Selective Population Protocols

The model of population protocols provides a universal platform to study distributed processes driven by pairwise interactions of anonymous agents. While population protocols present an elegant and robust model for randomized distributed computation, their efficiency wanes when tackling issues that require more focused communication or the execution of multiple processes. To address this issue, we propose a new, selective variant of population protocols by introducing a partition of the state space and the corresponding conditional selection of responders. We demonstrate on several examples that the new model offers a natural environment, complete with tools and a high-level description, to facilitate more efficient solutions. In particular, we provide fixed-state stable and efficient solutions to two central problems: leader election and majority computation, both with confirmation. This constitutes a separation result, as achieving stable and efficient majority computation requires $\Omega(\log n)$ states in standard population protocols, even when the leader is already determined. Additionally, we explore the computation of the median using the comparison model, where the operational state space of agents is fixed, and the transition function determines the order between (arbitrarily large) hidden keys associated with interacting agents. Our findings reveal that the computation of the median of $n$ numbers requires $\Omega(n)$ time. Moreover, we demonstrate that the problem can be solved in $O(n\log n)$ time, both in expectation and with high probability, in standard population protocols. In contrast, we establish that a feasible solution in selective population protocols can be achieved in $O(\log^4 n)$ time.