We give a simple characterization of which functions can be computed deterministically by anonymous processes in disconnected 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". 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：我们给出一个简洁的表征，用以描述在断连动态网络中，匿名进程能够确定性计算的函数，该表征取决于网络中的领导者数量。此外，我们提供了高效分布式算法，用于在假设对网络知之甚少或完全未知的情况下计算所有这些函数。我们的每个算法均有两个版本：一个版本在输出正确结果后终止，另一个更快的版本则稳定在正确输出上但不显式终止。值得注意的是，这些是首批运行时间与进程数及网络参数（我们称之为“动态断连度”）均呈线性关系的确定性算法。我们还给出了匹配的下界，证明对于任意固定数量的领导者，所有算法均为渐近最优的。尽管现有匿名动态网络文献大多依赖经典的群体分布技术，我们的工作采用了近期引入的名为“历史树”的组合结构，并在此方向拓展了其理论。在众多贡献中，我们的结果在匿名动态网络的两个基础性流行问题上取得了决定性进展：无领导者平均共识（即计算各进程输入数值的均值）与多领导者计数（即确定网络中的确切进程数）。事实上，我们的方法统一并改进了匿名网络领域的多条独立研究路线，包括Nedic等人（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。