We consider leader election in clique networks, where $n$ nodes are connected by point-to-point communication links. For the synchronous clique under simultaneous wake-up, i.e., where all nodes start executing the algorithm in round $1$, we show a tradeoff between the number of messages and the amount of time. More specifically, we show that any deterministic algorithm with a message complexity of $n f(n)$ requires $\Omega\left(\frac{\log n}{\log f(n)+1}\right)$ rounds, for $f(n) = \Omega(\log n)$. Our result holds even if the node IDs are chosen from a relatively small set of size $\Theta(n\log n)$, as we are able to avoid using Ramsey's theorem. We also give an upper bound that improves over the previously-best tradeoff. Our second contribution for the synchronous clique under simultaneous wake-up is to show that $\Omega(n\log n)$ is in fact a lower bound on the message complexity that holds for any deterministic algorithm with a termination time $T(n)$. We complement this result by giving a simple deterministic algorithm that achieves leader election in sublinear time while sending only $o(n\log n)$ messages, if the ID space is of at most linear size. We also show that Las Vegas algorithms (that never fail) require $\Theta(n)$ messages. For the synchronous clique under adversarial wake-up, we show that $\Omega(n^{3/2})$ is a tight lower bound for randomized $2$-round algorithms. Finally, we turn our attention to the asynchronous clique: Assuming adversarial wake-up, we give a randomized algorithm that achieves a message complexity of $O(n^{1 + 1/k})$ and an asynchronous time complexity of $k+8$. For simultaneous wake-up, we translate the deterministic tradeoff algorithm of Afek and Gafni to the asynchronous model, thus partially answering an open problem they pose.

翻译：我们考虑团网络中的领导者选举问题，其中 $n$ 个节点通过点对点通信链路相连。对于同步唤醒下的同步团，即所有节点在第 $1$ 轮同时开始执行算法，我们展示了消息数量与时间开销之间的权衡。具体而言，我们证明任何消息复杂度为 $n f(n)$ 的确定性算法需要 $\Omega\left(\frac{\log n}{\log f(n)+1}\right)$ 轮，其中 $f(n) = \Omega(\log n)$。即使节点ID取自规模为 $\Theta(n\log n)$ 的相对较小集合，我们的结果依然成立，因为我们能够避免使用拉姆齐定理。我们还给出了一个改进先前最佳权衡的上界。我们在同步唤醒下同步团的第二个贡献是证明 $\Omega(n\log n)$ 实际上是任何终止时间为 $T(n)$ 的确定性算法消息复杂度的下界。我们通过给出一个简单的确定性算法来补充这一结果，该算法在亚线性时间内实现领导者选举，同时仅发送 $o(n\log n)$ 条消息，前提是ID空间的大小最多为线性。我们还表明，拉斯维加斯算法（从不失败）需要 $\Theta(n)$ 条消息。对于对抗性唤醒下的同步团，我们证明 $\Omega(n^{3/2})$ 是随机化 $2$ 轮算法的紧致下界。最后，我们将注意力转向异步团：在对抗性唤醒假设下，我们给出一个随机化算法，其消息复杂度为 $O(n^{1 + 1/k})$，异步时间复杂度为 $k+8$。对于同步唤醒，我们将Afek和Gafni的确定性权衡算法转化为异步模型，从而部分回答了他们提出的一个开放问题。