## Abstract

Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the defacto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph G = (V, E) with n vertices, m edges and T triangles, our first result is an algorithm that outputs a (1 + ε)-approximation to T, with asymptotically optimal round and total space complexity provided any S ≥ max (^{√}m, n^{2}/m) space per machine and assuming T = Ω(pm/n). Our result gives a quadratic improvement on the bound on T over previous works. We also provide a simple extension of our result to counting any subgraph of k size for constant k ≥ 1. Our second result is an O_{δ}(log log n)-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the arboricity α of the input graph. We extend this result to exactly counting k-cliques for any constant k. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most 5 can be implemented in the MPC model in Õ_{δ}(√log n) rounds, O(n^{δ}) space per machine and O(mα^{3}) total space. In addition to our theoretical results, we simulate our triangle counting algorithms in real-world graphs obtained from the Stanford Network Analysis Project (SNAP) database. Our results show that both our approximate and exact counting algorithms exhibit improvements in terms of round complexity and approximation ratio, respectively, compared to two previous widely used algorithms for these problems.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022 |

Editors | Amit Chakrabarti, Chaitanya Swamy |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772495 |

DOIs | |

State | Published - 1 Sep 2022 |

Externally published | Yes |

Event | 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 - Virtual, Urbana-Champaign, United States Duration: 19 Sep 2022 → 21 Sep 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 245 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 |
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Country/Territory | United States |

City | Virtual, Urbana-Champaign |

Period | 19/09/22 → 21/09/22 |

### Funding

Funders | Funder number |
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Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung | P400P2_191122/1 |

Directorate for Computer and Information Science and Engineering | 1733808, 1741137, 1740751 |

## Keywords

- approximation algorithms
- clique counting
- massively parallel computation
- subgraph counting
- triangle counting