Nearly-Tight Analysis for 2-Choice and 3-Majority Consensus Dynamics

We present improved analyses for two of the well-studied randomized dynamics of stabilizing consensus, namely 2-choice and 3-majority. The resulting bounds are tight up to logarithmic factors. In the latter case, this answers an open question of Becchetti et al. [SODA'16]. Consider a distributed system of n nodes, each initially holding an opinion in \1, 2, ...., k\ . The system should converge to a setting where all (non-corrupted) nodes hold the same opinion. This consensus opinion should be valid, meaning that it should be among the initially supported opinions, and the (fast) convergence should happen even in the presence of a malicious adversary who can corrupt a bounded number of nodes per round and in particular modify their opinions. Two of the well-studied distributed algorithms for this problem work as follows: In both of these dynamics, each node gathers the opinion of three nodes and sets its new opinion equal to the majority of this set. In the 2-choice dynamics, the three nodes are the node itself and two random nodes and ties are broken towards the node's own opinion. In the 3-majority dynamics, the three nodes are selected at random and ties are broken randomly. Becchetti et al. [SODA'16] showed that the 3-majority dynamics converges to consensus in O((k^2\sq√rtlog n + klog n)(k+log n)) rounds, even in the presence of a limited adversary, for k bounded by some polynomial of n. We prove that, even with a stronger adversary, the convergence happens within O(klog n) rounds, both for 3-majority and 2-choice. For 3-majority, this bound is known to be optimal for k=\tildeO (√ ). More generally, we prove that 3-majority converges always in Õ (n^2/3 ) time, improving on a Õ (n^3/4 ) upper bound of Berenbrink et al. [PODC'17].