Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T15:27:34.622Z Has data issue: false hasContentIssue false
  • English
  • Français

Inversions in Split Trees and Conditional Galton–Watson Trees

Part of: Combinatorial probability

Published online by Cambridge University Press:  31 October 2018

XING SHI CAI ,
CECILIA HOLMGREN ,
SVANTE JANSON ,
TONY JOHANSSON  and
FIONA SKERMAN
XING SHI CAI
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
CECILIA HOLMGREN
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
TONY JOHANSSON
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
FIONA SKERMAN
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study I(T), the number of inversions in a tree T with its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of I(T) have explicit formulas involving the k-total common ancestors of T (an extension of the total path length). Then we consider Xn, the normalized version of I(Tn), for a sequence of trees Tn. For fixed Tn's, we prove a sufficient condition for Xn to converge in distribution. As an application, we identify the limit of Xn for complete b-ary trees. For Tn being split trees [16], we show that Xn converges to the unique solution of a distributional equation. Finally, when Tn's are conditional Galton–Watson trees, we show that Xn converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that significantly strengthen and broaden previous work by Panholzer and Seitz [46].

MSC classification

Primary: 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 335 - 364
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was partially supported by two grants from the Knut and Alice Wallenberg Foundation, a grant from the Swedish Research Council, and the Swedish Foundations' starting grant from the Ragnar Söderberg Foundation.

§

Work partially conducted while affiliated with the Heilbronn Institute for mathematical research at the University of Bristol.

References

[1] Addario-Berry, L., Devroye, L. and Janson, S. (2013) Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees. Ann. Probab. 41 1072–1087.Google Scholar
[2] Albert, M., Holmgren, C., Johansson, T. and Skerman, F. (2018) Permutations in binary trees and split trees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018), Vol. 110 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum für Informatik.Google Scholar
[3] Aldous, D. (1991) The continuum random tree I. Ann. Probab. 19 1–28.Google Scholar
[4] Aldous, D. (1991) The continuum random tree II: An overview. In Stochastic Analysis: Proceedings of the Durham Symposium, 1990, Vol. 167 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 23–70.Google Scholar
[5] Aldous, D. (1993) The continuum random tree III. Ann. Probab. 21 248–289.Google Scholar
[6] Baeza-Yates, R. A. (1987) Some average measures in m-ary search trees. Inform. Process. Lett. 25 375–381.Google Scholar
[7] Bender, E. A. (1973) Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15 91–111.Google Scholar
[8] Bienaymé, I. J. (1845) De la loi de multiplication et de la durée des familles. Société Philomatique Paris, 1845. Reprinted in D. G. Kendall (1975), The genealogy of genealogy branching processes before (and after) 1873, Bull. London Math. Soc. 7 225–253.Google Scholar
[9] Broutin, N. and Holmgren, C. (2012) The total path length of split trees. Ann. Appl. Probab. 22 1745–1777.Google Scholar
[10] Cai, X. S. and Devroye, L. (2017) A study of large fringe and non-fringe subtrees in conditional Galton–Watson trees. Latin American Journal of Probability and Mathematical Statistics XIV 579–611.Google Scholar
[11] Chauvin, B. and Pouyanne, N. (2004) m-ary search trees when m ≥ 27: A strong asymptotics for the space requirements. Random Struct. Alg. 24 133–154.Google Scholar
[12] Coffman, E. G. Jr., and Eve, J. (1970) File structures using hashing functions. Commun. Assoc. Comput. Mach. 13 427–432.Google Scholar
[13] Conrad, K. Probability distributions and maximum entropy. http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdfGoogle Scholar
[14] Cramer, G. (1750) Introduction à l'Analyse des Lignes Courbes Algébriques, Frères Cramer et Claude Philibert.Google Scholar
[15] Devroye, L. (1993) On the expected height of fringe-balanced trees. Acta Inform. 30 459–466.Google Scholar
[16] Devroye, L. (1999) Universal limit laws for depths in random trees. SIAM J. Comput. 28 409–432.Google Scholar
[17] Devroye, L., Holmgren, C. and Sulzbach, H. (2017) The heavy path approach to Galton–Watson trees with an application to Apollonian networks. arXiv:1701.02527Google Scholar
[18] DLMF (2017) NIST Digital Library of Mathematical Functions (Olver, F. W. J.et al., eds), Release 1.0.15 of 2017-06-01. http://dlmf.nist.gov/Google Scholar
[19] Drmota, M., Iksanov, A., Moehle, M. and Roesler, U. (2009) A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Struct. Alg. 34 319–336.Google Scholar
[20] Durrett, R. (2010) Probability: Theory and Examples, fourth edition, Vol. 31 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.Google Scholar
[21] Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, third edition, Wiley.Google Scholar
[22] Finkel, R. A. and Bentley, J. L. (1974) Quad trees a data structure for retrieval on composite keys. Acta Informatica 4 1–9.Google Scholar
[23] Flajolet, P., Poblete, P., and Viola, A. (1998) On the analysis of linear probing hashing. Algorithmica 22 490–515.Google Scholar
[24] Flajolet, P., Roux, M. and Vallée, B. (2010) Digital trees and memoryless sources: From arithmetics to analysis. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), DMTCS Proc. AM, pp. 233–260.Google Scholar
[25] Gessel, I. M., Sagan, B. E. and Yeh, Y. N. (1995) Enumeration of trees by inversions. J. Graph Theory 19 435–459.Google Scholar
[26] Gittenberger, B. (2003) A note on: ‘State spaces of the snake and its tour: Convergence of the discrete snake’. J. Theoret. Probab. 16 1063–1067 (2004), 2003.Google Scholar
[27] Gut, A. (2013) Probability: A Graduate Course, second edition, Springer Texts in Statistics, Springer.Google Scholar
[28] Hoare, C. A. R. (1962) Quicksort. Comput. J. 5 10–15.Google Scholar
[29] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[30] Holmgren, C. (2012) Novel characteristic of split trees by use of renewal theory. Electron. J. Probab. 17 5, 1–27.Google Scholar
[31] Janson, S. (2003) The Wiener index of simply generated random trees. Random Struct. Alg. 22 337–358.Google Scholar
[32] Janson, S. (2012) Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation: Extended abstract. In 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proc. AQ, pp. 479–490.Google Scholar
[33] Janson, S. (2016) Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees. Random Struct. Alg. 48 57–101.Google Scholar
[34] Janson, S. (2017) Random recursive trees and preferential attachment trees are random split trees. arXiv:1706.05487Google Scholar
[35] Janson, S. and Chassaing, P. (2004) The center of mass of the ISE and the Wiener index of trees. Electron. Comm. Probab. 9 178–187.Google Scholar
[36] Janson, S. and Marckert, J.-F. (2005) Convergence of discrete snakes. J. Theoret. Probab. 18 615–647.Google Scholar
[37] Knuth, D. E. (1998) The Art of Computer Programming, Vol. 3, Addison-Wesley.Google Scholar
[38] Kortchemski, I. (2017) Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees. Probab. Theory Rel. Fields 168 1–40.Google Scholar
[39] Louchard, G. and Prodinger, H. (2003) The number of inversions in permutations: A saddle point approach. J. Integer Seq. 6 03.2.8.Google Scholar
[40] Mahmoud, H. M. and Pittel, B. (1989) Analysis of the space of search trees under the random insertion algorithm. J. Algorithms 10 52–75.Google Scholar
[41] Mallows, C. L. and Riordan, J. (1968) The inversion enumerator for labeled trees. Bull. Amer. Math. Soc. 74 92–94.Google Scholar
[42] Margolius, B. H. (2001) Permutations with inversions. J. Integer Seq. 4 01.2.4.Google Scholar
[43] Neininger, R. (2001) On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Struct. Alg. 19 498–524.Google Scholar
[44] Neininger, R. and Rüschendorf, L. (1999) On the internal path length of d-dimensional quad trees. Random Struct. Alg. 15 25–41.Google Scholar
[45] Neininger, R. and Rüschendorf, L. (2004) A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418.Google Scholar
[46] Panholzer, A. and Seitz, G. (2012) Limiting distributions for the number of inversions in labelled tree families. Ann. Combin. 16 847–870.Google Scholar
[47] Pyke, R. (1965) Spacings (with discussion). J. Roy. Statist. Soc. Ser. B 27 395–449.Google Scholar
[48] Régnier, M. and Jacquet, P. (1989) New results on the size of tries. IEEE Trans. Inform. Theory 35 203–205.Google Scholar
[49] Rösler, U. (1991) A limit theorem for ‘Quicksort’. RAIRO Inform. Théor. Appl. 25 85–100.Google Scholar
[50] Rösler, U. (2001) On the analysis of stochastic divide and conquer algorithms. Algorithmica 29 238–261.Google Scholar
[51] Rösler, U. and Rüschendorf, L. (2001) The contraction method for recursive algorithms. Algorithmica 29 3–33.Google Scholar
[52] Sachkov, V. N. (1997) Probabilistic Methods in Combinatorial Analysis (translated from the Russian), Vol. 56 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.Google Scholar
[53] Smith, P. J. (1995) A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Amer. Statist. 49 217–218.Google Scholar
[54] Walker, A. and Wood, D. (1976) Locally balanced binary trees. Comput. J. 19 322–325.Google Scholar
[55] Watson, H. W. and Galton, F. (1875) On the probability of the extinction of families. J. Anthropological Institute of Great Britain and Ireland 4 138–144.Google Scholar