Optimal leader election in multi-hop radio networks

We present two optimal randomized leader election algorithms for multi-hop radio networks, which run in expected time asymptotically equal to the time required to broadcast one message to the entire network. We first observe that, under certain assumptions, a simulation approach of Bar-Yehuda, Golreich and Itai (1991) can be used to obtain an algorithm that for directed and undirected networks elects a leader in $O(D \log\frac{n}{D} + \log^2 n)$ expected time, where $n$ is the number of the nodes and $D$ is the eccentricity or the diameter of the network. We then extend this approach and present a second algorithm, which operates on undirected multi-hop radio networks with collision detection and elects a leader in $O(D + \log n)$ expected run-time. This algorithm in fact operates on the beep model, a strictly weaker model in which nodes can only communicate via beeps or silence. Both of these algorithms are optimal; no optimal expected-time algorithms for these models have been previously known. We further apply our techniques to design an algorithm that is quicker to achieve leader election with high probability. We give an algorithm for the model without collision detection which always runs in time $O((D \log\frac{n}{D} + \log^2 n)\cdot \sqrt{\log n})$, and succeeds with high probability. While non-optimal, and indeed slightly slower than the algorithm of Ghaffari and Haeupler (2013), it has the advantage of working in directed networks; it is the fastest known leader election algorithm to achieve a high-probability bound in such circumstances.