Computation in Anonymous Directed Dynamic Networks: Exact Worst-Case Termination Time and Network-Wise Intractability

Abstract

We determine the exact round complexity of Counting in anonymous directed dynamic networks. Agents have no identifiers, but $\ell\geq1$ of them are designated as leaders. Every agent knows $\ell$ and its own leader status, while the population size $n$ is unknown and has to be determined by the agents. Communication proceeds in synchronous rounds; the directed communication graph may change arbitrarily and unpredictably from one round to the next, but is assumed to remain strongly connected. We solve Counting with a distributed algorithm that makes every agent output $n$ and halt after a number of rounds equal to $\begin{cases} n&\text{if }n=\ell\\ 2n-1&\text{if }n>\ell\text{ and }\ell=1\\ 2n-2&\text{if }n>\ell\text{ and }\ell\geq2. \end{cases}$ The algorithm works even for networks with loops and parallel arcs and even when agents learn their outdegree only after sending. We prove that this bound is worst-case tight over directed multigraphs for every $n$ and $\ell$ with $1\leq\ell\leq n$. Moreover, the local computation is polynomial (provided that each communication multigraph has at most $2^{\operatorname{poly}(n)}$ arcs): in every round, each agent solves at most one linear system with up to $n$ variables. By standard techniques, solving Counting also enables the evaluation of any computable function of the multiset of agents' input values within $2n-1$ communication rounds. The bound above optimizes the worst case over all networks. For an individual network, we show that even deciding whether an agent already has enough local information to determine $n$ is $\mathsf{coNP}$-complete (provided that the numbers of parallel arcs are encoded in binary). Thus, worst-case-optimal Counting admits polynomial local computation, whereas exact network-wise optimization is intractable under standard complexity assumptions.