[论文解读] Graph Spanners in the Message-Passing Model

Manuel Fernández, David P. Woodruff+1 more|arXiv (Cornell University)|Nov 14, 2019

Advanced Graph Theory Research被引用 2

一句话总结

本文提出了在多个站点间划分边的分布式消息传递模型中计算图聚类的首个通信高效算法。它为加法和乘法聚类建立了紧致的通信复杂度界限，揭示了允许与不允许边复制的模型之间的差异，并在通信成本与聚类质量之间提供了近乎最优的权衡。

ABSTRACT

Graph spanners are sparse subgraphs which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been extensively studied. We study the problem of computing a graph spanner when the edges of the input graph are distributed across two or more sites in an arbitrary, possibly worst-case partition, and the goal is for the sites to minimize the communication used to output a spanner. We assume the message-passing model of communication, for which there is a point-to-point link between all pairs of sites as well as a coordinator who is responsible for producing the output. We stress that the subset of edges that each site has is not related to the network topology, which is fixed to be point-to-point. While this model has been extensively studied for related problems such as graph connectivity, it has not been systematically studied for graph spanners. We present the first tradeoffs for total communication versus the quality of the spanners computed, for two or more sites, as well as for additive and multiplicative notions of distortion. We show separations in the communication complexity when edges are allowed to occur on multiple sites, versus when each edge occurs on at most one site. We obtain nearly tight bounds (up to polylog factors) for the communication of additive $2$-spanners in both the with and without duplication models, multiplicative $(2k-1)$-spanners in the with duplication model, and multiplicative $3$ and $5$-spanners in the without duplication model. Our lower bound for multiplicative $3$-spanners employs biregular bipartite graphs rather than the usual Erd\H{o}s girth conjecture graphs and may be of wider interest.

研究动机与目标

• 填补在多站点消息传递系统中对分布式图聚类计算通信复杂度理解的空白。
• 研究边在各站点间分布的方式如何影响具有有界失真度的聚类计算的通信成本。
• 在允许和不允许边复制的模型中，建立通信开销与聚类质量（加法或乘法失真）之间的权衡关系。
• 为多种聚类类型（包括加法2-聚类和乘法(2k−1)-聚类）提供近乎紧致的通信复杂度上下界。
• 提出基于双正则二分图的新下界技术，提供了超越传统基于围长构造的全新洞见。

提出的方法

• 在包含协调者和多个站点的消息传递设置中建模问题，每个站点持有输入图中边的子集。
• 设计在站点与协调者之间通信最少的分布式算法，以计算满足加法和乘法失真保证的聚类。
• 采用一种协调模型，站点通过点对点链路交换消息，协作构建聚类而不集中所有数据。
• 提出一种基于双正则二分图的新下界技术，用于证明乘法3-聚类的通信复杂度极限。
• 在两种模型中分析通信复杂度：一种允许边复制（边存储在多个站点），另一种强制唯一边存储。
• 利用聚类的结构特性和图的稀疏性，推导出在多对数因子内的紧致界限。

实验结果

研究问题

• RQ1 在消息传递模型中，当边被分布在多个站点时，计算图聚类所需的最小通信量是多少？
• RQ2 边在各站点间存在复制时，如何影响具有加法或乘法失真的聚类计算的通信复杂度？
• RQ3 能否为特定聚类类型（如加法2-聚类和乘法(2k−1)-聚类）建立近乎紧致的通信复杂度界限？
• RQ4 可以开发何种新颖的下界技术以确立聚类计算的通信复杂度极限？
• RQ5 对于各种聚类类型，允许复制与不允许复制的模型在通信需求上存在哪些差异？

主要发现

• 本文在允许和不允许边复制的模型中，为计算加法2-聚类建立了近乎紧致的通信复杂度界限（多对数因子内）。
• 对于乘法(2k−1)-聚类，本文在允许复制的模型中提供了近乎最优的通信界限。
• 在不允许复制的模型中，对乘法3-聚类和5-聚类也实现了近乎紧致的界限。
• 提出了一种基于双正则二分图的新下界技术，其结果比依赖Erd\'os围长猜想的先前方法更强且更具普遍性。
• 研究揭示了在允许和不允许边复制的模型之间存在严格的通信复杂度分离，尤其体现在乘法聚类上。
• 结果表明，在不允许复制的模型中，通信成本显著更高，尤其对于具有小乘法失真的聚类。