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Fully Dynamic Four-Vertex Subgraph Counting

Authors Kathrin Hanauer , Monika Henzinger , Qi Cheng Hua

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Author Details

Kathrin Hanauer
  • Faculty of Computer Science, University of Vienna, Austria
Monika Henzinger
  • Faculty of Computer Science, University of Vienna, Austria
Qi Cheng Hua
  • Faculty of Computer Science, University of Vienna, Austria

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 101019564, "The Design of Modern Fully Dynamic Data Structures (MoDynStruct)"), as well as from the Austrian Science Fund (FWF) and netIDEE SCIENCE project P 33775-N.

Acknowledgements

The authors want to thank Leonhard Paul Sidl for careful proofreading.

Cite As Get BibTex

Kathrin Hanauer, Monika Henzinger, and Qi Cheng Hua. Fully Dynamic Four-Vertex Subgraph Counting. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SAND.2022.18

Abstract

This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamic graph. Specifically, our algorithms maintain the number of paths of length three in deterministic amortized O(m^{1/2}) update time, and any other connected four-vertex subgraph which is not a clique in deterministic amortized update time O(m^{2/3}). Queries can be answered in constant time. We also study the query times for subgraphs containing an arbitrary edge that is supplied only with the query as well as the case where only subgraphs containing a vertex s that is fixed beforehand are considered. For length-3 paths, paws, 4-cycles, and diamonds our bounds match or are not far from (conditional) lower bounds: Based on the OMv conjecture we show that any dynamic algorithm that detects the existence of paws, diamonds, or 4-cycles or that counts length-3 paths takes update time Ω(m^{1/2-δ}).
Additionally, for 4-cliques and all connected induced subgraphs, we show a lower bound of Ω(m^{1-δ}) for any small constant δ > 0 for the amortized update time, assuming the static combinatorial 4-clique conjecture holds. This shows that the O(m) algorithm by Eppstein et al. [David Eppstein et al., 2012] for these subgraphs cannot be improved by a polynomial factor.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Dynamic Graph Algorithms
  • Subgraph Counting
  • Motif Search

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