Massively Parallel Algorithms for Small Subgraph Counting

Amartya Shankha Biswas*, Talya Eden*, Quanquan C. Liu*, Ronitt Rubinfeld*, Slobodan Mitrović*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

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, n2/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 Oδ(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 Õδ(√log n) rounds, O(nδ) space per machine and O(mα3) 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 languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022
EditorsAmit Chakrabarti, Chaitanya Swamy
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772495
DOIs
StatePublished - 1 Sep 2022
Externally publishedYes
Event25th 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 202221 Sep 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume245
ISSN (Print)1868-8969

Conference

Conference25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022
Country/TerritoryUnited States
CityVirtual, Urbana-Champaign
Period19/09/2221/09/22

Funding

FundersFunder number
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen ForschungP400P2_191122/1
Directorate for Computer and Information Science and Engineering1733808, 1741137, 1740751

    Keywords

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

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