Pub Date : 2024-07-23DOI: 10.4230/LIPIcs.AofA.2024.6
M. Albenque, Éric Fusy, Zéphyr Salvy
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 $sqrt{n/log(n)}$ and $sqrt{n}$.
{"title":"Phase Transition for Tree-Rooted Maps","authors":"M. Albenque, Éric Fusy, Zéphyr Salvy","doi":"10.4230/LIPIcs.AofA.2024.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2024.6","url":null,"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 $sqrt{n/log(n)}$ and $sqrt{n}$.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"122 19","pages":"6:1-6:14"},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141812113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-30DOI: 10.4230/LIPIcs.AofA.2022.11
P. Jacquet, S. Janson
We present an analysis of the depth-first search algorithm in a random digraph model with independent outdegrees having a geometric distribution. The results include asymptotic results for the depth profile of vertices, the height (maximum depth) and average depth, the number of trees in the forest, the size of the largest and second-largest trees, and the numbers of arcs of different types in the depth-first jungle. Most results are first order. For the height we show an asymptotic normal distribution. 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.
{"title":"Depth-First Search Performance in a Random Digraph with Geometric Degree Distribution","authors":"P. Jacquet, S. Janson","doi":"10.4230/LIPIcs.AofA.2022.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.11","url":null,"abstract":"We present an analysis of the depth-first search algorithm in a random digraph model with independent outdegrees having a geometric distribution. The results include asymptotic results for the depth profile of vertices, the height (maximum depth) and average depth, the number of trees in the forest, the size of the largest and second-largest trees, and the numbers of arcs of different types in the depth-first jungle. Most results are first order. For the height we show an asymptotic normal distribution. 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.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117020269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-21DOI: 10.4230/LIPIcs.AofA.2022.15
Andreas Nessmann
In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic functions, and convergence between the discrete and continuous cases is shown. 2012 ACM Subject Classification Theory of computation → Random walks and Markov chains; Mathematics of computing → Markov processes; Mathematics of computing → Generating functions; Mathematics of computing → Combinatorics Acknowledgements I would like to thank Kilian Raschel for introducing me to this topic as well as for a lot of valuable input and many fruitful discussions. Also, I would like to thank the anonymous reviewers for their valuable remarks.
{"title":"Polyharmonic Functions in the Quarter Plane","authors":"Andreas Nessmann","doi":"10.4230/LIPIcs.AofA.2022.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.15","url":null,"abstract":"In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic functions, and convergence between the discrete and continuous cases is shown. 2012 ACM Subject Classification Theory of computation → Random walks and Markov chains; Mathematics of computing → Markov processes; Mathematics of computing → Generating functions; Mathematics of computing → Combinatorics Acknowledgements I would like to thank Kilian Raschel for introducing me to this topic as well as for a lot of valuable input and many fruitful discussions. Also, I would like to thank the anonymous reviewers for their valuable remarks.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125118285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-22DOI: 10.4230/LIPIcs.AofA.2022.6
G. Chapuy, B. Louf, Harriet Walsh
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 which appears naturally in the context of Hurwitz numbers, enumerating certain 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 topological objects, Hurwitz 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.
{"title":"Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps","authors":"G. Chapuy, B. Louf, Harriet Walsh","doi":"10.4230/LIPIcs.AofA.2022.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.6","url":null,"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 which appears naturally in the context of Hurwitz numbers, enumerating certain 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 topological objects, Hurwitz 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.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"5 Suppl 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126759772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-15DOI: 10.4230/LIPIcs.AofA.2022.5
Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, M. Wallner, Guan Yu
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.
{"title":"Enumeration of d-Combining Tree-Child Networks","authors":"Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, M. Wallner, Guan Yu","doi":"10.4230/LIPIcs.AofA.2022.5","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.5","url":null,"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.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130515318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-02DOI: 10.4230/LIPIcs.AofA.2022.2
Etienne Bellin
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). 2012 ACM Subject Classification Mathematics of computing → Random graphs Acknowledgements I am grateful to Igor Kortchemski for his careful reading of the manuscript and for telling me Frederic Chapoton’s suggestion to consider canonical tricolourations of random trees. I am also grateful to the anonymous referees and their useful remarks.
{"title":"On the Independence Number of Random Trees via Tricolourations","authors":"Etienne Bellin","doi":"10.4230/LIPIcs.AofA.2022.2","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.2","url":null,"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). 2012 ACM Subject Classification Mathematics of computing → Random graphs Acknowledgements I am grateful to Igor Kortchemski for his careful reading of the manuscript and for telling me Frederic Chapoton’s suggestion to consider canonical tricolourations of random trees. I am also grateful to the anonymous referees and their useful remarks.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"318 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125832353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-04DOI: 10.4230/LIPIcs.AofA.2022.4
F. Burghart
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 (J. Austral. Math. Soc.11, 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 for binary caterpillar trees and complete binary trees.
{"title":"A Modification of the Random Cutting Model","authors":"F. Burghart","doi":"10.4230/LIPIcs.AofA.2022.4","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.4","url":null,"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 (J. Austral. Math. Soc.11, 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 for binary caterpillar trees and complete binary trees.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126654335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-29DOI: 10.4230/LIPIcs.AofA.2022.9
Zhicheng Gao
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 2 d and prescribed ℓ leading coefficients always exist provided that ℓ is slightly less than d/ 2.
{"title":"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","doi":"10.4230/LIPIcs.AofA.2022.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.9","url":null,"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 2 d and prescribed ℓ leading coefficients always exist provided that ℓ is slightly less than d/ 2.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132902526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-29DOI: 10.4230/LIPIcs.AofA.2020.10
M. Drmota, M. Noy, Benedikt Stufler
The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.
{"title":"Cut Vertices in Random Planar Maps","authors":"M. Drmota, M. Noy, Benedikt Stufler","doi":"10.4230/LIPIcs.AofA.2020.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.10","url":null,"abstract":"The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116928936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-15DOI: 10.4230/LIPIcs.AofA.2020.4
V. Berthé, E. Cesaratto, F. Paccaut, P. Rotondo, M. Safe, B. Vallée
This article is devoted to the study of two arithmetical sources associated with classical partitions, that are both defined through the mediant of two fractions. The Stern-Brocot source is associated with the sequence of all the mediants, while the Sturm source only keeps mediants whose denominator is "not too large". Even though these sources are both of zero Shannon entropy, with very similar Renyi entropies, their probabilistic features yet appear to be quite different. We then study how they influence the behaviour of tries built on words they emit, and we notably focus on the trie depth. The paper deals with Analytic Combinatorics methods, and Dirichlet generating functions, that are usually used and studied in the case of good sources with positive entropy. To the best of our knowledge, the present study is the first one where these powerful methods are applied to a zero-entropy context. In our context, the generating function associated with each source is explicit and related to classical functions in Number Theory, as the ζ function, the double ζ function or the transfer operator associated with the Gauss map. We obtain precise asymptotic estimates for the mean value of the trie depth that prove moreover to be quite different for each source. Then, these sources provide explicit and natural instances which lead to two unusual and different trie behaviours.
{"title":"Two Arithmetical Sources and Their Associated Tries","authors":"V. Berthé, E. Cesaratto, F. Paccaut, P. Rotondo, M. Safe, B. Vallée","doi":"10.4230/LIPIcs.AofA.2020.4","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.4","url":null,"abstract":"This article is devoted to the study of two arithmetical sources associated with classical partitions, that are both defined through the mediant of two fractions. The Stern-Brocot source is associated with the sequence of all the mediants, while the Sturm source only keeps mediants whose denominator is \"not too large\". Even though these sources are both of zero Shannon entropy, with very similar Renyi entropies, their probabilistic features yet appear to be quite different. We then study how they influence the behaviour of tries built on words they emit, and we notably focus on the trie depth. The paper deals with Analytic Combinatorics methods, and Dirichlet generating functions, that are usually used and studied in the case of good sources with positive entropy. To the best of our knowledge, the present study is the first one where these powerful methods are applied to a zero-entropy context. In our context, the generating function associated with each source is explicit and related to classical functions in Number Theory, as the ζ function, the double ζ function or the transfer operator associated with the Gauss map. We obtain precise asymptotic estimates for the mean value of the trie depth that prove moreover to be quite different for each source. Then, these sources provide explicit and natural instances which lead to two unusual and different trie behaviours.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132350899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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