Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts the opinion of that neighbour. The voting is said to be completed when all vertices hold the same opinion. On many graph classes including regular graphs, irrespective of the expansion properties, pull voting requires Ω(n) expected time steps to complete, even if initially there are only two distinct opinions with the minority opinion being sufficiently large. In this paper we consider a related process which we call two-sample voting. In this process every vertex chooses two random neighbors in each step. If the opinions of these neighbors coincide, then the vertex revises its opinion according to the chosen sample. Otherwise, it keeps its own opinion. We consider the performance of this process in the case where two different opinions reside on vertices of some (arbitrary) sets A and B, respectively. Here, |A| + |B| = n is the number of vertices of the graph. We show that there is a constant K such that if the initial imbalance between the two opinions is ν 0 = (|A| − |B|)/n ≥ K (1/d) + (d/n), then with high probability two sample voting completes in a random d regular graph in O(log n) steps and the initial majority opinion wins. We also show the same performance for any regular graph, if ν 0 ≥ Kλ 2 , where λ 2 is the second largest eigenvalue of the transition matrix. In the graphs we consider, standard pull voting requires Ω(n) steps, and the minority can still win with probability |B|/n. Our results hold even if an adversary is able to rearrange the opinions in each step, and has complete knowledge of the graph structure.
For random d-regular graphs, REF proves that all nodes agree on the majority color in O(log n) rounds, provided that the bias is ω(n · 1/d + d/n).
The Power of Two Choices in Distributed Voting
ArXiv
ArXiv
Mathematics Computer Science