Sublinear-time distributed algorithms for detecting small cliques and even cycles

In this paper we give sublinear-time distributed algorithms in the 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 O~ (n^{5 / 6}) to 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 Ω~(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 δ∈ (0 , 1 / 2) such that for anyk, a lower bound of Ω(n^1/2+δ) 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 Ω~ (n^2-Θ(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-Θ(1/k)) rounds, nearly matching the lower bound.