Computing in Anonymous Dynamic Networks Is Linear

We give the first linear-time terminating 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 \emph{every} function that is computable deterministically 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 \emph{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.