New Clocks, Fast Line Formation and Self-Replication Population Protocols

The model of population protocols was developed to study distributed processes based on pairwise interactions between anonymous agents drawn from a large population of size n. The interacting pairs of agents are chosen by the random scheduler and their states are adjusted adequately by the predefined transition function which governs the considered process. The state space of agents is fixed (limited to a constant size) and the size n is not known, i.e., not hard-coded in the transition function. We assume that a population protocol starts in the predefined initial configuration of agents’ states representing the input, and it concludes in the final configuration of states representing the output of the considered problem. The sequential time complexity of such protocols refers to the number of interactions required to stabilise the relevant protocol in the final configuration. We also define the parallel time as the sequential time complexity divided by n. In this paper we consider a variant of the standard population protocol model in which agents are allowed to be connected by edges, known as the constructors model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n) interactions on average which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any nontrivial rigid network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient solutions are expected to operate in O(poly logn) parallel time. The contributions of this paper are manifold. • We propose and analyse a novel type of phase clocks allowing to count parallel time Θ(n logn) in the constructors model. This new type of clocks can be also implemented in the standard population protocol model assuming a unique leader is available. • The new clock enables a nearly optimal O(n logn) parallel time spanning line construction which improves dramatically on the best previously known O(n) parallel time solution. • We define a probabilistic version of bubble-sort in which random comparisons are allowed only between adjacent numbers in the sequence being sorted. We show that rather surprisingly this probabilistic bubblesort requires O(n) comparisons in expectation, i.e., on the same level as its deterministic counterpart. • We propose the first self-replication protocol allowing to reproduce a strand (line-segment carrying information) of length k in parallel time O(n(k + log n)). This result is based on the probabilistic bubble-sort argument. This protocol permits also simultaneous replication where l copies of the strand can be obtained in time O(n(k + logn) log l). All protocols in this paper operate with high probability defined as 1 − n−η, for a constant η > 0.