61 Search Results for "Ward, Mark Daniel"

Volume

LIPIcs, Volume 225

33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

AofA 2022, June 20-24, 2022, Philadelphia, PA, USA

Editors: Mark Daniel Ward

Volume

LIPIcs, Volume 110

29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

AofA 2018, June 25-29, 2018, Uppsala, Sweden

Editors: James Allen Fill and Mark Daniel Ward

Document
DOI: 10.4230/LIPIcs.CP.2024.20
Constraint Modelling with LLMs Using In-Context Learning

Authors: Kostis Michailidis, Dimos Tsouros, and Tias Guns

Published in: LIPIcs, Volume 307, 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)

Abstract
Constraint Programming (CP) allows for the modelling and solving of a wide range of combinatorial problems. However, modelling such problems using constraints over decision variables still requires significant expertise, both in conceptual thinking and syntactic use of modelling languages. In this work, we explore the potential of using pre-trained Large Language Models (LLMs) as coding assistants, to transform textual problem descriptions into concrete and executable CP specifications. We present different transformation pipelines with explicit intermediate representations, and we investigate the potential benefit of various retrieval-augmented example selection strategies for in-context learning. We evaluate our approach on 2 datasets from the literature, namely NL4Opt (optimisation) and Logic Grid Puzzles (satisfaction), and a heterogeneous set of exercises from a CP course. The results show that pre-trained LLMs have promising potential for initialising the modelling process, with retrieval-augmented in-context learning significantly enhancing their modelling capabilities.
Cite as

Kostis Michailidis, Dimos Tsouros, and Tias Guns. Constraint Modelling with LLMs Using In-Context Learning. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 20:1-20:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{michailidis_et_al:LIPIcs.CP.2024.20,
  author =	{Michailidis, Kostis and Tsouros, Dimos and Guns, Tias},
  title =	{{Constraint Modelling with LLMs Using In-Context Learning}},
  booktitle =	{30th International Conference on Principles and Practice of Constraint Programming (CP 2024)},
  pages =	{20:1--20:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-336-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{307},
  editor =	{Shaw, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2024.20},
  URN =		{urn:nbn:de:0030-drops-207053},
  doi =		{10.4230/LIPIcs.CP.2024.20},
  annote =	{Keywords: Constraint Modelling, Constraint Acquisition, Constraint Programming, Large Language Models, In-Context Learning, Natural Language Processing, Named Entity Recognition, Retrieval-Augmented Generation, Optimisation}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.AofA.2022
LIPIcs, Volume 225, AofA 2022, Complete Volume

Authors: Mark Daniel Ward

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

Abstract
LIPIcs, Volume 225, AofA 2022, Complete Volume
Cite as

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. 1-300, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{ward:LIPIcs.AofA.2022,
  title =	{{LIPIcs, Volume 225, AofA 2022, Complete Volume}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{1--300},
  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},
  URN =		{urn:nbn:de:0030-drops-160857},
  doi =		{10.4230/LIPIcs.AofA.2022},
  annote =	{Keywords: LIPIcs, Volume 225, AofA 2022, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.AofA.2022.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Mark Daniel Ward

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

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

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. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ward:LIPIcs.AofA.2022.0,
  author =	{Ward, Mark Daniel},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{0:i--0:xvi},
  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.2022.0},
  URN =		{urn:nbn:de:0030-drops-160866},
  doi =		{10.4230/LIPIcs.AofA.2022.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.AofA.2022.1
Building Sources of Zero Entropy: Rescaling and Inserting Delays (Invited Talk)

Authors: Ali Akhavi, Fréderic Paccaut, and Brigitte Vallée

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

Abstract
Most of the natural sources that intervene in Information Theory have a positive entropy. They are well studied. The paper aims in building, in an explicit way, natural instances of sources with zero entropy. Such instances are obtained by slowing down sources of positive entropy, with processes which rescale sources or insert delays. These two processes - rescaling or inserting delays - are essentially the same; they do not change the fundamental intervals of the source, but only the "depth" at which they will be used, or the "speed" at which they are divided. However, they modify the entropy and lead to sources with zero entropy. The paper begins with a "starting" source of positive entropy, and uses a natural class of rescalings of sublinear type. In this way, it builds a class of sources of zero entropy that will be further analysed. As the starting sources possess well understood probabilistic properties, and as the process of rescaling does not change its fundamental intervals, the new sources keep the memory of some important probabilistic features of the initial source. Thus, these new sources may be thoroughly analysed, and their main probabilistic properties precisely described. We focus in particular on two important questions: exhibiting asymptotical normal behaviours à la Shannon-MacMillan-Breiman; analysing the depth of the tries built on the sources. In each case, we obtain a parameterized class of precise behaviours. The paper deals with the analytic combinatorics methodology and makes a great use of generating series.
Cite as

Ali Akhavi, Fréderic Paccaut, and Brigitte Vallée. Building Sources of Zero Entropy: Rescaling and Inserting Delays (Invited Talk). 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. 1:1-1:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{akhavi_et_al:LIPIcs.AofA.2022.1,
  author =	{Akhavi, Ali and Paccaut, Fr\'{e}deric and Vall\'{e}e, Brigitte},
  title =	{{Building Sources of Zero Entropy: Rescaling and Inserting Delays}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{1:1--1:28},
  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.2022.1},
  URN =		{urn:nbn:de:0030-drops-160879},
  doi =		{10.4230/LIPIcs.AofA.2022.1},
  annote =	{Keywords: Information Theory, Probabilistic analysis of sources, Sources with zero-entropy, Analytic combinatorics, Dirichlet generating functions, Transfer operator, Trie structure, Continued fraction expansion, Rice method, Quasi-power Theorem}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.2
On the Independence Number of Random Trees via Tricolourations

Authors: Etienne Bellin

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

Abstract
We are interested in the independence number of large random simply generated trees and related parameters, such as their matching number or the kernel dimension of their adjacency matrix. We express these quantities using a canonical tricolouration, which is a way to colour the vertices of a tree with three colours. As an application we obtain limit theorems in L^p for the renormalised independence number in large simply generated trees (including large size-conditioned Bienaymé-Galton-Watson trees).
Cite as

Etienne Bellin. On the Independence Number of Random Trees via Tricolourations. 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. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bellin:LIPIcs.AofA.2022.2,
  author =	{Bellin, Etienne},
  title =	{{On the Independence Number of Random Trees via Tricolourations}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{2:1--2:14},
  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.2022.2},
  URN =		{urn:nbn:de:0030-drops-160886},
  doi =		{10.4230/LIPIcs.AofA.2022.2},
  annote =	{Keywords: Independence number, simply generated tree, Galton-Watson tree, tricolouration}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.3
Fragmentation Processes Derived from Conditioned Stable Galton-Watson Trees

Authors: Gabriel Berzunza Ojeda and Cecilia Holmgren

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

Abstract
We study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree 𝐭_n conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index α ∈ (1,2]. This fragmentation process is analogous to that introduced in the works of Aldous, Evans and Pitman (1998), who considered the case of Cayley trees. Our main result establishes that, after rescaling, the fragmentation process of 𝐭_n converges as n → ∞ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an α-stable Lévy tree of index α ∈ (1,2]. We further establish that the latter can be constructed by considering the partitions of the unit interval induced by the normalized α-stable Lévy excursion with a deterministic drift studied by Miermont (2001). In particular, this extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.
Cite as

Gabriel Berzunza Ojeda and Cecilia Holmgren. Fragmentation Processes Derived from Conditioned Stable Galton-Watson Trees. 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. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berzunzaojeda_et_al:LIPIcs.AofA.2022.3,
  author =	{Berzunza Ojeda, Gabriel and Holmgren, Cecilia},
  title =	{{Fragmentation Processes Derived from Conditioned Stable Galton-Watson Trees}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{3:1--3:14},
  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.2022.3},
  URN =		{urn:nbn:de:0030-drops-160898},
  doi =		{10.4230/LIPIcs.AofA.2022.3},
  annote =	{Keywords: Additive coalescent, fragmentation, Galton-Watson trees, spectrally positive stable L\'{e}vy processes, stable L\'{e}vy tree, Prim’s algorithm}
}
Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.5
Enumeration of d-Combining Tree-Child Networks

Authors: Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu

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

Abstract
Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks which are tree-child networks with every reticulation node having exactly two parents. In this paper, we extend these studies to d-combining tree-child networks where every reticulation node has now d ≥ 2 parents. Moreover, we also give results and conjectures on the distributional behavior of the number of reticulation nodes of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves.
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Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu. Enumeration of d-Combining Tree-Child Networks. 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. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chang_et_al:LIPIcs.AofA.2022.5,
  author =	{Chang, Yu-Sheng and Fuchs, Michael and Liu, Hexuan and Wallner, Michael and Yu, Guan-Ru},
  title =	{{Enumeration of d-Combining Tree-Child Networks}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{5:1--5:13},
  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.2022.5},
  URN =		{urn:nbn:de:0030-drops-160914},
  doi =		{10.4230/LIPIcs.AofA.2022.5},
  annote =	{Keywords: Phylogenetic network, tree-child network, d-combining tree-child network, exact enumeration, asymptotic enumeration, reticulation node, limit law, stretched exponential}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.6
Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps

Authors: Guillaume Chapuy, Baptiste Louf, and Harriet Walsh

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

Abstract
We study the asymptotic behaviour of random integer partitions under a new probability law that we introduce, the Plancherel-Hurwitz measure. This distribution, which has a natural definition in terms of Young tableaux, is a deformation of the classical Plancherel measure. It appears naturally in the enumeration of Hurwitz maps, or equivalently transposition factorisations in symmetric groups. We study a regime in which the number of factors in the underlying factorisations grows linearly with the order of the group, and the corresponding maps are of high genus. We prove that the limiting behaviour exhibits a new, twofold, phenomenon: the first part becomes very large, while the rest of the partition has the standard Vershik-Kerov-Logan-Shepp limit shape. As a consequence, we obtain asymptotic estimates for unconnected Hurwitz numbers with linear Euler characteristic, which we use to study random Hurwitz maps in this regime. This result can also be interpreted as the return probability of the transposition random walk on the symmetric group after linearly many steps.
Cite as

Guillaume Chapuy, Baptiste Louf, and Harriet Walsh. Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps. 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. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chapuy_et_al:LIPIcs.AofA.2022.6,
  author =	{Chapuy, Guillaume and Louf, Baptiste and Walsh, Harriet},
  title =	{{Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{6:1--6:12},
  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.2022.6},
  URN =		{urn:nbn:de:0030-drops-160921},
  doi =		{10.4230/LIPIcs.AofA.2022.6},
  annote =	{Keywords: Random partitions, limit shapes, transposition factorisations, map enumeration, Hurwitz numbers, RSK algorithm, giant components}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.7
Universal Properties of Catalytic Variable Equations

Authors: Michael Drmota and Eva-Maria Hainzl

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

Abstract
Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions the dominant singularity of the solution function has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square-root singularity appears, and non-linear catalytic equations, where we - usually - have a singularity of type 3/2.
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Michael Drmota and Eva-Maria Hainzl. Universal Properties of Catalytic Variable Equations. 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. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{drmota_et_al:LIPIcs.AofA.2022.7,
  author =	{Drmota, Michael and Hainzl, Eva-Maria},
  title =	{{Universal Properties of Catalytic Variable Equations}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{7:1--7:15},
  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.2022.7},
  URN =		{urn:nbn:de:0030-drops-160930},
  doi =		{10.4230/LIPIcs.AofA.2022.7},
  annote =	{Keywords: catalytic equation, singular expansion, univeral asymptotics}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.8
Partial Match Queries in Quad- K-d Trees

Authors: Amalia Duch and Conrado Martínez

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

Abstract
Quad-K-d trees [Bereckzy et al., 2014] are a generalization of several well-known hierarchical K-dimensional data structures. They were introduced to provide a unified framework for the analysis of associative queries and to investigate the trade-offs between the cost of different operations and the memory needs (each node of a quad-K-d tree has arity 2^m for some m, 1 ≤ m ≤ K). Indeed, we consider here partial match - one of the fundamental associative queries - for several families of quad-K-d trees including, among others, relaxed K-d trees and quadtrees. In particular, we prove that the expected cost of a random partial match P̂_n that has s out of K specified coordinates in a random quad-K-d tree of size n is P̂_n ∼ β⋅ n^α where α and β are constants given in terms of K and s as well as additional parameters that characterize the specific family of quad-K-d trees under consideration. Additionally, we derive a precise asymptotic estimate for the main order term of P_{n,𝐪} - the expected cost of a fixed partial match in a random quad-K-d tree of size n. The techniques and procedures used to derive the mentioned costs extend those already successfully applied to derive analogous results in quadtrees and relaxed K-d trees; our results show that the previous results are just particular cases, and states the validity of the conjecture made in [Duch et al., 2016] to a wider variety of multidimensional data structures.
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Amalia Duch and Conrado Martínez. Partial Match Queries in Quad- K-d Trees. 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. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{duch_et_al:LIPIcs.AofA.2022.8,
  author =	{Duch, Amalia and Mart{\'\i}nez, Conrado},
  title =	{{Partial Match Queries in Quad- K-d Trees}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{8:1--8:16},
  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.2022.8},
  URN =		{urn:nbn:de:0030-drops-160949},
  doi =		{10.4230/LIPIcs.AofA.2022.8},
  annote =	{Keywords: Quadtree, Partial match queries, Associative queries, Multidimensional search, Analysis of algorithms}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.9
Improved Error Bounds for the Number of Irreducible Polynomials and Self-Reciprocal Irreducible Monic Polynomials with Prescribed Coefficients over a Finite Field

Authors: Zhicheng Gao

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

Abstract
A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field. The improved bounds imply that self-reciprocal irreducible monic polynomials with degree 2d and prescribed 𝓁 leading coefficients always exist provided that 𝓁 is slightly less than d/2.
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Zhicheng Gao. Improved Error Bounds for the Number of Irreducible Polynomials and Self-Reciprocal Irreducible Monic Polynomials with Prescribed Coefficients over a Finite Field. 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. 9:1-9:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gao:LIPIcs.AofA.2022.9,
  author =	{Gao, Zhicheng},
  title =	{{Improved Error Bounds for the Number of Irreducible Polynomials and Self-Reciprocal Irreducible Monic Polynomials with Prescribed Coefficients over a Finite Field}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{9:1--9:13},
  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.2022.9},
  URN =		{urn:nbn:de:0030-drops-160958},
  doi =		{10.4230/LIPIcs.AofA.2022.9},
  annote =	{Keywords: finite fields, irreducible polynomials, prescribed coefficients, generating functions, Weil bounds, self-reciprocal}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.10
Uncovering a Random Tree

Authors: Benjamin Hackl, Alois Panholzer, and Stephan Wagner

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

Abstract
We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this extended abstract: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase. Lastly, the largest connected component, for which we also observe a phase transition.
Cite as

Benjamin Hackl, Alois Panholzer, and Stephan Wagner. Uncovering a Random Tree. 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. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hackl_et_al:LIPIcs.AofA.2022.10,
  author =	{Hackl, Benjamin and Panholzer, Alois and Wagner, Stephan},
  title =	{{Uncovering a Random Tree}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{10:1--10:17},
  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.2022.10},
  URN =		{urn:nbn:de:0030-drops-160962},
  doi =		{10.4230/LIPIcs.AofA.2022.10},
  annote =	{Keywords: Labeled tree, uncover process, functional central limit theorem, limiting distribution, phase transition}
}
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