Fast computation by population protocols with a leader

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 LOG-SPACE 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 Theta(1) interactions per time unit. Open problems are discussed, together with simulation results that suggest the possibility of removing the initial-leader assumption.