We study the message complexity of leader election in synchronous networks of diameter two. Our main contribution is a refined analysis of the randomized algorithm proposed by Chatterjee et al. [DC, 2020]. In their work, the authors established a lower bound of $Ω(n)$ messages ($n$ is the number of nodes in the network) and presented a randomized algorithm that elects a leader in ${O}(1)$ rounds using $O(n \log^3 n)$ messages with high probability. In this paper, we improve their $\polylog n$ gap in the message bound by providing a tighter analysis of their algorithm, reducing the message complexity to $O(n\log n)$, while preserving the $O(1)$-round complexity and high-probability correctness guarantee.

翻译：我们研究同步直径二网络中领导者选举的消息复杂度。主要贡献是对Chatterjee等人[DC, 2020]提出的随机化算法进行了更精细的分析。在他们的工作中，作者建立了$Ω(n)$条消息的下界（$n$为网络节点数），并提出了一种随机化算法，该算法能以高概率在${O}(1)$轮内使用$O(n \log^3 n)$条消息完成领导者选举。本文通过提供更紧致的算法分析，将消息复杂度降低至$O(n \log n)$，同时保持$O(1)$轮复杂度与高概率正确性保证，从而缩小了消息边界中的$\polylog n$差距。