c$-chain that has stretch factor $\Omega(n^{(1-\varepsilon)(d-1)/d})$; on the other hand, the stretch factor of any $c$-chain is $O((n-1)^{(d-1)/d})$; for every $c>1$, we can test whether an $n$-vertex chain in $\mathbb{R}^d$ is a $c$-chain in $O(n^{3-1/d}\ {\rm polylog}\ n)$ expected time and $O(n\log n)$ space.

中文翻译：

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关于多边形链的拉伸系数

SIAM 离散数学杂志，第 35 卷，第 3 期，第 1592-1614 页，2021 年 1 月。
设 $P=(p_1, p_2, \dots, p_n)$ 是 $\mathbb{R}^d$ 中的多边形链。$P$的拉伸因子是$P$的总长度与其端点距离的比值，$\sum_{i = 1}^{n-1} |p_i p_{i+1}|/| p_1 p_n|$。对于参数 $c \geq 1$，如果 $|p_ip_j|+|p_jp_k|，我们称 $P$ 为 $c$-chain \leq c|p_ip_k|$ 对于每个三元组 $(i,j,k)$，$1 \leq i<j<k \leq n$。拉伸因子是一个全局属性：它衡量 $P$ 与直线的接近程度，它涉及 $P$ 的所有顶点；另一方面，作为 $c$-chain 是一个指纹属性：它只取决于链的 $O(1)$ 顶点的子集。我们研究了 $c$-chain 属性如何影响平面中的拉伸因子：（i）我们证明对于每个 $\varepsilon > 0$，对于足够大的常数 $c=c(\varepsilon)$，有一个非交叉的 $c$-chain 具有拉伸因子 $\Omega(n^{1/2-\varepsilon})$；(ii) 另一方面，对于每个常数 $c\geq 1$，$c$-chain $P$ 的拉伸因子是 $O(n^{1/2})$，无论 $P$是穿越还是非穿越；(iii) 我们给出了一个随机算法，该算法可以确定，对于 $\mathbb{R}^2$ 中具有 $n$ 个顶点的多边形链 $P$，$P$ 的最小 $c\geq 1$ $O(n^{2.5}\ {\rm polylog}\ n)$ 预期时间和 $O(n\log n)$ 空间中的 $c$-chain。这些结果推广到 $\mathbb{R}^d$。对于每个维度 $d\geq 2$ 和每个 $\varepsilon>0$，我们构造一个非交叉的 $c$-chain，它具有拉伸因子 $\Omega(n^{(1-\varepsilon)(d-1)/ d})$; 另一方面，任何 $c$-chain 的拉伸因子是 $O((n-1)^{(d-1)/d})$; 每$c>1$，