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Lower Bounds for Induced Cycle Detection in Distributed Computing

Authors François Le Gall, Masayuki Miyamoto



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François Le Gall
  • Graduate School of Mathematics, Nagoya University, Japan
Masayuki Miyamoto
  • Graduate School of Mathematics, Nagoya University, Japan

Acknowledgements

The authors are grateful to Keren Censor-Hillel, Orr Fischer, Pierre Fraigniaud, Dean Leitersdorf and Rotem Oshman for helpful discussions and comments, and Shin-ichi Minato for his support.

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François Le Gall and Masayuki Miyamoto. Lower Bounds for Induced Cycle Detection in Distributed Computing. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 58:1-58:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.58

Abstract

The distributed subgraph detection asks, for a fixed graph H, whether the n-node input graph contains H as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently. In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the n-node input graph contains H as an induced subgraph or not. We first show a Ω̃(n) lower bound for detecting the existence of an induced k-cycle for any k ≥ 4 in the CONGEST model. This lower bound is tight for k = 4, and shows that the induced variant of k-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for 5 ≤ k ≤ 7, this Ω̃(n) lower bound cannot be improved via the two-party communication framework. We then show how to prove stronger lower bounds for larger values of k. More precisely, we show that detecting an induced k-cycle for any k ≥ 8 requires Ω̃(n^{2-Θ{(1/k)}}) rounds in the CONGEST model, nearly matching the known upper bound Õ(n^{2-Θ{(1/k)}}) of the general k-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman [DISC 2019]. Finally, we investigate the case where H is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed computing models
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Distributed computing
  • Lower bounds
  • Subgraph detection

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