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.

翻译：种群协议模型提供了一个通用平台，用于研究由匿名智能体间随机成对交互驱动的分布式过程。种群协议的时间复杂度指达到最终配置所需的交互次数。近期研究聚焦于并行时间，其定义时间复杂度除以n，若协议在并行时间O(poly log n)内稳定则视为高效。此类协议的计算缺陷包括：接近最终稳定时有效交互比例下降（抑制并行效率）、常数空间种群协议的计算能力局限于Presburger算术中的半线性谓词（反映时空权衡），以及无限计算（影响多阶段协议）。针对这些缺陷，我们通过施加状态空间的基本结构，并在随机交互对选择过程中引入条件概率选择，提出新型选择性种群协议变体。研究表明，此类协议能够计算比半线性谓词更复杂的函数（即超越Presburger算术）。在比较模型下，我们首次对种群协议中的中位数计算展开非平凡研究：该模型中智能体的操作状态空间固定，转移函数决定交互智能体关联（可能较大的）隐藏键的顺序。我们证明：n个数中位数的计算需要Ω(n)并行时间，且该问题在标准种群协议中可在期望值与高概率下以O(n log n)并行时间求解。最后，我们证明选择性种群协议可实现O(log⁴ n)并行时间的中位数计算。