Near Optimal Leader Election in Multi-Hop Radio Networks

We present distributed randomized leader election protocols for multi-hop radio networks that elect a leader in almost the same time TBC required for broadcasting a message. For the setting without collision detection, our algorithm runs with high probability in O(D log n D + log n) · min{log logn, log n D } rounds on any n-node network with diameter D. Since TBC = Θ(D log n D + log n) is a lower bound, our upper bound is optimal up to a factor of at most log logn and the extra logn factor on the additive term. This algorithm is furthermore the first O(n) time algorithm for this setting. Our algorithm improves over a 25 year old simulation approach of Bar-Yehuda, Goldreich and Itai with a O(TBC logn) 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 with high probability and O(log logn) 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 a distributed randomized leader election algorithm for the setting with collision detection (even with single-bit messages) that with high probability runs inO(D+logn log logn)· min{log logn, log n D } = O(D+logn) ·O(log logn) rounds. This round complexity is optimal up to O(log logn) factors and it improves over a deterministic algorithm that requires Θ(n) rounds independently of the diameter 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 O(TBC logn) running time had become a bottleneck, preventing optimal algorithms. Breaking the simulation barrier for leader election in this paper has subsequently led to the development of near optimal protocols for these communication primitives.