团的高效分布式列举
Tight Distributed Listing of Cliques
AI总结：
本文提出针对所有$p \geq 4$的$K_p$-列举的$\tilde{O}(n^{1 - 2/p})$-轮分布式算法，完全弥合了上界与下界差距，证明在CONGEST和CONGESTED CLIQUE模型中轮复杂度相同，并匹配$K_4$检测下界。
摘要：
近年来，在分布式计算的CONGEST模型中，理解子图发现问题复杂度格局方面取得了很大进展。然而，迄今为止，该领域中已知的紧界非常少。对于三角形（即3-团）列举，Chang等人[SODA 2019, PODC 2019]构造了一个最优的$\tilde{O}(n^{1/3})$-轮分布式算法。Eden等人[DISC 2019]和Censor-Hillel等人[PODC 2020]的近期工作展示了对于每个$p \geq 4$的$K_p$-列举的次线性算法，但仍然在上界与已知下界之间留下了显著差距。在本文中，我们完全弥合了这一差距。我们证明对于每个$p \geq 4$，存在一个$\tilde{O}(n^{1 - 2/p})$-轮分布式算法，在通信网络中列举所有$p$-团$K_p$。我们的算法在多项式对数因子内是最优的，这归因于Fischer等人[SPAA 2018]的$\tilde{\Omega}(n^{1 - 2/p})$-轮下界，该下界即使在CONGESTED CLIQUE模型中也成立。结合Chang等人[SODA 2019, PODC 2019]的三角形列举算法，我们的结果因此表明，对于所有$p$，$K_p$-列举的轮复杂度在CONGEST和CONGESTED CLIQUE模型中都是相同的，均为$\tilde{\Theta}(n^{1 - 2/p})$轮。对于$p=4$，我们的结果还匹配了Czumaj和Konrad [DISC 2018]关于$K_4$-\textit{检测}的$\tilde{\Omega}(n^{1/2})$下界，这意味着在CONGEST模型中，$K_4$的检测和列举的轮复杂度是等价的。
英文摘要：
Much progress has recently been made in understanding the complexity landscape of subgraph finding problems in the CONGEST model of distributed computing. However, so far, very few tight bounds are known in this area. For triangle (i.e., 3-clique) listing, an optimal $\tilde{O}(n^{1/3})$-round distributed algorithm has been constructed by Chang et al.~[SODA 2019, PODC 2019]. Recent works of Eden et al.~[DISC 2019] and of Censor-Hillel et al.~[PODC 2020] have shown sublinear algorithms for $K_p$-listing, for each $p \geq 4$, but still leaving a significant gap between the upper bounds and the known lower bounds of the problem. In this paper, we completely close this gap. We show that for each $p \geq 4$, there is an $\tilde{O}(n^{1 - 2/p})$-round distributed algorithm that lists all $p$-cliques $K_p$ in the communication network. Our algorithm is \emph{optimal} up to a polylogarithmic factor, due to the $\tildeΩ(n^{1 - 2/p})$-round lower bound of Fischer et al.~[SPAA 2018], which holds even in the CONGESTED CLIQUE model. Together with the triangle-listing algorithm by Chang et al.~[SODA 2019, PODC 2019], our result thus shows that the round complexity of $K_p$-listing, for all $p$, is the same in both the CONGEST and CONGESTED CLIQUE models, at $\tildeΘ(n^{1 - 2/p})$ rounds. For $p=4$, our result additionally matches the $\tildeΩ(n^{1/2})$ lower bound for $K_4$-\emph{detection} by Czumaj and Konrad [DISC 2018], implying that the round complexities for detection and listing of $K_4$ are equivalent in the CONGEST model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
