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 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：我们给出了一个简洁刻画，描述了在动态网络中由匿名进程可确定性计算的函数，该刻画取决于网络中领导者的数量。此外，我们提供了高效分布式算法，用于计算所有此类函数，且这些算法只需对网络有最少了解或完全无了解。每个算法有两个版本：一个版本在输出正确结果后终止，另一个执行更快的版本则在输出正确结果后稳定（不显式终止）。值得注意的是，这是首批运行时间与进程数量及我们称为"动态不连通度"的网络参数成线性关系的确定性算法（这意味着我们的动态网络不必始终连通）。我们还给出了匹配的下界，证明对于任何固定数量的领导者，所有算法都是渐近最优的。尽管现有关于匿名动态网络的文献大多依赖经典的"质量分布"技术，我们的工作采用了一种名为"历史树"的新型组合结构，该结构本身值得独立关注。在其他贡献中，我们的结果为匿名动态网络中的两个基本问题取得了决定性进展：无领导者"平均共识"（即计算分布在各进程中输入数字的均值）和多领导者"计数"（即确定网络中进程的精确数量）。我们的贡献不仅为历史树的应用开辟了一条有前景的研究方向，还表明匿名动态网络中的计算在实践上是可行的，且远未达到先前推测的苛刻程度。