Sublinear-Time Distributed Algorithms for Detecting Small Cliques and Even Cycles.

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.