We give a simple characterization of the functions that 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 classic mass-distribution techniques, our work makes use of a novel combinatorial structure called "history tree", which is of independent interest. Among other contributions, our results make conclusive 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). Our contribution not only opens a promising line of research on applications of history trees, but also demonstrates that computation in anonymous dynamic networks is practically feasible and far less demanding than previously conjectured.

翻译：我们给出了一个简单的特征描述，用于刻画在匿名动态网络中可由匿名进程确定性计算的函数，该特征取决于网络中领导者的数量。此外，我们提供了高效的分布式算法来计算所有此类函数，这些算法假设对网络具有最小或无先验知识。我们的每个算法都有两个版本：一个版本以正确输出终止，另一个更快的版本则稳定在正确输出上而不显式终止。值得注意的是，这些是首批运行时间随进程数量以及我们称为“动态非连通性”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一定需要始终保持连通）。我们还提供了匹配的下界，表明对于任意固定数量的领导者，我们的所有算法都是渐近最优的。虽然现有关于匿名动态网络的大多数文献依赖于经典的质心分布技术，但我们的工作利用了一种称为“历史树”的新型组合结构，该结构本身具有独立的研究价值。在我们的贡献中，研究结果在匿名动态网络的两个流行基本问题上取得了决定性进展：无领导者平均共识（即计算分布在进程间的输入数值的均值）和多领导者计数（即确定网络中进程的确切数量）。我们的贡献不仅开辟了历史树应用研究的有前景的新方向，而且证明了匿名动态网络中的计算实际上是可行的，并且远比先前推测的要求更低。