Population Protocols Are Fast

A population protocol describes a set of state change rules for a population of $n$ indistinguishable finite-state agents (automata), undergoing random pairwise interactions. Within this very basic framework, it is possible to resolve a number of fundamental tasks in distributed computing, including: leader election, aggregate and threshold functions on the population, such as majority computation, and plurality consensus. For the first time, we show that solutions to all of these problems can be obtained \emph{quickly} using finite-state protocols. For any input, the designed finite-state protocols converge under a fair random scheduler to an output which is correct with high probability in expected $O(\mathrm{poly} \log n)$ parallel time. In the same setting, we also show protocols which always reach a valid solution, in expected parallel time $O(n^{\varepsilon)$,} where the number of states of the interacting automata depends only on the choice of $\varepsilon>0$. The stated time bounds hold for \emph{any} semi-linear predicate computable in the population protocol framework. The key ingredient of our result is the decentralized design of a hierarchy of phase-clocks, which tick at different rates, with the rates of adjacent clocks separated by a factor of $\Theta(\log n)$. The construction of this clock hierarchy relies on a new protocol composition technique, combined with an adapted analysis of a self-organizing process of oscillatory dynamics. This clock hierarchy is used to provide nested synchronization primitives, which allow us to view the population in a global manner and design protocols using a high-level imperative programming language with a (limited) capacity for loops and branching instructions.