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 subgraph detection for two classes of graphs: 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 listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time.
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 Omega~(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 Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity.
For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(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 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018].
Distributed Computing
Subgraph Freeness
CONGEST
Networks~Network algorithms
Theory of computation~Distributed algorithms
Theory of computation~Lower bounds and information complexity
15:1-15:16
Regular Paper
A full version of the paper is available at https://www.cs.tau.ac.il/~roshman/papers/DISC19_EFFKO19.pdf.
Talya
Eden
Talya Eden
Electrical Engineering Department, Tel-Aviv University, Israel
The Israel Science Foundation grant No.1146/18. Talya Eden is greatful to the Azrieli Foundation for an award of an Azrieli Fellowship.
Nimrod
Fiat
Nimrod Fiat
Computer Science Department, Tel-Aviv University, Israel
The Israel Science Foundation grant No.1146/18 and the Kadar family award.
Orr
Fischer
Orr Fischer
Computer Science Department, Tel-Aviv University, Israel
The Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11) and the Laura Schwarz-Kipp Institute of Computer Networks.
Fabian
Kuhn
Fabian Kuhn
Computer Science Department, University of Freiburg, Germany
Rotem
Oshman
Rotem Oshman
Computer Science Department, Tel-Aviv University, Israel
The Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).
10.4230/LIPIcs.DISC.2019.15
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Talya Eden, Nimrod Fiat, Orr Fischer, Fabian Kuhn, and Rotem Oshman
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