This paper concerns designing distributed algorithms that are {\em singularly optimal}, i.e., algorithms that are {\em simultaneously} time and message {\em optimal}, for the fundamental leader election problem in {\em asynchronous} networks. Kutten et al. (JACM 2015) presented a singularly near optimal randomized leader election algorithm for general {\em synchronous} networks that ran in $O(D)$ time and used $O(m \log n)$ messages (where $D$, $m$, and $n$ are the network's diameter, number of edges and number of nodes, respectively) with high probability.\footnote{Throughout, "with high probability" means "with probability at least $1-1/n^c$, for constant $c$."} Both bounds are near optimal (up to a logarithmic factor), since $\Omega(D)$ and $\Omega(m)$ are the respective lower bounds for time and messages for leader election even for synchronous networks and even for (Monte-Carlo) randomized algorithms. On the other hand, for general asynchronous networks, leader election algorithms are only known that are either time or message optimal, but not both. Kutten et al. (DISC 2020) presented a randomized asynchronous leader election algorithm that is singularly near optimal for \emph{complete networks}, but left open the problem for general networks. This paper shows that singularly near optimal (up to polylogarithmic factors) bounds can be achieved for general {\em asynchronous} networks. We present a randomized singularly near optimal leader election algorithm that runs in $O(D + \log^2n)$ time and $O(m\log^2 n)$ messages with high probability. Our result is the first known distributed leader election algorithm for asynchronous networks that is near optimal with respect to both time and message complexity and improves over a long line of results including the classical results of Gallager et al. (ACM TOPLAS, 1983), Peleg (JPDC, 1989), and Awerbuch (STOC 89).

翻译：本文涉及设计分布式的算法, 以美元( D) 时间运行, 并使用 $( m ) 信息( 美元, 美元, 美元) 的算法, 网络直径, 边缘和网络节点的数量, 概率很高。 Kutten et al. (JACM 2015) 为普通 Excial 同步 网络提供了一个极近于最佳随机化的领导选举算法( 以O( D) 时间运行为美元( 美元), 并且使用 $( log n) 的信息( 美元, 美元, 美元) 信息是网络的 Oral 。 网络和网络的节点( Ormal 网络), “ 概率高” 意味着“ 概率至少为1-1/ n 美元, 美元, 恒定 美元 。 ” 两种线点都接近( 美元), 预言( D) 和 预言( m) 预言( Orent i) 预言), 预言( On- hold) 预言, 预言。