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DOI: 10.4230/LIPIcs.AofA.2024.10

A Bijection for the Evolution of B-Trees

Authors: Fabian Burghart and Stephan Wagner

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract

A B-tree is a type of search tree where every node (except possibly for the root) contains between m and 2m keys for some positive integer m, and all leaves have the same distance to the root. We study sequences of B-trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with B-trees.

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Fabian Burghart and Stephan Wagner. A Bijection for the Evolution of B-Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burghart_et_al:LIPIcs.AofA.2024.10,
  author =	{Burghart, Fabian and Wagner, Stephan},
  title =	{{A Bijection for the Evolution of B-Trees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.10},
  URN =		{urn:nbn:de:0030-drops-204451},
  doi =		{10.4230/LIPIcs.AofA.2024.10},
  annote =	{Keywords: B-trees, histories, increasing trees, bijection, asymptotic enumeration, tree statistics}
}

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DOI: 10.4230/LIPIcs.AofA.2022.4

A Modification of the Random Cutting Model

Authors: Fabian Burghart

Published in: LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

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.

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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)

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@InProceedings{burghart:LIPIcs.AofA.2022.4,
  author =	{Burghart, Fabian},
  title =	{{A Modification of the Random Cutting Model}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-230-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{225},
  editor =	{Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.4},
  URN =		{urn:nbn:de:0030-drops-160903},
  doi =		{10.4230/LIPIcs.AofA.2022.4},
  annote =	{Keywords: Random cutting model, Random separation of graphs, Percolation}
}

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Documents authored by Burghart, Fabian

A Bijection for the Evolution of B-Trees

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A Modification of the Random Cutting Model

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