Near Optimal Leader Election in Multi-Hop Radio Networks

We design leader election protocols for multi-hop radio networks that elect a leader in almost the same time TBC that it takes for broadcasting one message (one ID). For the setting without collision detection our algorithm runs whp. in O(D log n D + log3 n) · min{log log n, log n D} rounds on any n-node network with diameter D. Since TBC = Θ(D log n D + log2 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. Our algorithm is furthermore the first O(n) time algorithm for this setting. Our algorithm improves over a 23 year old simulation approach of Bar-Yehuda, Goldreich and Itai with a O(TBC 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 TBC 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 whp. 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 TBC. This is achieved by going away from the simulation approach. We also give an O(D + log n log log n) · min{log log n, log n D} = O(D + log n) · O(log log n)2 leader election algorithm for the setting with collision detection (even with single-bit messages). This is optimal up to log log n factors and improves over a deterministic algorithm that requires Θ(n) rounds independently of D. Our almost optimal leader election protocols are especially important because countless communication protocols in radio networks use leader election as a crucial first step to solve various, seemingly unrelated, communication primitives such as gathering, multiple unicasts or multiple broadcasts. Even though leader election seems easier than these tasks, its best-known