This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in networks. Our main result is a randomized distributed leader election algorithm for asynchronous complete networks that is essentially (up to a polylogarithmic factor) singularly optimal. Our algorithm uses $O(n)$ messages with high probability and runs in $O(\log^2 n)$ time (with high probability) to elect a unique leader. The $O(n)$ message complexity should be contrasted with the $Ω(n \log n)$ lower bounds for the deterministic message complexity of leader election algorithms (regardless of time), proven by Korach, Moran, and Zaks (TCS, 1989) for asynchronous algorithms and by Afek and Gafni (SIAM J. Comput., 1991) for synchronous networks. Hence, our result also separates the message complexities of randomized and deterministic leader election. More importantly, our (randomized) time complexity of $O(\log^2 n)$ for obtaining the optimal $O(n)$ message complexity is significantly smaller than the long-standing $\tildeΘ(n)$ time complexity obtained by Afek and Gafni and by Singh (SIAM J. Comput., 1997) for message optimal (deterministic) election in asynchronous networks. In synchronous complete networks, Afek and Gafni showed an essentially singularly optimal deterministic algorithm with $O(\log n)$ time and $O(n \log n)$ messages. Ramanathan et al. (Distrib. Comput. 2007) used randomization to improve the message complexity, and showed a randomized algorithm with $O(n)$ messages and $O(\log n)$ time (with failure probability $O(1 / \log^{Ω(1)}n)$). Our second result is a tightly singularly optimal randomized algorithm, with $O(1)$ time and $O(n)$ messages, for this setting, whose time bound holds with certainty and message bound holds with high probability.

24 pages. Full version of paper accepted at DISC 2020