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.
@InProceedings{verbitsky_et_al:LIPIcs.ESA.2023.100, author = {Verbitsky, Oleg and Zhukovskii, Maksim}, title = {{Canonization of a Random Graph by Two Matrix-Vector Multiplications}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {100:1--100:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.100}, URN = {urn:nbn:de:0030-drops-187536}, doi = {10.4230/LIPIcs.ESA.2023.100}, annote = {Keywords: Graph Isomorphism, canonical labeling, random graphs, walk matrix, color refinement, linear time} }
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