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A Central Limit Theorem for Almost Local Additive Tree Functionals

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Abstract

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Svante Janson recently proved a central limit theorem for additive functionals of conditioned Galton–Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are “almost local” in a certain sense, thus covering a wider range of functionals. The notion of almost local functional intuitively means that the toll function can be approximated well by considering only a neighbourhood of the root. Our main result is illustrated by several explicit examples including natural graph-theoretic parameters such as the number of independent sets, the number of matchings, and the number of dominating sets. We also cover a functional stemming from a tree reduction procedure that was studied by Hackl, Heuberger, Kropf, and Prodinger.

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References

  1. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  2. Bród, D., Skupień, Z.: Trees with extremal numbers of dominating sets. Australas. J. Comb. 35, 273–290 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Drmota, M.: Random Trees: An Interplay Between Combinatorics and Probability. Springer, Berlin (2009)

    Book  Google Scholar 

  4. Hackl, B., Heuberger, C., Kropf, S., Prodinger, H.: Fringe analysis of plane trees related to cutting and pruning. Aequ. Math. 92(2), 311–353 (2018)

    Article  MathSciNet  Google Scholar 

  5. Holmgren, C., Janson, S.: Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electron. J. Probab. 20(4), 1–51 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Holmgren, C., Janson, S., Šileikis, M.: Multivariate normal limit laws for the numbers of fringe subtrees in \(m\)-ary search trees and preferential attachment trees. Electron. J. Comb. 24(2), 49 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Janson, S.: Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29(2), 139–179 (2006)

    Article  MathSciNet  Google Scholar 

  8. Janson, S.: Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9, 103–252 (2012)

    Article  MathSciNet  Google Scholar 

  9. Janson, S.: Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees. Random Struct. Algorithms 48(1), 57–101 (2016)

    Article  MathSciNet  Google Scholar 

  10. Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York), 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  11. Kesten, H.: Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22(4), 425–487 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Kirschenhofer, P., Prodinger, H., Tichy, R.F.: Fibonacci numbers of graphs. III. Planted plane trees. Fibonacci Numbers and Their Applications (Patras 1984) of Applied mathematics, vol. 28, pp. 105–120. Reidel, Dordrecht (1986)

    Chapter  Google Scholar 

  13. Klazar, M.: Twelve countings with rooted plane trees. Eur. J. Comb. 18(2), 195–210 (1997)

    Article  MathSciNet  Google Scholar 

  14. Prodinger, H., Tichy, R.F.: Fibonacci numbers of graphs. Fibonacci Q. 20(1), 16–21 (1982)

    MathSciNet  MATH  Google Scholar 

  15. Ralaivaosaona, D., Šileikis, M., Wagner, S.: Asymptotic normality of almost local functionals in conditioned Galton–Watson trees. In: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, vol. 110 of LIPIcs. Leibniz International Proceedings in Informatics, Art. No. 33, pp. 17. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, (2018)

  16. Ralaivaosaona, D., Wagner, S.: A central limit theorem for additive functionals of increasing trees. Comb. Probab. Comput. 28(4), 618–637 (2019)

    Article  MathSciNet  Google Scholar 

  17. Takács, L.: Conditional limit theorems for branching processes. J. Appl. Math. Stoch. Anal. 4(4), 263–292 (1991)

    Article  MathSciNet  Google Scholar 

  18. Wagner, S.: Correlation of graph-theoretical indices. SIAM J. Discrete Math. 21(1), 33–46 (2007)

    Article  MathSciNet  Google Scholar 

  19. Wagner, S.: On the number of matchings of a tree. Eur. J. Comb. 28(4), 1322–1330 (2007)

    Article  MathSciNet  Google Scholar 

  20. Wagner, S.: Central limit theorems for additive tree parameters with small toll functions. Comb. Probab. Comput. 24(1), 329–353 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank Jim Fill for useful remarks and the anonymous referees for very detailed reports and helpful comments which helped to improve the exposition.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

    Dimbinaina Ralaivaosaona & Stephan Wagner

  2. The Czech Academy of Sciences, Institute of Computer Science, Pod Vodárenskou věží 2, 182 07, Prague, Czech Republic

    Matas Šileikis

Authors
  1. Dimbinaina Ralaivaosaona
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  2. Matas Šileikis
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  3. Stephan Wagner
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Correspondence to Stephan Wagner.

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The first author was partially supported by the Division for Research Development (DRD) of Stellenbosch University. The second author was supported by the Czech Science Foundation, Grant Number GJ16-07822Y, with institutional support RVO:67985807. The third author was supported by the National Research Foundation of South Africa, Grant 96236. An extended abstract of this paper appeared in the Proceedings of the 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018, see [15].

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Ralaivaosaona, D., Šileikis, M. & Wagner, S. A Central Limit Theorem for Almost Local Additive Tree Functionals. Algorithmica 82, 642–679 (2020). https://doi.org/10.1007/s00453-019-00622-4

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