LIPIcs.ESA.2023.100.pdf
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Canonization of a Random Graph by Two Matrix-Vector Multiplications
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31st Annual European Symposium on Algorithms (ESA 2023)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2023-08-30
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Oleg Verbitsky and Maksim Zhukovskii. Canonization of a Random Graph by Two Matrix-Vector Multiplications. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 100:1-100:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.100
Abstract
We show that a canonical labeling of a random n-vertex graph can be obtained by assigning to each vertex x the triple (w₁(x),w₂(x),w₃(x)), where w_k(x) is the number of walks of length k starting from x. This takes time 𝒪(n²), where n² is the input size, by using just two matrix-vector multiplications. The linear-time canonization of a random graph is the classical result of Babai, Erdős, and Selkow. For this purpose they use the well-known combinatorial color refinement procedure, and we make a comparative analysis of the two algorithmic approaches.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Random graphs
- Mathematics of computing → Graph algorithms
Keywords
- Graph Isomorphism
- canonical labeling
- random graphs
- walk matrix
- color refinement
- linear time
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