  We present distributed randomized leader election protocols for multi-hop
radio networks that elect a leader in almost the same time $T_{BC}$ required
for broadcasting a message. For the setting without collision detection, our
algorithm runs with high probability in $O(D \log \frac{n}{D} + \log^3 n)
\min\{\log\log n,\log \frac{n}{D}\}$ rounds on any $n$-node network with
diameter $D$. Since $T_{BC} = \Theta(D \log \frac{n}{D} + \log^2 n)$ is a lower
bound, our upper bound is optimal up to a factor of at most $\log \log n$ and
the extra $\log n$ factor on the additive term. This algorithm is furthermore
the first $O(n)$ time algorithm for this setting.
  Our algorithms improve over a 25 year old simulation approach of Bar-Yehuda,
Goldreich and Itai with a $O(T_{BC} \log n)$ running time: In 1987 they
designed a fast broadcast protocol and subsequently in 1989 they showed how it
can be used to simulate one round of a single-hop network that has collision
detection in $T_{BC}$ time. The prime application of this simulation was to
simulate Willards single-hop leader election protocol, which elects a leader in
$O(\log n)$ rounds with high probability and $O(\log \log n)$ rounds in
expectation. While it was subsequently shown that Willards bounds are tight, it
was unclear whether the simulation approach is optimal. Our results break this
barrier and essentially remove the logarithmic slowdown over the broadcast time
$T_{BC}$ by going away from the simulation approach.
  We also give a distributed randomized leader election algorithm for the
setting with collision detection that runs in $O(D + \log n \log \log n) \cdot
\min\{\log \log n, \log \frac{n}{D}\}$ rounds. This round complexity is optimal
up to $O(\log \log n)$ factors and improves over a deterministic algorithm that
requires $\Theta(n)$ rounds independently of the diameter $D$.
