Detecting Cliques in CONGEST Networks
The problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph H, the input network contains a subgraph isomorphic to H or not. We investigate this problem for H being a clique K_l in the classical distributed CONGEST model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links.
Our first and main result is a lower bound, showing that detecting K_l requires Omega(sqrt{n} / b) communication rounds, for every 4 <=l <=sqrt{n}, and Omega(n / (l b)) rounds for every l >= sqrt{n}, where b is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for K_4 detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework.
Lower bounds
CONGEST
subgraph detection
two-party communication
Theory of computation~Distributed algorithms
Networks~Network algorithms
16:1-16:15
Regular Paper
Artur
Czumaj
Artur Czumaj
Department of Computer Science and Centre for Discrete Mathematics and its Applications (DIMAP), University of Warwick, UK
Research partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP), by EPSRC award EP/D063191/1, by EPSRC award EP/N011163/1, and by an IBM Faculty Award.
Christian
Konrad
Christian Konrad
Department of Computer Science, University of Bristol, UK
Most of work on this paper has been carried out while the author was at the University of Warwick, where he was supported by the Centre for Discrete Mathematics and its Applications (DIMAP) and by EPSRC award EP/N011163/1.
10.4230/LIPIcs.DISC.2018.16
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Artur Czumaj and Christian Konrad
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