Volume
LIPIcs, Volume 302
35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)
Event
AofA 2024, June 17-21, 2024, University of Bath, UKEditors
Publication Details
- published at: 2024-07-18
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-329-4
- DBLP: db/conf/aofa/aofa2024
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Complete Volume
LIPIcs, Volume 302, AofA 2024, Complete Volume
Abstract
LIPIcs, Volume 302, AofA 2024, Complete Volume
Cite as
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. 1-458, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@Proceedings{mailler_et_al:LIPIcs.AofA.2024,
title = {{LIPIcs, Volume 302, AofA 2024, Complete Volume}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {1--458},
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},
URN = {urn:nbn:de:0030-drops-204341},
doi = {10.4230/LIPIcs.AofA.2024},
annote = {Keywords: LIPIcs, Volume 302, AofA 2024, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
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. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mailler_et_al:LIPIcs.AofA.2024.0,
author = {Mailler, C\'{e}cile and Wild, Sebastian},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {0:i--0:xii},
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.2024.0},
URN = {urn:nbn:de:0030-drops-204353},
doi = {10.4230/LIPIcs.AofA.2024.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Fringe Trees for Random Trees with Given Vertex Degrees
Abstract
We prove that the number of fringe subtrees, isomorphic to a given tree, in uniformly random trees with given vertex degrees, asymptotically follows a normal distribution. As an application, we establish the same asymptotic normality for random simply generated trees (conditioned Galton-Watson trees). Our approach relies on an extension of Gao and Wormald’s (2004) theorem to the multivariate setting.
Cite as
Gabriel Berzunza Ojeda, Cecilia Holmgren, and Svante Janson. Fringe Trees for Random Trees with Given Vertex Degrees. 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. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{berzunzaojeda_et_al:LIPIcs.AofA.2024.1,
author = {Berzunza Ojeda, Gabriel and Holmgren, Cecilia and Janson, Svante},
title = {{Fringe Trees for Random Trees with Given Vertex Degrees}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {1:1--1:13},
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.2024.1},
URN = {urn:nbn:de:0030-drops-204369},
doi = {10.4230/LIPIcs.AofA.2024.1},
annote = {Keywords: Conditioned Galton-Watson trees, fringe trees, simply generated trees, uniformly random trees with given vertex degrees}
}
Document
Enumeration and Succinct Encoding of AVL Trees
Abstract
We use a novel decomposition to create succinct data structures - supporting a wide range of operations on static trees in constant time - for a variety of tree classes, extending results of Munro, Nicholson, Benkner, and Wild. Motivated by the class of AVL trees, we further derive asymptotics for the information-theoretic lower bound on the number of bits needed to store tree classes whose generating functions satisfy certain functional equations. In particular, we prove that AVL trees require approximately 0.938 bits per node to encode.
Cite as
Jeremy Chizewer, Stephen Melczer, J. Ian Munro, and Ava Pun. Enumeration and Succinct Encoding of AVL 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. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chizewer_et_al:LIPIcs.AofA.2024.2,
author = {Chizewer, Jeremy and Melczer, Stephen and Munro, J. Ian and Pun, Ava},
title = {{Enumeration and Succinct Encoding of AVL Trees}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {2:1--2:12},
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.2024.2},
URN = {urn:nbn:de:0030-drops-204376},
doi = {10.4230/LIPIcs.AofA.2024.2},
annote = {Keywords: AVL trees, analytic combinatorics, succinct data structures, enumeration}
}
Document
Maximal Number of Subword Occurrences in a Word
Abstract
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We then give upper and lower bounds of minimal subword entropy for words of fixed length in a fixed alphabet, and also showing that minimal subword entropy per letter has a limit value. A better upper bound of minimal subword entropy for a binary alphabet is then given by looking at certain families of periodic words. We also give some conjectures based on experimental observations.
Cite as
Wenjie Fang. Maximal Number of Subword Occurrences in a Word. 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. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fang:LIPIcs.AofA.2024.3,
author = {Fang, Wenjie},
title = {{Maximal Number of Subword Occurrences in a Word}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {3:1--3:12},
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.2024.3},
URN = {urn:nbn:de:0030-drops-204387},
doi = {10.4230/LIPIcs.AofA.2024.3},
annote = {Keywords: Subword occurrence, subword entropy, enumeration, periodic words}
}
Document
Sparsification of Phylogenetic Covariance Matrices of k-Regular Trees
Abstract
Consider a tree T = (V,E) with root ∘ and an edge length function 𝓁:E → ℝ_+. The phylogenetic covariance matrix of T is the matrix C with rows and columns indexed by L, the leaf set of T, with entries C(i,j): = ∑_{e ∈ [i∧ j,o]}𝓁(e), for each i,j ∈ L. Recent work [Gorman & Lladser 2023] has shown that the phylogenetic covariance matrix of a large but random binary tree T is significantly sparsified, with overwhelmingly high probability, under a change-of-basis to the so-called Haar-like wavelets of T. Notably, this finding enables manipulating the spectrum of covariance matrices of large binary trees without the necessity to store them in computer memory but instead performing two post-order traversals of the tree [Gorman & Lladser 2023]. Building on the methods of the aforesaid paper, this manuscript further advances their sparsification result to encompass the broader class of k-regular trees, for any given k ≥ 2. This extension is achieved by refining existing asymptotic formulas for the mean and variance of the internal path length of random k-regular trees, utilizing hypergeometric function properties and identities.
Cite as
Sean Svihla and Manuel E. Lladser. Sparsification of Phylogenetic Covariance Matrices of k-Regular 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. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{svihla_et_al:LIPIcs.AofA.2024.4,
author = {Svihla, Sean and Lladser, Manuel E.},
title = {{Sparsification of Phylogenetic Covariance Matrices of k-Regular Trees}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {4:1--4:17},
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.2024.4},
URN = {urn:nbn:de:0030-drops-204399},
doi = {10.4230/LIPIcs.AofA.2024.4},
annote = {Keywords: cophenetic matrix, Haar-like wavelets, hierarchical data, hypergeometric functions, metagenomics, phylogenetic covariance matrix, sparsification, ultrametric matrix}
}
Document
Bit-Array-Based Alternatives to HyperLogLog
Abstract
We present a family of algorithms for the problem of estimating the number of distinct items in an input stream that are simple to implement and are appropriate for practical applications. Our algorithms are a logical extension of the series of algorithms developed by Flajolet and his coauthors starting in 1983 that culminated in the widely used HyperLogLog algorithm. These algorithms divide the input stream into M substreams and lead to a time-accuracy tradeoff where a constant number of bits per substream are saved to achieve a relative accuracy proportional to 1/√M. Our algorithms use just one or two bits per substream. Their effectiveness is demonstrated by a proof of approximate normality, with explicit expressions for standard errors that inform parameter settings and allow proper quantitative comparisons with other methods. Hypotheses about performance are validated through experiments using a realistic input stream, with the conclusion that our algorithms are more accurate than HyperLogLog when using the same amount of memory, and they use two-thirds as much memory as HyperLogLog to achieve a given accuracy.
Cite as
Svante Janson, Jérémie Lumbroso, and Robert Sedgewick. Bit-Array-Based Alternatives to HyperLogLog. 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. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{janson_et_al:LIPIcs.AofA.2024.5,
author = {Janson, Svante and Lumbroso, J\'{e}r\'{e}mie and Sedgewick, Robert},
title = {{Bit-Array-Based Alternatives to HyperLogLog}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {5:1--5:19},
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.2024.5},
URN = {urn:nbn:de:0030-drops-204402},
doi = {10.4230/LIPIcs.AofA.2024.5},
annote = {Keywords: Cardinality estimation, sketching, Hyperloglog}
}
Document
Phase Transition for Tree-Rooted Maps
Abstract
We introduce a model of tree-rooted planar maps weighted by their number of 2-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest 2-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings √{n/log(n)} and √n.
Cite as
Marie Albenque, Éric Fusy, and Zéphyr Salvy. Phase Transition for Tree-Rooted Maps. 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. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{albenque_et_al:LIPIcs.AofA.2024.6,
author = {Albenque, Marie and Fusy, \'{E}ric and Salvy, Z\'{e}phyr},
title = {{Phase Transition for Tree-Rooted Maps}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {6:1--6:14},
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.2024.6},
URN = {urn:nbn:de:0030-drops-204413},
doi = {10.4230/LIPIcs.AofA.2024.6},
annote = {Keywords: Asymptotic Enumeration, Planar maps, Random trees, Phase transition}
}
Document
Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models
Abstract
Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of q-enumeration (this is a model where objects with a parameter of value k have a Gibbs measure/Boltzmann weight q^k). For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value q = q_c, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime (q > q_c), and it is a Boltzmann distribution in the subcritical regime (0 < q < q_c). We apply our results to fundamental statistics of lattice paths and quarter-plane walks. We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons.
Cite as
Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner. Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models. 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. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{banderier_et_al:LIPIcs.AofA.2024.7,
author = {Banderier, Cyril and Kuba, Markus and Wagner, Stephan and Wallner, Michael},
title = {{Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {7:1--7:18},
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.2024.7},
URN = {urn:nbn:de:0030-drops-204427},
doi = {10.4230/LIPIcs.AofA.2024.7},
annote = {Keywords: Composition schemes, q-enumeration, generating functions, Gibbs distribution, phase transitions, limit laws, Mittag-Leffler distribution, chi distribution, Boltzmann distribution}
}
Document
Galled Tree-Child Networks
Abstract
We propose the class of galled tree-child networks which is obtained as intersection of the classes of galled networks and tree-child networks. For the latter two classes, (asymptotic) counting results and stochastic results have been proved with very different methods. We show that a counting result for the class of galled tree-child networks follows with similar tools as used for galled networks, however, the result has a similar pattern as the one for tree-child networks. In addition, we also consider the (suitably scaled) numbers of reticulation nodes of random galled tree-child networks and show that they are asymptotically normal distributed. This is in contrast to the limit laws of the corresponding quantities for galled networks and tree-child networks which have been both shown to be discrete.
Cite as
Yu-Sheng Chang, Michael Fuchs, and Guan-Ru Yu. Galled Tree-Child Networks. 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. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chang_et_al:LIPIcs.AofA.2024.8,
author = {Chang, Yu-Sheng and Fuchs, Michael and Yu, Guan-Ru},
title = {{Galled Tree-Child Networks}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {8:1--8:13},
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.2024.8},
URN = {urn:nbn:de:0030-drops-204439},
doi = {10.4230/LIPIcs.AofA.2024.8},
annote = {Keywords: Phylogenetic Network, galled Network, tree-child Network, asymptotic Enumeration, Limit Law, Lagrange Inversion}
}
Document
On Fluctuations of Complexity Measures for the FIND Algorithm
Abstract
The FIND algorithm (also called Quickselect) is a fundamental algorithm to select ranks or quantiles within a set of data. It was shown by Grübel and Rösler that the number of key comparisons required by FIND as a process of the quantiles α ∈ [0,1] in a natural probabilistic model converges after normalization in distribution within the càdlàg space D[0,1] endowed with the Skorokhod metric. We show that the process of the residuals in the latter convergence after normalization converges in distribution to a mixture of Gaussian processes in D[0,1] and identify the limit’s conditional covariance functions. A similar result holds for the related algorithm QuickVal. Our method extends to other cost measures such as the number of swaps (key exchanges) required by FIND or cost measures which are based on key comparisons but take into account that the cost of a comparison between two keys may depend on their values, an example being the number of bit comparisons needed to compare keys given by their bit expansions.
Cite as
Jasper Ischebeck and Ralph Neininger. On Fluctuations of Complexity Measures for the FIND Algorithm. 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. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ischebeck_et_al:LIPIcs.AofA.2024.9,
author = {Ischebeck, Jasper and Neininger, Ralph},
title = {{On Fluctuations of Complexity Measures for the FIND Algorithm}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {9:1--9:15},
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.2024.9},
URN = {urn:nbn:de:0030-drops-204440},
doi = {10.4230/LIPIcs.AofA.2024.9},
annote = {Keywords: FIND, Quickselect, rank selection, probabilistic analysis of algorithms, weak convergence, functional limit theorem}
}
Document
A Bijection for the Evolution of B-Trees
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.
Cite as
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}
}
Document
Tree Walks and the Spectrum of Random Graphs
Abstract
It is a classic result in spectral theory that the limit distribution of the spectral measure of random graphs G(n,p) converges to the semicircle law in case np tends to infinity with n. The spectral measure for random graphs G(n,c/n) however is less understood. In this work, we combine and extend two combinatorial approaches by Bauer and Golinelli (2001) and Enriquez and Menard (2016) and approximate the moments of the spectral measure by counting walks that span trees.
Cite as
Eva-Maria Hainzl and Élie de Panafieu. Tree Walks and the Spectrum of Random Graphs. 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. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hainzl_et_al:LIPIcs.AofA.2024.11,
author = {Hainzl, Eva-Maria and de Panafieu, \'{E}lie},
title = {{Tree Walks and the Spectrum of Random Graphs}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {11:1--11:15},
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.2024.11},
URN = {urn:nbn:de:0030-drops-204466},
doi = {10.4230/LIPIcs.AofA.2024.11},
annote = {Keywords: Spectrum of random matrices, generating functions}
}
Document
Asymptotics of Weighted Reflectable Walks in A₂
Abstract
Lattice walks are used to model various physical phenomena. In particular, walks within Weyl chambers connect directly to representation theory via the Littelmann path model. We derive asymptotics for centrally weighted lattice walks within the Weyl chamber corresponding to A₂ by using tools from analytic combinatorics in several variables (ACSV). We find universality classes depending on the weights of the walks, in line with prior results on the weighted Gouyou-Beauchamps model. Along the way, we identify a type of singularity within a multivariate rational generating function that is not yet covered by the theory of ACSV. We conjecture asymptotics for this type of singularity.
Cite as
Torin Greenwood and Samuel Simon. Asymptotics of Weighted Reflectable Walks in A₂. 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. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{greenwood_et_al:LIPIcs.AofA.2024.12,
author = {Greenwood, Torin and Simon, Samuel},
title = {{Asymptotics of Weighted Reflectable Walks in A₂}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {12:1--12:14},
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.2024.12},
URN = {urn:nbn:de:0030-drops-204472},
doi = {10.4230/LIPIcs.AofA.2024.12},
annote = {Keywords: Lattice walks, Weyl chambers, asymptotics weights, analytic combinatorics in several variables}
}
Document
On the Number of Distinct Fringe Subtrees in Binary Search Trees
Abstract
A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to (c₁n)/(log n) for a constant c₁ ≈ 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.
Cite as
Stephan Wagner. On the Number of Distinct Fringe Subtrees in Binary Search 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. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{wagner:LIPIcs.AofA.2024.13,
author = {Wagner, Stephan},
title = {{On the Number of Distinct Fringe Subtrees in Binary Search Trees}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {13:1--13:11},
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.2024.13},
URN = {urn:nbn:de:0030-drops-204482},
doi = {10.4230/LIPIcs.AofA.2024.13},
annote = {Keywords: Fringe subtrees, binary search trees, tree compression, minimal DAG, asymptotics}
}
Document
Early Typical Vertices in Subcritical Random Graphs of Preferential Attachment Type
Abstract
We study the size of the connected component of early typical vertices in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The principal tools in our analysis are, first, a coupling of the neighbourhood of a typical vertex in the graph to a killed branching random walk and, second, an asymptotic result for the number of particles absorbed at the killing barrier in this branching random walk.
Cite as
Peter Mörters and Nick Schleicher. Early Typical Vertices in Subcritical Random Graphs of Preferential Attachment Type. 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. 14:1-14:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{morters_et_al:LIPIcs.AofA.2024.14,
author = {M\"{o}rters, Peter and Schleicher, Nick},
title = {{Early Typical Vertices in Subcritical Random Graphs of Preferential Attachment Type}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {14:1--14:10},
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.2024.14},
URN = {urn:nbn:de:0030-drops-204493},
doi = {10.4230/LIPIcs.AofA.2024.14},
annote = {Keywords: Inhomogeneous random graphs, preferential attachment, networks, subcritical behaviour, size of components, connectivity, coupling, branching random walk, random tree}
}
Document
Asymptotics of Relaxed k-Ary Trees
Abstract
A relaxed k-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree k. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed k-ary tree with n nodes for n → ∞. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term e^{c n^{1/3}} appears in all these cases. We also derive the recurrences for compacted k-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.
Cite as
Manosij Ghosh Dastidar and Michael Wallner. Asymptotics of Relaxed k-Ary 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. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ghoshdastidar_et_al:LIPIcs.AofA.2024.15,
author = {Ghosh Dastidar, Manosij and Wallner, Michael},
title = {{Asymptotics of Relaxed k-Ary Trees}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {15:1--15:13},
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.2024.15},
URN = {urn:nbn:de:0030-drops-204506},
doi = {10.4230/LIPIcs.AofA.2024.15},
annote = {Keywords: Asymptotic enumeration, stretched exponential, Airy function, directed acyclic graph, Dyck paths, compacted trees, minimal automata}
}
Document
Matching Algorithms in the Sparse Stochastic Block Model
Abstract
In sparse Erdős-Rényi graphs, it is known that a linear-time algorithm of Karp and Sipser achieves near-optimal matching sizes asymptotically almost surely, giving a law-of-large numbers for the matching numbers of such graphs in terms of solutions to an ODE [Jonathan Aronson et al., 1998]. We provide an extension of this analysis, identifying broad ranges of stochastic block model parameters for which the Karp-Sipser algorithm achieves near-optimal matching sizes, but demonstrating that it cannot perform optimally on general stochastic block model instances. We also consider the problem of constructing a matching online, in which the vertices of one half of a bipartite stochastic block model arrive one-at-a-time, and must be matched as they arrive. We show that, when the expected degrees in all communities are equal, the competitive ratio lower bound of 0.837 found by Mastin and Jaillet for the Erdős-Rényi case [Andrew Mastin and Patrick Jaillet, 2013] is achieved by a simple greedy algorithm, and this competitive ratio is optimal. We then propose and analyze a linear-time online matching algorithm with better performance in general stochastic block models.
Cite as
Anna Brandenberger, Byron Chin, Nathan S. Sheffield, and Divya Shyamal. Matching Algorithms in the Sparse Stochastic Block Model. 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. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{brandenberger_et_al:LIPIcs.AofA.2024.16,
author = {Brandenberger, Anna and Chin, Byron and Sheffield, Nathan S. and Shyamal, Divya},
title = {{Matching Algorithms in the Sparse Stochastic Block Model}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {16:1--16:21},
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.2024.16},
URN = {urn:nbn:de:0030-drops-204515},
doi = {10.4230/LIPIcs.AofA.2024.16},
annote = {Keywords: Matching Algorithms, Online Matching, Stochastic Block Model}
}
Document
Lexicographic Unranking Algorithms for the Twelvefold Way
Abstract
The Twelvefold Way represents Rota’s classification, addressing the most fundamental enumeration problems and their associated combinatorial counting formulas. These distinct problems are connected to enumerating functions defined from a set of elements denoted by 𝒩 into another one 𝒦. The counting solutions for the twelve problems are well known. We are interested in unranking algorithms. Such an algorithm is based on an underlying total order on the set of structures we aim at constructing. By taking the rank of an object, i.e. its number according to the total order, the algorithm outputs the structure itself after having built it. One famous total order is the lexicographic order: it is probably the one that is the most used by people when one wants to order things. While the counting solutions for Rota’s classification have been known for years it is interesting to note that three among the problems have yet no lexicographic unranking algorithm. In this paper we aim at providing algorithms for the last three cases that remain without such algorithms. After presenting in detail the solution for set partitions associated with the famous Stirling numbers of the second kind, we explicitly explain how to adapt the algorithm for the two remaining cases. Additionally, we propose a detailed and fine-grained complexity analysis based on the number of bitwise arithmetic operations.
Cite as
Amaury Curiel and Antoine Genitrini. Lexicographic Unranking Algorithms for the Twelvefold Way. 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. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{curiel_et_al:LIPIcs.AofA.2024.17,
author = {Curiel, Amaury and Genitrini, Antoine},
title = {{Lexicographic Unranking Algorithms for the Twelvefold Way}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {17:1--17:14},
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.2024.17},
URN = {urn:nbn:de:0030-drops-204522},
doi = {10.4230/LIPIcs.AofA.2024.17},
annote = {Keywords: Twelvefold Way, Set partitions, Unranking, Lexicographic order}
}
Document
Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves
Abstract
The Colijn-Plazzotta ranking is a certain bijection between the unlabeled binary rooted trees and the positive integers, such that the integer associated with a tree is determined from the integers associated with the two immediate subtrees of its root. Letting a_n denote the minimal Colijn-Plazzotta rank among all trees with a specified number of leaves n, the sequence {a_n} begins 1, 2, 3, 4, 6, 7, 10, 11, 20, 22, 28, 29, 53, 56, 66, 67 (OEIS A354970). Here we show that a_n ∼ 2 [2^{P(log₂ n)}]ⁿ, where P varies as a periodic function dependent on {log₂ n} and satisfies 1.24602 < 2^{P(log₂ n)} < 1.33429.
Cite as
Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg. Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves. 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. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{doboli_et_al:LIPIcs.AofA.2024.18,
author = {Doboli, Michael R. and Hwang, Hsien-Kuei and Rosenberg, Noah A.},
title = {{Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {18:1--18:14},
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.2024.18},
URN = {urn:nbn:de:0030-drops-204530},
doi = {10.4230/LIPIcs.AofA.2024.18},
annote = {Keywords: Colijn-Plazzotta ranking, recurrences, unlabeled trees}
}
Document
Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Case Study
Abstract
Making use of a newly developed package in the computer algebra system SageMath, we show how to perform a full asymptotic analysis by means of the Mellin transform with explicit error bounds. As an application of the method, we answer a question of Bóna and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.
Cite as
Benjamin Hackl and Stephan Wagner. Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Case Study. 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. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hackl_et_al:LIPIcs.AofA.2024.19,
author = {Hackl, Benjamin and Wagner, Stephan},
title = {{Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Case Study}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {19:1--19:15},
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.2024.19},
URN = {urn:nbn:de:0030-drops-204549},
doi = {10.4230/LIPIcs.AofA.2024.19},
annote = {Keywords: binomial sum, Mellin transform, asymptotics, explicit error bounds, B-terms}
}
Document
Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm
Abstract
In the hardcore model, certain vertices in a graph are active: the active vertices must form an independent set. We extend this to a multicoloured version: instead of simply being active or not, the active vertices are assigned a colour; active vertices of the same colour must not be adjacent.
This models a scenario in which two neighbouring resources may interfere when active - eg, short-range radio communication. However, there are multiple channels (colours) available; they only interfere if both use the same channel. Other applications include routing in fibreoptic networks.
We analyse Glauber dynamics. Vertices update their status at random times, at which a uniform colour is proposed: the vertex is assigned that colour if it is available; otherwise, it is set inactive.
We find conditions for fast mixing of these dynamics. We also use them to model a queueing system: vertices only serve customers in their queue whilst active. The mixing estimates are applied to establish positive recurrence of the queue lengths, and bound their expectation in equilibrium.
Cite as
Sam Olesker-Taylor. Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm. 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. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{oleskertaylor:LIPIcs.AofA.2024.20,
author = {Olesker-Taylor, Sam},
title = {{Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {20:1--20:14},
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.2024.20},
URN = {urn:nbn:de:0030-drops-204558},
doi = {10.4230/LIPIcs.AofA.2024.20},
annote = {Keywords: mixing time, queueing theory, hardcore model, proper colourings, independent set, data transmission, randomised algorithms, routing, scheduling, multihop wireless networks}
}
Document
Binary Search Trees of Permuton Samples
Abstract
Binary search trees (BST) are a popular type of structure when dealing with ordered data. They allow efficient access and modification of data, with their height corresponding to the worst retrieval time. From a probabilistic point of view, BSTs associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform permutation.
We consider here the case where the input comes from i.i.d. random points in the plane with law μ, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree only depends on the behavior of the measure μ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures μ. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.
Cite as
Benoît Corsini, Victor Dubach, and Valentin Féray. Binary Search Trees of Permuton Samples. 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. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{corsini_et_al:LIPIcs.AofA.2024.21,
author = {Corsini, Beno\^{i}t and Dubach, Victor and F\'{e}ray, Valentin},
title = {{Binary Search Trees of Permuton Samples}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {21:1--21:13},
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.2024.21},
URN = {urn:nbn:de:0030-drops-204562},
doi = {10.4230/LIPIcs.AofA.2024.21},
annote = {Keywords: Binary search trees, random permutations, permutons}
}
Document
The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension
Abstract
It is celebrated that a simple random walk on ℤ and ℤ² returns to the initial vertex v infinitely many times during infinitely many transitions, which is said recurrent, while it returns to v only finite times on ℤ^d for d ≥ 3, which is said transient. It is also known that a simple random walk on a growing region on ℤ^d can be recurrent depending on growing speed for any fixed d. This paper shows that a simple random walk on {0,1,…,N}ⁿ with an increasing n and a fixed N can be recurrent depending on the increasing speed of n. Precisely, we are concerned with a specific model of a random walk on a growing graph (RWoGG) and show a phase transition between the recurrence and transience of the random walk regarding the growth speed of the graph. For the proof, we develop a pausing coupling argument introducing the notion of weakly less homesick as graph growing (weakly LHaGG).
Cite as
Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai. The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension. 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. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kumamoto_et_al:LIPIcs.AofA.2024.22,
author = {Kumamoto, Shuma and Kijima, Shuji and Shirai, Tomoyuki},
title = {{The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {22:1--22:15},
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.2024.22},
URN = {urn:nbn:de:0030-drops-204577},
doi = {10.4230/LIPIcs.AofA.2024.22},
annote = {Keywords: Random walk, dynamic graph, recurrence, transience, coupling}
}
Document
The Alternating Normal Form of Braids and Its Minimal Automaton
Abstract
The alternating normal form of braids is a well-known normal form on standard braid monoids. This normal form is regular: the language it identifies with is regular. We give a characterisation of the minimal automaton of this language and compute its size, both in terms of number of states and of transitions, depending on the number of generators of the monoid.
Cite as
Vincent Jugé and June Roupin. The Alternating Normal Form of Braids and Its Minimal Automaton. 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. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{juge_et_al:LIPIcs.AofA.2024.23,
author = {Jug\'{e}, Vincent and Roupin, June},
title = {{The Alternating Normal Form of Braids and Its Minimal Automaton}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {23:1--23:15},
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.2024.23},
URN = {urn:nbn:de:0030-drops-204587},
doi = {10.4230/LIPIcs.AofA.2024.23},
annote = {Keywords: Automata, braids, enumeration, normal forms}
}
Document
Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences
Abstract
In the asymptotic analysis of regular sequences as defined by Allouche and Shallit, it is usually advisable to study their summatory function because the original sequence has a too fluctuating behaviour. It might be that the process of taking the summatory function has to be repeated if the sequence is fluctuating too much. In this paper we show that for all regular sequences except for some degenerate cases, repeating this process finitely many times leads to a "nice" asymptotic expansion containing periodic fluctuations whose Fourier coefficients can be computed using the results on the asymptotics of the summatory function of regular sequences by the first two authors of this paper.
In a recent paper, Hwang, Janson, and Tsai perform a thorough investigation of divide-and-conquer recurrences. These can be seen as 2-regular sequences. By considering them as the summatory function of their forward difference, the results on the asymptotics of the summatory function of regular sequences become applicable. We thoroughly investigate the case of a polynomial toll function.
Cite as
Clemens Heuberger, Daniel Krenn, and Tobias Lechner. Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences. 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. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{heuberger_et_al:LIPIcs.AofA.2024.24,
author = {Heuberger, Clemens and Krenn, Daniel and Lechner, Tobias},
title = {{Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {24:1--24:14},
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.2024.24},
URN = {urn:nbn:de:0030-drops-204597},
doi = {10.4230/LIPIcs.AofA.2024.24},
annote = {Keywords: Regular sequence, Divide-and-Conquer Recurrence, Summatory Function, Asymptotic Analysis}
}
Document
Patricia’s Bad Distributions
Abstract
The height of a random PATRICIA tree built from independent, identically distributed infinite binary strings with arbitrary diffuse probability distribution μ on {0,1}^ℕ is studied. We show that the expected height grows asymptotically sublinearly in the number of leaves for any such μ, but can be made to exceed any specific sublinear growth rate by choosing μ appropriately.
Cite as
Louigi Addario-Berry, Pat Morin, and Ralph Neininger. Patricia’s Bad Distributions. 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. 25:1-25:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{addarioberry_et_al:LIPIcs.AofA.2024.25,
author = {Addario-Berry, Louigi and Morin, Pat and Neininger, Ralph},
title = {{Patricia’s Bad Distributions}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {25:1--25:8},
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.2024.25},
URN = {urn:nbn:de:0030-drops-204600},
doi = {10.4230/LIPIcs.AofA.2024.25},
annote = {Keywords: PATRICIA tree, random tree, height of tree, analysis of algorithms}
}
Document
Limit Laws for Critical Dispersion on Complete Graphs
Abstract
We consider a synchronous process of particles moving on the vertices of a graph G, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, M particles are placed on a vertex of G. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time.
In this work we study the case where G is the complete graph on n vertices and the number of particles is M = n/2+α n^{1/2} + o(n^{1/2}), α ∈ ℝ. This choice of M corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by n^{-1/2}, converges in p-th mean, as n → ∞ and for any p ∈ ℝ, to a continuous and almost surely positive random variable T_α. We find that T_α is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that 𝔼[T₀] = π^{3/2}/√7, and furthermore we formulate explicit asymptotics when |α| gets large that quantify the transition into and out of the critical window. We also study the random variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by nln(n), it converges to 2/7 in probability.
Cite as
Umberto De Ambroggio, Tamás Makai, Konstantinos Panagiotou, and Annika Steibel. Limit Laws for Critical Dispersion on Complete Graphs. 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. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{deambroggio_et_al:LIPIcs.AofA.2024.26,
author = {De Ambroggio, Umberto and Makai, Tam\'{a}s and Panagiotou, Konstantinos and Steibel, Annika},
title = {{Limit Laws for Critical Dispersion on Complete Graphs}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {26:1--26:12},
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.2024.26},
URN = {urn:nbn:de:0030-drops-204617},
doi = {10.4230/LIPIcs.AofA.2024.26},
annote = {Keywords: Random processes on graphs, diffusion processes, stochastic differential equations, martingale inequalities}
}
Document
Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls
Abstract
Galled trees appear in problems concerning admixture, horizontal gene transfer, hybridization, and recombination. Building on a recursive enumerative construction, we study the asymptotic behavior of the number of rooted binary unlabeled (normal) galled trees as the number of leaves n increases, maintaining a fixed number of galls g. We find that the exponential growth with n of the number of rooted binary unlabeled normal galled trees with g galls has the same value irrespective of the value of g ≥ 0. The subexponential growth, however, depends on g; it follows c_g n^{2g-3/2}, where c_g is a constant dependent on g. Although for each g, the exponential growth is approximately 2.4833ⁿ, summing across all g, the exponential growth is instead approximated by the much larger 4.8230ⁿ.
Cite as
Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls. 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. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{agranattamir_et_al:LIPIcs.AofA.2024.27,
author = {Agranat-Tamir, Lily and Fuchs, Michael and Gittenberger, Bernhard and Rosenberg, Noah A.},
title = {{Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {27:1--27:14},
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.2024.27},
URN = {urn:nbn:de:0030-drops-204626},
doi = {10.4230/LIPIcs.AofA.2024.27},
annote = {Keywords: galled trees, generating functions, phylogenetics, unlabeled trees}
}
Document
Sharpened Localization of the Trailing Point of the Pareto Record Frontier
Abstract
For d ≥ 2 and i.i.d. d-dimensional observations X^{(1)}, X^{(2)}, … with independent Exponential(1) coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 25:Paper No. 92, 24 pp., 2020) of the boundary (relative to the closed positive orthant), or "frontier", F_n of the closed Pareto record-setting (RS) region RS_n := {0 ≤ x ∈ R^d: x ⊀ X^(i) for all 1 ≤ i ≤ n} at time n, where 0 ≤ x means that 0 ≤ x_j for 1 ≤ j ≤ d and x ≺ y means that x_j < y_j for 1 ≤ j ≤ d. With x_+ : = ∑_{j = 1}^d x_j = ‖x‖₁, let
F_n^- := min{x_+: x ∈ F_n} and F_n^+ : = max{x_+: x ∈ F_n}.
Almost surely, there are for each n unique vectors λ_n ∈ F_n and τ_n ∈ F_n such that F_n^+ = (λ_n)_+ and F_n^- = (τ_n)_+; we refer to λ_n and τ_n as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of F^+, but somewhat crude information about F^-, namely, that for any ε > 0 and c_n → ∞ we have
P(F_n^- - ln n ∈ (- (2 + ε) ln ln ln n, c_n)) → 1
(describing typical behavior) and almost surely
limsup (F_n^- - ln n)/(ln ln n) ≤ 0 and liminf (F_n^- - ln n)/(ln ln ln n) ∈ [-2, -1].
In this extended abstract we use the theory of generators (minima of F_n) together with the first- and second-moment methods to improve considerably the trailing-point location results to
F_n^- - (ln n - ln ln ln n) ⟶P -ln(d - 1)
(describing typical behavior) and, for d ≥ 3, almost surely
limsup [F_n^- -(ln n - ln ln ln n)] ≤ -ln(d - 2) + ln 2
and liminf [F_n^- -(ln n - ln ln ln n)] ≥ -ln d - ln 2.
Cite as
James Allen Fill, Daniel Q. Naiman, and Ao Sun. Sharpened Localization of the Trailing Point of the Pareto Record Frontier. 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. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fill_et_al:LIPIcs.AofA.2024.28,
author = {Fill, James Allen and Naiman, Daniel Q. and Sun, Ao},
title = {{Sharpened Localization of the Trailing Point of the Pareto Record Frontier}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {28:1--28:21},
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.2024.28},
URN = {urn:nbn:de:0030-drops-204631},
doi = {10.4230/LIPIcs.AofA.2024.28},
annote = {Keywords: Multivariate records, Pareto records, generators, interior generators, minima, maxima, record-setting region, frontier, current records, boundary-crossing probabilities, first moment method, second moment method, orthants}
}
Document
Statistics of Parking Functions and Labeled Forests
Abstract
In this paper we obtain some new results on the enumeration of parking functions and labeled forests. We introduce new statistics both for parking functions and for labeled forests that are connected to each other by means of a bijection. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics.
Cite as
Stephan Wagner and Mei Yin. Statistics of Parking Functions and Labeled Forests. 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. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{wagner_et_al:LIPIcs.AofA.2024.29,
author = {Wagner, Stephan and Yin, Mei},
title = {{Statistics of Parking Functions and Labeled Forests}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {29:1--29:14},
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.2024.29},
URN = {urn:nbn:de:0030-drops-204648},
doi = {10.4230/LIPIcs.AofA.2024.29},
annote = {Keywords: parking function, labeled forest, generating function, Pollak’s circle argument, bijection}
}
Document
Depth-First Search Performance in Random Digraphs
Abstract
We present an analysis of the depth-first search algorithm in a random digraph model with independent outdegrees having an arbitrary distribution with finite variance. The results include asymptotics for the distribution of the stack index and depths of the search. The search yields a series of trees of finite size before and after the exploration of a giant tree. Our analysis mainly concerns the giant tree. Most results are first order. This analysis proposed by Donald Knuth in his next to appear volume of The Art of Computer Programming gives interesting insight in one of the most elegant and efficient algorithm for graph analysis due to Tarjan.
Cite as
Philippe Jacquet and Svante Janson. Depth-First Search Performance in Random Digraphs. 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. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{jacquet_et_al:LIPIcs.AofA.2024.30,
author = {Jacquet, Philippe and Janson, Svante},
title = {{Depth-First Search Performance in Random Digraphs}},
booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
pages = {30:1--30:15},
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.2024.30},
URN = {urn:nbn:de:0030-drops-204655},
doi = {10.4230/LIPIcs.AofA.2024.30},
annote = {Keywords: Depth First Search, random digraph, Analysis of Algorithms}
}