Selective Population Protocols

The model of {\em population protocols} provides a universal platform to study distributed processes driven by random pairwise interactions of anonymous agents. The {\em time complexity} of population protocols refers to the number of interactions required to reach a final configuration. More recently, the focus is on the {\em parallel time} defined as the time complexity divided by $n,$ where a given protocol is {\em efficient} if it stabilises in parallel time $O(\mbox{poly}\log n)$. Among computational deficiencies of such protocols are depleting fraction of {\em meaningful interactions} closing in on the final stabilisation (suppressing parallel efficiency), computation power of constant-space population protocols limited to semi-linear predicates in Presburger arithmetic (reflecting on time-space trade offs), and indefinite computation (impacting multi-stage protocols). With these deficiencies in mind, we propose a new {\em selective} variant of population protocols by imposing an elementary structure on the state space, together with a conditional probabilistic choice during random interacting pair selection. We show that such protocols are capable of computing functions more complex than semi-linear predicates, i.e., beyond Presburger arithmetic. We provide the first non-trivial study on median computation (in population protocols) in a comparison model where the operational state space of agents is fixed and the transition function decides on the order between (potentially large) hidden keys associated with the interacting agents. We show that computation of the median of $n$ numbers requires $\Omega(n)$ parallel time and the problem can be solved in $O(n\log n)$ parallel time in expectation and whp in standard population protocols. Finally, we show $O(\log^4 n)$ parallel time median computation in selective population protocols.