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：我们给出一个简洁刻画，阐明在动态网络中，匿名进程可根据网络中领导者的数量确定性计算哪些函数。此外，我们提供了高效分布式算法，能在对网络知之甚少或一无所知的前提下计算所有此类函数。每个算法均有两个版本：一个以正确输出终止，另一个更快版本可在无显式终止的情况下稳定在正确输出上。值得注意的是，这些是首批运行时间与进程数量及网络参数（称为“动态不连通性”，即动态网络无需始终连通）均成线性关系的确定性算法。我们还提供了匹配的下界证明，表明针对任意固定领导者数量，所有算法均渐近最优。尽管现有匿名动态网络文献多依赖经典质量分布技术，但本文采用了近期引入的名为“历史树”的组合结构，并拓展了其理论新方向。我们的成果在匿名动态网络两个基础性问题上取得决定性进展：无领导平均共识（即计算各进程输入数值的均值）与多领导计数（即精确确定网络中进程数量）。事实上，本文方法统一并改进了匿名网络领域的多个独立研究路线，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。