论文标题
关于集团的分布式清单
On Distributed Listing of Cliques
论文作者
论文摘要
我们显示了$ \ tilde {o}（n^{p/（p+2）}）$ - 在\ colest模型中\ emph {listing}的$ k_p $（带有$ p $ nodes的集团）的圆形算法，用于所有$ p = 4，p = 4，p = 4，p \ geq 6 $。对于$ p = 5 $，我们显示$ \ tilde {o}（n^{3/4}）$ - 圆形算法。 对于$ p = 4 $和$ p = 5 $，我们的结果在$ o（n^{5/6+o（1）}）的先前最先前的成果上，分别为$ O（N^{21/22+o（1）}）$。 [DISC 2019]。对于所有$ p \ geq 6 $，我们的是第一个用于$ k_p $上市的子线性圆形算法。 我们利用Chang等人最近的扩展器分解算法。 [SODA 2019]以良好的混合时间创建簇。我们算法中的三个主要新颖性是：（1）我们仔细迭代清单过程，并在集群外的簇和支撑物中的最低度值耦合，（2）所有列表都在集群中完成，这需要在群集中完成新的技术，从而将有关\ emph $ emph的信息带入cluster的信息。在每个群集中，我们使用一种稀疏感知列表算法，该算法比一般清单算法快，我们可以允许群集使用，因为我们确保在迭代过程中对图形进行稀疏。 作为我们算法的副产品，我们显示了$ k_p $ listing的\ emph {optimal} sparsity-awawawawawawawawawawawawawawawawawawawawawawawe算法，该算法以$ \tildeθ（1 + m/n^{1 + 2/p}）运行。以前，Pandurangan等人。 [SPAA 2018]，Chang等。 [SODA 2019]和Censor-Hillel等。 [TCS 2020]显示了$ p = 3 $的稀疏感知算法，但我们的算法是第一个以$ p \ geq 4 $的稀疏感知算法。
We show an $\tilde{O}(n^{p/(p+2)})$-round algorithm in the \congest model for \emph{listing} of $K_p$ (a clique with $p$ nodes), for all $p =4, p\geq 6$. For $p = 5$, we show an $\tilde{O}(n^{3/4})$-round algorithm. For $p=4$ and $p=5$, our results improve upon the previous state-of-the-art of $O(n^{5/6+o(1)})$ and $O(n^{21/22+o(1)})$, respectively, by Eden et al. [DISC 2019]. For all $p\geq 6$, ours is the first sub-linear round algorithm for $K_p$ listing. We leverage the recent expander decomposition algorithm of Chang et al. [SODA 2019] to create clusters with a good mixing time. Three key novelties in our algorithm are: (1) we carefully iterate our listing process with coupled values of min-degree within the clusters and arboricity outside the clusters, (2) all the listing is done within the cluster, which necessitates new techniques for bringing into the cluster the information about \emph{all} edges that can potentially form $K_p$ instances with the cluster edges, and (3) within each cluster we use a sparsity-aware listing algorithm, which is faster than a general listing algorithm and which we can allow the cluster to use since we make sure to sparsify the graph as the iterations proceed. As a byproduct of our algorithm, we show an \emph{optimal} sparsity-aware algorithm for $K_p$ listing, which runs in $\tildeΘ(1 + m/n^{1 + 2/p})$ rounds in the \clique model. Previously, Pandurangan et al. [SPAA 2018], Chang et al. [SODA 2019], and Censor-Hillel et al. [TCS 2020] showed sparsity-aware algorithms for the case of $p = 3$, yet ours is the first such sparsity aware algorithm for $p \geq 4$.
