In this paper, we study the online Euclidean spanners problem for points in $\mathbb{R}^d$. Suppose we are given a sequence of $n$ points $(s_1,s_2,\ldots, s_n)$ in $\mathbb{R}^d$, where point $s_i$ is presented in step~$i$ for $i=1,\ldots, n$. The objective of an online algorithm is to maintain a geometric $t$-spanner on $S_i=\{s_1,\ldots, s_i\}$ for each step~$i$. First, we establish a lower bound of $\Omega(\varepsilon^{-1}\log n / \log \varepsilon^{-1})$ for the competitive ratio of any online $(1+\varepsilon)$-spanner algorithm, for a sequence of $n$ points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a $(1+\varepsilon)$-spanner with competitive ratio $O(\varepsilon^{-1}\log n / \log \varepsilon^{-1})$. Next, we design online algorithms for sequences of points in $\mathbb{R}^d$, for any constant $d\ge 2$, under the $L_2$ norm. We show that previously known incremental algorithms achieve a competitive ratio $O(\varepsilon^{-(d+1)}\log n)$. However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of $\varepsilon$. We describe an online Steiner $(1+\varepsilon)$-spanner algorithm with competitive ratio $O(\varepsilon^{(1-d)/2} \log n)$. As a counterpart, we show that the dependence on $n$ cannot be eliminated in dimensions $d \ge 2$. In particular, we prove that any online spanner algorithm for a sequence of $n$ points in $\mathbb{R}^d$ under the $L_2$ norm has competitive ratio $\Omega(f(n))$, where $\lim_{n\rightarrow \infty}f(n)=\infty$. Finally, we provide improved lower bounds under the $L_1$ norm: $\Omega(\varepsilon^{-2}/\log \varepsilon^{-1})$ in the plane and $\Omega(\varepsilon^{-d})$ in $\mathbb{R}^d$ for $d\geq 3$.

中文翻译：

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在线欧几里得扳手

在本文中，我们研究了 $\mathbb{R}^d$ 中点的在线欧几里得扳手问题。假设我们在 $\mathbb{R}^d$ 中给定了 $n$ 个点 $(s_1,s_2,\ldots, s_n)$ 的序列，其中点 $s_i$ 在 step~$i$ 中表示为 $i =1,\ldots, n$。在线算法的目标是在 $S_i=\{s_1,\ldots, s_i\}$ 上为每一步~$i$ 维护一个几何 $t$-spanner。首先，我们为任何在线$(1+\varepsilon)$-的竞争比率建立$\Omega(\varepsilon^{-1}\log n / \log \varepsilon^{-1})$的下界扳手算法，用于一维 $n$ 点的序列。我们证明了这个界限是紧的，并且有一个在线算法可以维持一个 $(1+\varepsilon)$-spanner 具有竞争比 $O(\varepsilon^{-1}\log n / \log \varepsilon^ {-1})$。接下来，我们为 $\mathbb{R}^d$ 中的点序列设计在线算法，对于任何常数 $d\ge 2$，在 $L_2$ 范数下。我们展示了先前已知的增量算法实现了竞争比率 $O(\varepsilon^{-(d+1)}\log n)$。但是，如果允许算法使用额外的点（Steiner 点），那么就可以大幅提高 $\varepsilon$ 方面的竞争比率。我们描述了一个在线 Steiner $(1+\varepsilon)$-spanner 算法，其竞争比为 $O(\varepsilon^{(1-d)/2} \log n)$。作为对应，我们表明在维度 $d \ge 2$ 中无法消除对 $n$ 的依赖。特别是，我们证明了在 $L_2$ 范数下 $\mathbb{R}^d$ 中的 $n$ 点序列的任何在线扳手算法都具有竞争比 $\Omega(f(n))$，其中 $ \lim_{n\rightarrow \infty}f(n)=\infty$。最后，我们在 $L_1$ 范数下提供改进的下限：