In this paper, we investigate the leader election problem in diameter-two networks. Recently, Chatterjee et al. [DC 2020] studied the leader election in diameter-two networks. They presented a $O(\log n)$-round deterministic {implicit} leader election algorithm which incurs optimal $O(n\log n)$ messages, but a drawback of their algorithm is that it requires knowledge of $n$. An important question -- whether it is possible to remove the assumption on the knowledge of $n$ was left open in their paper. Another interesting open question raised in their paper is whether {\em explicit} leader election can be solved in $\tilde{O}(n)$ messages deterministically. In this paper, we give an affirmative answer to them. Further, we solve the {\em broadcast problem}, another fundamental problem in distributed computing, deterministically in diameter-two networks with $\tilde{O}(n)$ messages and $\tilde{O}(1)$ rounds without the knowledge of $n$. In fact, we address all the open questions raised by Chatterjee et al. for the deterministic leader election problem in diameter-two networks. To the best of our knowledge, this is the first $\tilde{O}(n)$ deterministic result for the explicit leader election in the diameter-two networks, that too without the knowledge of $n$.

翻译：本文研究直径二网络中的领导者选举问题。近期，Chatterjee等人[DC 2020]探讨了直径二网络中的领导者选举问题，提出了一种$O(\log n)$轮的确定性隐含领导者选举算法，该算法实现了最优$O(n\log n)$的消息复杂度，但其缺陷在于需要预先获知网络规模$n$。该论文中留下一个开放性问题——能否消除对网络规模$n$已知的假设。另一个有趣的开放问题是：能否以确定性方式用$\tilde{O}(n)$条消息解决显式领导者选举问题。本文对这两个问题给出了肯定答案。此外，我们还解决了分布式计算中的另一基础问题——广播问题，在无需获知$n$的条件下，以$\tilde{O}(n)$条消息和$\tilde{O}(1)$轮实现了直径二网络的确定性广播。实际上，本文针对Chatterjee等人提出的所有关于直径二网络确定性领导者选举的开放问题均给出了解决方案。据我们所知，这是首个在直径二网络中实现显式领导者选举的$\tilde{O}(n)$确定性结果，且无需预先获知网络规模$n$。