We give a simple characterization of which functions can be computed deterministically by anonymous processes in dynamic networks, depending on the number of leaders in the network. In addition, we provide efficient distributed algorithms for computing all such functions assuming minimal or no knowledge about the network. Each of our algorithms comes in two versions: one that terminates with the correct output and a faster one that stabilizes on the correct output without explicit termination. Notably, these are the first deterministic algorithms whose running times scale linearly with both the number of processes and a parameter of the network which we call "dynamic disconnectivity" (meaning that our dynamic networks do not necessarily have to be connected at all times). We also provide matching lower bounds, showing that all our algorithms are asymptotically optimal for any fixed number of leaders. While most of the existing literature on anonymous dynamic networks relies on classical mass-distribution techniques, our work makes use of a recently introduced combinatorial structure called "history tree", also developing its theory in new directions. Among other contributions, our results make definitive progress on two popular fundamental problems for anonymous dynamic networks: leaderless Average Consensus (i.e., computing the mean value of input numbers distributed among the processes) and multi-leader Counting (i.e., determining the exact number of processes in the network). In fact, our approach unifies and improves upon several independent lines of research on anonymous networks, including Nedic et al., IEEE Trans. Automat. Contr. 2009; Olshevsky, SIAM J. Control Optim. 2017; Kowalski-Mosteiro, ICALP 2019, SPAA 2021; Di Luna-Viglietta, FOCS 2022.

翻译：我们给出了一个简单刻画，用于说明在动态网络中，匿名进程能够确定性计算的函数取决于网络中的领导数量。此外，我们提供了高效的分布式算法，假定对网络只有极少的了解或完全不了解，即可计算所有此类函数。我们的每个算法均有两个版本：一个版本在终止时输出正确结果，另一个更快的版本则稳定于正确结果而无需显式终止。值得注意的是，这些是首批运行时间随进程数量及我们称之为"动态断连性"的网络参数线性扩展的确定性算法（这意味着我们的动态网络不必始终连接）。我们还提供了匹配的下界，证明对于任意固定数量的领导，我们的所有算法均具有渐近最优性。尽管现有关于匿名动态网络的文献大多依赖于经典的质量分布技术，但我们的工作利用了近年引入的名为"历史树"的组合结构，并发展了其理论新方向。在其他贡献中，我们的结果在匿名动态网络的两个经典基础问题上取得了决定性进展：无领导平均共识（即计算分布在进程中输入数值的均值）和多领导计数（即确定网络中进程的确切数量）。事实上，我们的方法统一并改进了多个关于匿名网络的独立研究路线，包括Nedic等（IEEE Trans. Automat. Contr. 2009）、Olshevsky（SIAM J. Control Optim. 2017）、Kowalski-Mosteiro（ICALP 2019, SPAA 2021）以及Di Luna-Viglietta（FOCS 2022）的工作。