We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D , randomized broadcasting requires Ω( D log n / D + log 2 n ) rounds, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be improved for the model with spontaneous transmissions, providing an O ( D log n log log n /log D + log O (1) n )-time broadcasting algorithm. In this article, we give a new and faster algorithm that completes broadcasting in O ( D log n /log D + log O (1) n ) time, succeeding with high probability. This yields the first optimal O ( D )-time broadcasting algorithm whenever n is polynomial in D . Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have used broadcasting as a subroutine and their complexity has been asymptotically strictly larger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O ( D log n / D min {log log n , log n / D } + log O (1) n )-time, succeeding with high probability. Our new algorithm again requires O ( D log n /log D + log O (1) n ) time, also succeeding with high probability.

中文翻译：

---

在无线电网络中利用自发传输进行广播和领导选举

我们研究两个基本的通信原语：广播和领袖选举在具有未知拓扑且没有碰撞检测机制的多跳无线电网络的经典模型中。近 20 年来，在无向网络中与n节点和直径D, 随机广播需要 Ω(D日志n/D+ 日志2 n) 轮，假设不允许未通知的节点进行通信（直到他们被通知）。就在最近，Haeupler 和 Wajc (PODC'2016) 表明，对于具有自发传输的模型，可以改进这一界限，提供○(D日志n日志n/日志D+ 日志 ○(1) n)-时间广播算法。在本文中，我们给出了一种新的更快的算法，可以在○(D日志n/日志D+ 日志 ○(1) n) 时间，以高概率成功。这产生了第一个最优○(D)-时间广播算法n是多项式在D. 此外，我们的方法可用于设计与我们的广播算法性能相匹配的新领导者选举算法。以前，所有快速随机领导者选举算法都使用广播作为子程序，其复杂度已渐近严格地大于广播的复杂度。特别是，先前已知最快的 Ghaffari 和 Haeupler 随机领导者选举算法（SODA'2013）需要○(D日志n/D最小 {log 日志n， 日志n/D} + 日志 ○(1) n)-时间，成功的概率很高。我们的新算法再次要求○(D日志n/日志D+ 日志 ○(1) n) 时间，也很有可能成功。