[论文解读] Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets

Shreyas Pai, Gopal Pandurangan+3 more|arXiv (Cornell University)|Jan 1, 2017

Complexity and Algorithms in Graphs参考文献 7被引用 1

一句话总结

本文提出了Congest模型中首个时间与消息效率的独立集算法，打破了2-和3-规则集长期存在的O(log n)轮复杂度屏障。在特定度数条件下，3-规则集实现O(log n / log log n)轮，2-规则集实现o(log n)轮，同时仅使用O(n log²n)条消息——接近线性，显著优于Luby的O(m)消息界限。

ABSTRACT

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Delta is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Theta(log n) time complexity barrier and the Theta(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A beta-ruling set is an independent set such that every node in the graph is at most beta hops from a node in the independent set. We present the following results: - Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O(log n/log log n) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon > 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. - Message Complexity: We show an Omega(n^2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^2 n) messages and runs in O(Delta log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).

研究动机与目标

• 解决Congest模型中对称性破缺算法在时间与消息效率方面的不足，特别是针对最大独立集（MIS）及其相关问题。
• 突破Luby算法在Congest模型中规则集问题长期存在的O(log n)轮复杂度屏障，尽管局部模型（Local model）已有进展。
• 设计消息高效的算法，因先前工作大多忽略消息复杂度，而Luby算法在m条边的图上使用O(m)条消息。
• 为2-规则集建立接近最优的O(n log²n)消息复杂度，显著优于O(m)的界限。
• 刻画Congest模型中规则集的时间与消息复杂度之间的权衡，尤其针对2-规则集。

提出的方法

• 设计一种针对3-规则集的随机化分布式算法，以高概率（whp）在O(log n / log log n)轮内完成，采用新颖的采样与剪枝技术。
• 引入分层聚类方法处理2-规则集，结合局部计算与通过随机采样实现的全局协调，使轮复杂度在Δ的广泛取值范围内低于O(log n)。
• 通过构建桥接图证明Congest模型中MIS的Ω(n²)消息复杂度下界，利用概率分析与超几何分布来界定桥接边的发现概率。
• 应用马尔可夫不等式表明，执行过程中发现桥接边的概率为o(1)，从而可将问题简化为更简单的基图进行分析。
• 利用算法在桥接图与基图上行为一致的性质（只要未发现桥接边），从而可借用已知简单图的下界结果。
• 构建一种随机化2-规则集算法，仅使用O(n log²n)条消息，运行时间为O(Δ log n)轮，实现近乎线性的消息复杂度，并在低度图中达到亚对数时间。

实验结果

研究问题

• RQ1 尽管局部模型已有进展，是否能在Congest模型中打破规则集问题的O(log n)轮复杂度屏障？
• RQ2 是否可能在Congest模型中为2-规则集设计出在Δ的广泛取值范围内轮复杂度为o(log n)的算法？
• RQ3 规则集在Congest模型中的最优消息复杂度是多少？能否使其接近n的线性函数？
• RQ4 能否刻画2-规则集的时间-消息权衡，特别是实现O(polylog n)轮与O(n polylog n)条消息的组合？
• RQ5 能否将局部模型中MIS的下界结果推广至Congest模型中的2-规则集？

主要发现

• 本文首次在Congest模型中提出3-规则集算法，实现O(log n / log log n)轮复杂度，打破了O(log n)的屏障。
• 对于2-规则集，算法实现o(log n)轮复杂度，具体为O(log ∆ · (log n)^{1/2+ε} + log n / log log n)，其中任意ε > 0，当Δ = 2^{(log n)^{1/2−ε}}时为亚对数复杂度。
• 提出一种随机化2-规则集算法，仅使用O(n log²n)条消息（高概率下），实现近乎线性的消息复杂度。
• 本文建立了在Congest模型中计算MIS（1-规则集）的Ω(n²)消息复杂度下界，即使对随机化算法也成立。
• 所提2-规则集算法的时间与消息复杂度分别为O(Δ log n)轮与O(n log²n)条消息，相较以往工作有显著改进。
• 结果表明，在Congest模型中可同时提升时间与消息复杂度，挑战了O(log n)轮与O(m)消息为固有极限的假设。