[Paper Review] The Power of Two Choices in Distributed Voting

Colin Cooper, Robert Elsässer⋆+1 more|arXiv (Cornell University)|Apr 29, 2014

Distributed systems and fault tolerance22 references11 citations

TL;DR

This paper introduces two-sample voting, a distributed consensus protocol where each node samples two neighbors and adopts their opinion only if both agree, otherwise retaining its own. It proves that with initial imbalance above a threshold, the process completes in O(log n) steps with high probability on regular and random d-regular graphs, ensuring the majority opinion wins even under adversarial interference.

ABSTRACT

Distributed voting is a fundamental topic in distributed computing. In pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts its opinion. The voting is completed when all vertices hold the same opinion. On many graph classes including regular graphs, pull voting requires $Θ(n)$ expected steps to complete, even if initially there are only two distinct opinions. In this paper we consider a related process which we call two-sample voting: every vertex chooses two random neighbours in each step. If the opinions of these neighbours 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 ?$ν_0 = (|A| - |B|)/n \geq K \sqrt{(1/d) + (d/n)}$, 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 $ν_0 \geq K λ_2$ 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 $|B|/n$.

Motivation & Objective

• Address the inefficiency of standard pull voting, which requires Ω(n) time on regular graphs and allows minority opinions to win with probability proportional to size.
• Design a distributed voting protocol that achieves consensus in logarithmic time while ensuring the initial majority opinion wins with high probability.
• Analyze the performance of the two-sample voting process under adversarial conditions, where an adversary can reassign opinions at each step.
• Establish theoretical bounds on the required initial imbalance for fast convergence and majority dominance in regular and random d-regular graphs.

Proposed method

• Propose two-sample voting: each vertex independently selects two random neighbors per round and updates its opinion only if both neighbors agree.
• Use a three-phase analysis: Phase I reduces imbalance from initial minority size to n/10; Phase II reduces from n/10 to ω = log n / log log n; Phase III reduces from ω to 0.
• Apply spectral graph theory, using the second largest eigenvalue λ₂ of the transition matrix to bound convergence time in regular graphs.
• For random d-regular graphs, use expansion properties and eigenvalue concentration to show that the minority vote diminishes rapidly.
• Employ probabilistic concentration bounds and drift analysis to show that the minority size decreases exponentially in phases, with high probability.
• Account for adversarial interference by modeling opinion changes as bounded perturbations and proving that the protocol remains robust as long as the adversary alters o(ν₀n) vertices.

Experimental results

Research questions

• RQ1 Can a distributed voting protocol achieve consensus in O(log n) steps on regular graphs, where standard pull voting requires Ω(n) steps?
• RQ2 What is the minimal initial imbalance ν₀ required for the majority opinion to win with high probability in two-sample voting?
• RQ3 How does the two-sample voting process perform under adversarial reassignment of opinions during execution?
• RQ4 Can the convergence time and majority dominance be guaranteed using spectral properties of the graph, such as the second eigenvalue λ₂?
• RQ5 Does the two-sample mechanism outperform standard pull voting in terms of both speed and correctness, even when the minority opinion is large?

Key findings

• Two-sample voting completes in O(log n) steps with high probability on d-regular graphs when the initial imbalance ν₀ ≥ K√((1/d) + (d/n)) for some constant K.
• For any regular graph, if ν₀ ≥ Kλ₂, then two-sample voting completes in O(log n) steps and the initial majority opinion wins with high probability.
• The minority vote is reduced from ω = log n / log log n to 0 in O(ω log ω) = O(log n) steps with high probability, under the same conditions.
• Even under adversarial interference that reassigns o(ν₀n) vertices per step, all but O(f) vertices adopt the majority opinion within O(log n) steps with high probability.
• Standard pull voting requires Ω(n) steps on regular graphs and allows the minority opinion to win with probability |B|/n, which is not acceptable for fast consensus.
• The two-sample mechanism ensures fast convergence and majority dominance even when the minority opinion is initially large, provided the imbalance exceeds a threshold dependent on graph structure.