Nucleus decompositions have been shown to be a useful tool for finding dense subgraphs. The coreness value of a clique represents its density based on the number of other cliques it is adjacent to. One useful output of nucleus decomposition is to generate a hierarchy among dense subgraphs at different resolutions. However, existing parallel algorithms for nucleus decomposition do not generate this hierarchy, and only compute the coreness values. This paper presents a scalable parallel algorithm for hierarchy construction, with practical optimizations, such as interleaving the coreness computation with hierarchy construction and using a concurrent union-find data structure in an innovative way to generate the hierarchy. We also introduce a parallel approximation algorithm for nucleus decomposition, which achieves much lower span in theory and better performance in practice. We prove strong theoretical bounds on the work and span (parallel time) of our algorithms. On a 30-core machine with two-way hyper-threading on real-world graphs, our parallel hierarchy construction algorithm achieves up to a 58.84x speedup over the state-of-the-art sequential hierarchy construction algorithm by Sariyuce et al. and up to a 30.96x self-relative parallel speedup. On the same machine, our approximation algorithm achieves a 3.3x speedup over our exact algorithm, while generating coreness estimates with a multiplicative error of 1.33x on average.

翻译：核分解已被证明是寻找稠密子图的有效工具。团的核值基于其相邻团的数量来表征密度。核分解的一个重要用途是在不同分辨率下生成稠密子图间的层次结构。然而，现有核分解的并行算法仅计算核值，并未生成这种层次结构。本文提出了一种可扩展的并行层次构建算法，并辅以实用优化技术，例如将核值计算与层次构建交错进行，以及创新性地利用并发联合-查找数据结构来生成层次结构。我们还引入了一种核分解的并行近似算法，该算法在理论上实现了更低的跨度，并在实践中展现出更优的性能。我们证明了算法在计算量和跨度（并行时间）方面的强理论界限。在配备超线程技术的30核机器上，针对真实世界图数据，我们的并行层次构建算法相比Sariyuce等人提出的最先进串行层次构建算法，实现了高达58.84倍的加速比，并取得了30.96倍的自相对并行加速比。在同一台机器上，我们的近似算法比精确算法实现了3.3倍的加速比，同时生成的核值估计平均乘法误差为1.33倍。