Time-Space Trade-offs in Population Protocols

Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Recent work has focused on the complexity of fundamental tasks in the population model, such as leader election (which requires convergence to a single agent in a special "leader" state), and majority (in which agents must converge to a decision as to which of two possible initial states had higher initial count). Known upper and lower bounds point towards an inherent trade-off between the time complexity of these protocols, and the space complexity, i.e. size of the memory available to each agent. In this paper, we explore this trade-off and provide new upper and lower bounds for these two fundamental tasks. First, we prove a new unified lower bound, which relates the space available per node with the time complexity achievable by the protocol: for instance, our result implies that any protocol solving either of these tasks for $n$ agents using $O( \log \log n )$ states must take $\Omega( n / \operatorname{polylog} n )$ expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs. On the positive side, we show that $O(\operatorname{polylog} n)$ convergence time can be achieved using $O( \log^2 n )$ space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between $O(\log \log n)$ and $\Theta( \log^2 n )$ state space size for both majority and leader election in population protocols. At the same time, we introduce several new tools and techniques, which should be applicable to other tasks and settings.