License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.2
URN: urn:nbn:de:0030-drops-88951
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8895/
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Addario-Berry, Louigi

Assumptionless Bounds for Random Trees (Keynote Speakers)

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LIPIcs-AofA-2018-2.pdf (0.2 MB)

Abstract

Let T be any Galton-Watson tree. Write vol(T) for the volume of T (the number of nodes), ht(T) for the height of T (the greatest distance of any node from the root) and wid(T) for the width of T (the greatest number of nodes at any level). We study the relation between vol(T), ht(T) and wid(T).
In the case when the offspring distribution p = (p_i, i >= 0) has mean one and finite variance, both ht(T) and wid(T) are typically of order vol(T)^{1/2}, and have sub-Gaussian upper tails on this scale. Heuristically, as the tail of the offspring distribution becomes heavier, the tree T becomes "shorter and bushier". I will describe a collection of work which can be viewed as justifying this heuristic in various ways In particular, I will explain how classical bounds on Lévy's concentration function for random walks may be used to show that for any offspring distribution, the random variable ht(T)/wid(T) has sub-exponential tails. I will also describe a more combinatorial approach to coupling random trees with different degree sequences which allows the heights of randomly sampled vertices to be compared.

BibTeX - Entry

@InProceedings{addarioberry:LIPIcs:2018:8895,
  author =	{Louigi Addario-Berry},
  title =	{{Assumptionless Bounds for Random Trees (Keynote Speakers)}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8895},
  URN =		{urn:nbn:de:0030-drops-88951},
  doi =		{10.4230/LIPIcs.AofA.2018.2},
  annote =	{Keywords: Random trees, simply generated trees}
}

Keywords: Random trees, simply generated trees
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018

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