In this paper we give sublinear-time distributed algorithms in the \(\mathsf {CONGEST}\) model for finding or listing cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be detected and listed in sublinear time, \(O(n^{5/6+o(1)})\) rounds and \(O(n^{73/75+o(1)})\) rounds, respectively. For even-length cycles, \(C_{2k}\), we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from \({\tilde{O}}(n^{5/6})\) to \({\tilde{O}}(n^{3/4})\) rounds. We also show two obstacles on proving lower bounds for \(C_{2k}\)-freeness: first, we use the new connection to extremal combinatorics to show that the current lower bound of \({\tilde{\varOmega }}(\sqrt{n})\) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant \(\delta \in (0,1/2)\) such that for any k, a lower bound of \(\varOmega (n^{1/2+\delta })\) on \(C_{2k}\)-freeness would imply new lower bounds in circuit complexity. We use the same technique to show a barrier for proving any polynomial lower bound on triangle-freeness. For general subgraphs, it was shown by Fischer et al. that for any fixed k, there exists a subgraph H of size k such that H-freeness requires \({\tilde{\varOmega }}(n^{2-\varTheta (1/k)})\) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in \(O(n^{2 - \varTheta (1/k)})\) rounds, nearly matching the lower bound.

在本文中，我们给出了\(\mathsf {CONGEST}\)模型中的亚线性时间分布式算法，用于查找或列出派系和偶数长度循环。我们首次证明了网络图中 4-cliques 和 5-cliques 的所有副本都可以在次线性时间内检测并列出，\(O(n^{5/6+o(1)})\)轮和\(O(n^{73/75+o(1)})\)轮，分别。对于偶数周期\(C_{2k}\)，我们给出了一种改进的亚线性时间算法，它利用了与极值组合的新联系。例如，对于 6 个周期，我们将运行时间从\({\tilde{O}}(n^{5/6})\) 改进为\({\tilde{O}}(n^{3/4 })\)回合。我们还展示了证明下界的两个障碍\(C_{2k}\) -freeness：首先，我们使用与极值组合的新连接来表明\({\tilde{\varOmega }}(\sqrt{n})\)的当前下界为6 周期自由度无法使用基于分区的 2 方通信复杂性降低来提高，该技术迄今为止已证明所有已知的子图检测下界都采用这种技术。其次，我们证明存在一些固定常数\(\delta \in (0,1/2)\)使得对于任何 k，\(\varOmega (n^{1/2+\delta } )\)在\(C_{2k}\)-freeness 意味着电路复杂性的新下限。我们使用相同的技术来展示证明三角形自由度的任何多项式下界的障碍。对于一般子图，它由 Fischer 等人展示。对于任何固定的k，存在大小为k的子图H使得H自由度需要\({\tilde{\varOmega }}(n^{2-\varTheta (1/k)})\)轮。这是否是紧密的，或者某些恒定大小的子图是否需要真正的二次时间来检测，这仍然是一个悬而未决的问题。我们证明，事实上，对于任何大小为k 的子图H，H自由度问题可以解决为\(O(n^{2 - \varTheta (1/k)})\)轮，几乎匹配下限。