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

Talya Eden, Nimrod Fiat, Orr Fischer*, Fabian Kuhn, Rotem Oshman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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(n5/6+o(1)) rounds and O(n73/75+o(1)) rounds, respectively. For even-length cycles, C2k, 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~ (n5 / 6) to O~ (n3 / 4) rounds. We also show two obstacles on proving lower bounds for C2k-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(n1/2+δ) on C2k-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~ (n2-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(n2-Θ(1/k)) rounds, nearly matching the lower bound.

Original languageEnglish
Pages (from-to)207-234
Number of pages28
JournalDistributed Computing
Issue number3
StatePublished - Jun 2022


FundersFunder number
Laura Schwarz-Kipp Institute of Computer Networks
Israel Science Foundation1146/18
Azrieli Foundation
Israeli Centers for Research Excellence4/11


    • Distributed computing
    • Expander decomposition
    • Subgraph freeness


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