  Let $P=\{p_0,\ldots,p_{n-1}\}$ be a set of points in $\mathbb{R}^d$, modeling
devices in a wireless network. A range assignment assigns a range $r(p_i)$ to
each point $p_i\in P$, thus inducing a directed communication graph $G_r$ in
which there is a directed edge $(p_i,p_j)$ iff $\textrm{dist}(p_i, p_j) \leq
r(p_i)$, where $\textrm{dist}(p_i,p_j)$ denotes the distance between $p_i$ and
$p_j$. The range-assignment problem is to assign the transmission ranges such
that $G_r$ has a certain desirable property, while minimizing the cost of the
assignment; here the cost is given by $\sum_{p_i\in P} r(p_i)^{\alpha}$, for
some constant $\alpha>1$ called the distance-power gradient.
  We introduce the online version of the range-assignment problem, where the
points $p_j$ arrive one by one, and the range assignment has to be updated at
each arrival. Following the standard in online algorithms, resources given out
cannot be taken away -- in our case this means that the transmission ranges
will never decrease. The property we want to maintain is that $G_r$ has a
broadcast tree rooted at the first point $p_0$. Our results include the
following.
  - For $d=1$, a 1-competitive algorithm does not exist. In particular, for
$\alpha=2$ any online algorithm has competitive ratio at least 1.57.
  - For $d=1$ and $d=2$, we analyze two natural strategies: Upon the arrival of
a new point $p_j$, Nearest-Neighbor increases the range of the nearest point to
cover $p_j$ and Cheapest Increase increases the range of the point for which
the resulting cost increase to be able to reach $p_j$ is minimal.
  - We generalize the problem to arbitrary metric spaces, where we present an
$O(\log n)$-competitive algorithm.
