Fast computation by population protocols with a leader - Angluin, Dana; Aspnes, James; Eisenstat, David (Distributed computing, 2008)
Fast algorithms are presented for performing computations in a probabilistic population model. This is a variant of the standard population protocol model, in which finite-state agents interact in pairs under the control of an adversary scheduler, where all pairs are equally likely to be chosen for each interaction. It is shown that when a unique leader agent is provided in the initial population, the population can simulate a virtual register machine with high probability in which standard arithmetic operations like comparison, addition, subtraction, and multiplication and division by constants can be simulated in O ( n log 5 n ) interactions using a simple register representation or in O ( n log 2 n ) interactions using a more sophisticated representation that requires an extra O ( n log O (1) n )-interaction initialization step. The central method is the extensive use of epidemics to propagate information from and to the leader, combined with an epidemic-based phase clock used to detect when these epidemics are likely to be complete. Applications include a reduction of the cost of computing a semilinear predicate to O ( n log 5 n ) interactions from the previously best-known bound of O ( n 2 log n ) interactions and simulation of a LOGSPACE Turing machine using O ( n log 2 n ) interactions per step after an initial O ( n log O (1) n )-interaction startup phase. These bounds on interactions translate into polylogarithmic time per step in a natural parallel model in which each agent participates in an expected Θ (1) interactions per time unit. Open problems are discussed, together with simulation results that suggest the possibility of removing the initial-leader assumption.