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.

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