We give the first linear-time counting algorithm for processes in anonymous 1-interval-connected dynamic networks with a leader. As a byproduct, we are able to compute in $3n$ rounds every function that is deterministically computable in such networks. If explicit termination is not required, the running time improves to $2n$ rounds, which we show to be optimal up to a small additive constant (this is also the first non-trivial lower bound for counting). As our main tool of investigation, we introduce a combinatorial structure called "history tree", which is of independent interest. This makes our paper completely self-contained, our proofs elegant and transparent, and our algorithms straightforward to implement. In recent years, considerable effort has been devoted to the design and analysis of counting algorithms for anonymous 1-interval-connected networks with a leader. A series of increasingly sophisticated works, mostly based on classical mass-distribution techniques, have recently led to a celebrated counting algorithm in $O({n^{4+ \epsilon}} \log^{3} (n))$ rounds (for $\epsilon>0$), which was the state of the art prior to this paper. 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：我们首次为具有领导者的匿名1-区间连通动态网络中的进程给出了线性时间计数算法。作为副产品，我们能够在$3n$轮内计算此类网络中确定性可计算的每一个函数。如果不需要显式终止，运行时间可优化至$2n$轮，我们证明这在一个小的加法常数内是最优的（这也是计数问题的首个非平凡下界）。作为主要研究工具，我们引入了一种名为“历史树”的组合结构，该结构具有独立的研究价值。这使得我们的论文完全自包含，证明过程优雅且透明，算法易于实现。近年来，大量工作致力于设计和分析具有领导者的匿名1-区间连通网络中的计数算法。一系列日益精妙的研究（主要基于经典质量分布技术）近期促成了一个著名的计数算法，其运行时间为$O({n^{4+ \epsilon}} \log^{3} (n))$轮（其中$\epsilon>0$），这是本文之前的最高水平。我们的贡献不仅开辟了关于历史树应用的有前景的研究方向，还表明匿名动态网络中的计算是实际可行的，且远不如先前推测的那样苛刻。