[论文解读] Efficient k-Party Voting with Two Choices.

Robert Elsässer⋆, Tom Friedetzky+3 more|arXiv (Cornell University)|Feb 15, 2016

Distributed systems and fault tolerance参考文献 12被引用 9

一句话总结

本文提出了一种针对完全图中k方共识的改进型双选投票协议，其中节点根据采样邻居的意见进行表态，并通过基于阶段的位机制传播意见。结果表明，当初始意见差距为Ω(√(n log n))时，该协议以高概率在O(k log n)步内收敛至多数意见，显著优于标准协议的性能。

ABSTRACT

We consider the problem of distributed $k$-party voting with two choices as well as a simple modification of this protocol in complete graphs. In the standard version, we are given a graph in which every node possesses one of $k$ different opinions at the beginning. In each step, every node chooses two neighbors uniformly at random. If the opinions of the two neighbors coincide, then this opinion is adopted. It is known that if $k=2$ and the difference between the two opinions is $\Omega(\sqrt{n \log n})$, then after $\mathcal{O}(\log n)$ steps, every node will possess the largest initial opinion, with high probability. We show that if $k =\mathcal{O}(n^\epsilon)$ for some small $\epsilon$, then this protocol converges to the initial majority within $\mathcal{O}(k\log{n})$ steps, with high probability, as long as the initial difference between the largest and second largest opinion is $\Omega(\sqrt{n \log n})$. Furthermore, there exist initial configurations where the $\Theta(k)$ bound on the run time is matched. If the initial difference is $\mathcal{O}(\sqrt{n})$, then the largest opinion may loose the vote with constant probability. To speed up our process, we consider the following variant of the two-choices protocol. The process is divided into several phases, and in the first step of a phase every node applies the two choices protocol. If a new opinion is adopted, the node remembers it by setting a certain bit to true. In the subsequent steps of that phase, each node samples one neighbor, and if the bit of this neighbor is set to true, then the node takes the opinion of this neighbor and sets its bit to true as well. At the end of the phase, the bits are reset to false. Then, the phases are repeated several times. We show that this modified protocol improves significantly over the standard two-choices protocol.

研究动机与目标

• 分析在每个节点有两次选择的k方设置下，标准双选投票协议的收敛时间。
• 识别标准协议在初始意见差距较小时无法收敛至多数意见的条件，特别是当初始差距较小时的情况。
• 设计并分析一种改进协议，通过基于位的阶段机制传播意见，以加速收敛。
• 在不同初始意见差异下，建立收敛时间与失败概率的理论边界。

提出的方法

• 在每一步中应用标准协议：每个节点随机采样两个邻居，并在两者意见一致时采纳其意见。
• 改进协议引入阶段机制：在每个阶段中，节点首先应用双选规则，若采纳了新意见，则将其“位”设为true。
• 在阶段剩余时间内，节点采样一个邻居，若邻居的位为true，则采纳其意见，并将自己的位设为true。
• 每个阶段结束时，所有位被重置为false，该过程在多个阶段中重复，以增强意见传播。
• 该协议利用位机制，比标准协议更高效地传播意见，尤其是在早期阶段。
• 理论分析采用概率方法，基于初始意见差异来界定收敛时间和失败概率。

实验结果

研究问题

• RQ1 当初始意见差距为Ω(√(n log n))时，标准双选协议在k方投票中的收敛时间是多少？
• RQ2 当初始多数意见差距为Ω(√(n log n))时，收敛时间如何随k变化？
• RQ3 当初始意见差距仅为O(√n)时，协议性能会发生什么变化？
• RQ4 基于阶段的位机制是否能显著提升收敛速度，优于标准协议？
• RQ5 在何种条件下，改进协议无法收敛至多数意见？

主要发现

• 当最大意见与第二大致意见之间的初始差距为Ω(√(n log n))时，标准双选协议以高概率在O(k log n)步内收敛至初始多数意见。
• 存在某些初始配置，使得标准协议的收敛时间为Θ(k)，证明了O(k log n)边界的紧致性。
• 若初始意见差距为O(√n)，则最大意见可能以常数概率输掉投票，表明存在一个失败阈值。
• 采用基于阶段的位传播机制的改进协议在收敛速度上显著优于标准协议。
• 改进协议通过在每个阶段内实现有针对性的多步传播，加速了多数意见的放大。
• 理论分析证实，在相同初始差距条件下，改进协议仍能以高概率收敛至多数意见。