Document Open Access Logo

An Output Sensitive Algorithm for Maximal Clique Enumeration in Sparse Graphs

Author George Manoussakis

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2017.27.pdf
  • Filesize: 434 kB
  • 8 pages

Document Identifiers

Subject Classification

Keywords
  • enumeration algorithms
  • maximal cliques
  • k-degenerate graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The degeneracy of a graph G is the smallest integer k such that every subgraph of G contains a vertex of degree at most k. Given an n-order k-degenerate graph G, we present an algorithm for enumerating all its maximal cliques. Assuming that c is the number of maximal cliques of G, our algorithm has setup time O(n(k^2+s(k+1))) and enumeration time cO((k+1)f(k+1)) where s(k+1) (resp. f(k+1)) is the preprocessing time (resp. enumeration time) for maximal clique enumeration in a general (k+1)-order graph. This is the first output sensitive algorithm whose enumeration time depends only on the degeneracy of the graph.

Cite As Get BibTex

George Manoussakis. An Output Sensitive Algorithm for Maximal Clique Enumeration in Sparse Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 27:1-27:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.IPEC.2017.27

Author Details

George Manoussakis

References

  1. V. Batagelj and M. Zaversnik. An 𝒪(m) algorithm for cores decomposition of networks. CoRR, cs.DS/0310049, 2003. URL: http://arxiv.org/abs/cs.DS/0310049.
  2. C. Bron and J. Kerbosch. Algorithm 457: Finding all cliques of an undirected graph. Commun. ACM, 16(9):575-577, 1973. URL: http://dx.doi.org/10.1145/362342.362367.
  3. F. Cazals and C. Karande. A note on the problem of reporting maximal cliques. Theoretical Computer Science, 407(1-3):564-568, 2008. Google Scholar
  4. L. Chang, J. X. Yu, and L. Qin. Fast maximal cliques enumeration in sparse graphs. Algorithmica, 66(1):173-186, 2013. Google Scholar
  5. N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14(1):210-223, 1985. Google Scholar
  6. C. Comin and R. Rizzi. An improved upper bound on maximal clique listing via rectangular fast matrix multiplication. arXiv preprint arXiv:1506.01082, 2015. Google Scholar
  7. A. Conte, R. Grossi, A. Marino, and L. Versari. Sublinear-space bounded-delay enumeration for massive network analytics: Maximal cliques. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 148, pages 1-148, 2016. Google Scholar
  8. David Eppstein, Maarten Löffler, and Darren Strash. Listing all maximal cliques in large sparse real-world graphs. ACM Journal of Experimental Algorithmics, 18, 2013. URL: http://dx.doi.org/10.1145/2543629.
  9. M. Farach. Optimal suffix tree construction with large alphabets. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science, FOCS '97, pages 137-, Washington, DC, USA, 1997. IEEE Computer Society. URL: http://dl.acm.org/citation.cfm?id=795663.796326.
  10. D. Gusfield. Algorithms on strings, trees and sequences: computer science and computational biology. Cambridge University Press, 1997. Google Scholar
  11. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. On generating all maximal independent sets. Inf. Process. Lett., 27(3):119-123, 1988. URL: http://dx.doi.org/10.1016/0020-0190(88)90065-8.
  12. D. R. Lick and A. T. White. d-degenerate graphs. Canad. J. Math., 22:1082-1096, 1970. URL: http://www.smc.math.ca/cjm/v22/p1082.
  13. K. Makino and T. Uno. New algorithms for enumerating all maximal cliques. In Scandinavian Workshop on Algorithm Theory, pages 260-272. Springer, 2004. Google Scholar
  14. Edward M. McCreight. A space-economical suffix tree construction algorithm. J. ACM, 23(2):262-272, 1976. URL: http://dx.doi.org/10.1145/321941.321946.
  15. J. W Moon and L. Moser. On cliques in graphs. Israel journal of Mathematics, 3(1):23-28, 1965. Google Scholar
  16. Etsuji Tomita, Akira Tanaka, and Haruhisa Takahashi. The worst-case time complexity for generating all maximal cliques and computational experiments. Theor. Comput. Sci., 363(1):28-42, 2006. URL: http://dx.doi.org/10.1016/j.tcs.2006.06.015.
  17. S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa. A new algorithm for generating all the maximal independent sets. SIAM Journal on Computing, 6(3):505-517, 1977. Google Scholar
  18. Esko Ukkonen. On-line construction of suffix trees. Algorithmica, 14(3):249-260, 1995. URL: http://dx.doi.org/10.1007/BF01206331.
  19. Peter Weiner. Linear pattern matching algorithms. In 14th Annual Symposium on Switching and Automata Theory, Iowa City, Iowa, USA, October 15-17, 1973, pages 1-11. IEEE Computer Society, 1973. URL: http://dx.doi.org/10.1109/SWAT.1973.13.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact