Smoothed Analysis of Population Protocols

Abstract

In this work, we initiate the study of smoothed analysis of population protocols. We consider a population protocol model where an adaptive adversary dictates the interactions between agents, but with probability p every such interaction may change into an interaction between two agents chosen uniformly at random. That is, p-fraction of the interactions are random, while (1 − p)-fraction are adversarial. The aim of our model is to bridge the gap between a uniformly random scheduler (which is too idealistic) and an adversarial scheduler (which is too strict).We focus on the fundamental problem of leader election in population protocols. We show that, for a population of size n, the leader election problem can be solved in O(p −2 n log 3 n) steps with high probability, using O((log 2 n) • (log(n/p))) states per agent, for all values of p ≤ 1. Although our result does not match the best known running time of O(n log n) for the uniformly random scheduler (p = 1), we are able to present a smooth transition between a running time of O(n polylog n) for p = 1 and an infinite running time for the adversarial scheduler (p = 0), where the problem cannot be solved. The key technical contribution of our work is a novel phase clock algorithm for our model. This is a key primitive for much-studied fundamental population protocol algorithms (leader election, majority), and we believe it is of independent interest.