  The topology of a wireless multi-hop network can be controlled by varying the
transmission power at each node. In this paper, we give a detailed analysis of
a cone-based distributed topology control algorithm. This algorithm, introduced
in [16], does not assume that nodes have GPS information available; rather it
depends only on directional information. Roughly speaking, the basic idea of
the algorithm is that a node $u$ transmits with the minimum power
$p_{u,\alpha}$ required to ensure that in every cone of degree $\alpha$ around
$u$, there is some node that $u$ can reach with power $p_{u,\alpha}$. We show
that taking $\alpha = 5\pi/6$ is a necessary and sufficient condition to
guarantee that network connectivity is preserved. More precisely, if there is a
path from $s$ to $t$ when every node communicates at maximum power, then, if
$\alpha <= 5\pi/6$, there is still a path in the smallest symmetric graph
$G_\alpha$ containing all edges $(u,v)$ such that $u$ can communicate with $v$
using power $p_{u,\alpha}$. On the other hand, if $\alpha > 5\pi/6$,
connectivity is not necessarily preserved. We also propose a set of
optimizations that further reduce power consumption and prove that they retain
network connectivity. Dynamic reconfiguration in the presence of failures and
mobility is also discussed. Simulation results are presented to demonstrate the
effectiveness of the algorithm and the optimizations.
