Document Open Access Logo

A Modification of the Random Cutting Model

Author Fabian Burghart

PDF
Thumbnail PDF

File

LIPIcs.AofA.2022.4.pdf
  • Filesize: 0.69 MB
  • 14 pages

Document Identifiers

Author Details

Fabian Burghart
  • Department of Mathematics, Uppsala University, Sweden

Funding

Partially supported by grants from the Knut and Alice Wallenberg Foundation, the Ragnar Söderberg Foundation, and the Swedish Research Council.

Acknowledgements

The author wishes to thank his academic advisors, Cecilia Holmgren and Svante Janson, for their generous support and many helpful remarks and discussions. The author also wishes to thank the anonymous referees for their helpful comments.

Cite As Get BibTex

Fabian Burghart. A Modification of the Random Cutting Model. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.AofA.2022.4

Abstract

We propose a modification to the random destruction of graphs: Given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon [Meir and Moon, 1970] and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions complete binary trees.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Random cutting model
  • Random separation of graphs
  • Percolation

Metrics

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

Related Versions

References

  1. OEIS Foundation Inc. (2021). The on-line encyclopedia of integer sequences. URL: http://oeis.org.
  2. L. Addario-Berry, N. Broutin, and C. Holmgren. Cutting down trees with a Markov chainsaw. Annals of Applied Probability, 24(6):2297-2339, 2014. Google Scholar
  3. N. Alon, I. Benjamini, and A. Stacey. Percolation on finite graphs and isoperimetric inequalities. The Annals of Probability, 32(3):1727-1745, 2004. Google Scholar
  4. J. Bertoin. Fires on trees. In Annales de l'IHP Probabilités et statistiques, volume 48, 909-921, 2012. Google Scholar
  5. G. Berzunza. The cut-tree of large trees with small heights. Random Structures & Algorithms, 51(3):404-427, 2017. Google Scholar
  6. G. Berzunza, X. S. Cai, and C. Holmgren. The k-cut model in deterministic and random trees. arXiv, 2019. URL: http://arxiv.org/abs/1907.02770.
  7. P. Billingsley. Convergence of Probability Measures. John Wiley & Sons, 1st edition, 1968. Google Scholar
  8. N. Broutin and M. Wang. Cutting down p-trees and inhomogeneous continuum random trees. Bernoulli, 23(4A):2380-2433, 2017. Google Scholar
  9. F. Burghart. A modification of the random cutting model, 2022. URL: http://arxiv.org/abs/2111.02968.
  10. X. S. Cai and C. Holmgren. Cutting resilient networks-complete binary trees. The Electronic Journal of Combinatorics, 26(4), 2019. Google Scholar
  11. X. S. Cai, C. Holmgren, L. Devroye, and F. Skerman. k-cut on paths and some trees. Electronic Journal of Probability, 24, 2019. Google Scholar
  12. V. Campos, V. Chvatal, L. Devroye, and P. Taslakian. Transversals in trees. Journal of graph theory, 73(1):32-43, 2013. Google Scholar
  13. P. Chassaing and R. Marchand. Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms, 2004. URL: http://arxiv.org/abs/math/0406094.
  14. L. Devroye. A note on the probability of cutting a Galton-Watson tree. Electronic Journal of Probability, 16:2001-2019, 2011. Google Scholar
  15. J. A. Fill, N. Kapur, and A. Panholzer. Destruction of very simple trees. Algorithmica, 46(3):345-366, 2006. Google Scholar
  16. P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. Google Scholar
  17. G. R. Grimmett. Percolation. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer, 2nd edition, 1999. Google Scholar
  18. C. Holmgren. Random records and cuttings in binary search trees. Combinatorics, Probability and Computing, 19(3):391-424, 2010. Google Scholar
  19. C. Holmgren. A weakly 1-stable distribution for the number of random records and cuttings in split trees. Advances in Applied Probability, 43(1):151-177, 2011. Google Scholar
  20. S. Janson. Random records and cuttings in complete binary trees. In Mathematics and Computer Science III, 241-253. Springer, 2004. Google Scholar
  21. S. Janson. Random cutting and records in deterministic and random trees. Random Structures & Algorithms, 29(2):139-179, 2006. Google Scholar
  22. D. E. Knuth and A. Schönhage. The expected linearity of a simple equivalence algorithm. Theoretical Computer Science, 6(3):281-315, 1978. Google Scholar
  23. M. Kuba and A. Panholzer. Analysis of the total costs for variants of the Union-Find algorithm. In Discrete Mathematics and Theoretical Computer Science, pages 283-294. Discrete Mathematics and Theoretical Computer Science, 2007. Google Scholar
  24. M. Kuba and A. Panholzer. Isolating a leaf in rooted trees via random cuttings. Annals of Combinatorics, 12(1):81-99, 2008. Google Scholar
  25. M. Kuba and A. Panholzer. Isolating nodes in recursive trees. Aequationes mathematicae, 76(3):258-280, 2008. Google Scholar
  26. M. Kuba and A. Panholzer. Multiple isolation of nodes in recursive trees. Online Journal of Analytic Combinatorics, 9(7), 2014. Google Scholar
  27. R. Lyons and Y. Peres. Probability on trees and networks. Cambridge University Press, 2017. Google Scholar
  28. A. Meir and J. W. Moon. Cutting down random trees. Journal of the Australian Mathematical Society, 11(3):313-324, 1970. Google Scholar
  29. A. Panholzer. Non-crossing trees revisited: cutting down and spanning subtrees. In Discrete Mathematics and Theoretical Computer Science, 265-276, 2003. Google Scholar
  30. A. Panholzer. Cutting down very simple trees. Quaestiones Mathematicae, 29(2):211-227, 2006. Google Scholar
  31. A. C.-C. Yao. On the average behavior of set merging algorithms. In Proceedings of the eighth annual ACM symposium on Theory of computing, 192-195, 1976. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact