  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 $d\geq 3$.
