eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
1
300
10.4230/LIPIcs.AofA.2022
article
LIPIcs, Volume 225, AofA 2022, Complete Volume
Ward, Mark Daniel
1
https://orcid.org/0000-0001-8795-746X
Purdue University, West Lafayette, IN, USA
LIPIcs, Volume 225, AofA 2022, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022/LIPIcs.AofA.2022.pdf
LIPIcs, Volume 225, AofA 2022, Complete Volume
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
0:i
0:xvi
10.4230/LIPIcs.AofA.2022.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Ward, Mark Daniel
1
https://orcid.org/0000-0001-8795-746X
Purdue University, West Lafayette, IN, USA
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.0/LIPIcs.AofA.2022.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
1:1
1:28
10.4230/LIPIcs.AofA.2022.1
article
Building Sources of Zero Entropy: Rescaling and Inserting Delays (Invited Talk)
Akhavi, Ali
1
2
Paccaut, Fréderic
3
Vallée, Brigitte
4
GREYC, Université de Caen-Normandie, France
LIPN, Université Sorbonne Paris Nord, France
LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, France
GREYC, CNRS and Université de Caen-Normandie, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.1/LIPIcs.AofA.2022.1.pdf
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
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
2:1
2:14
10.4230/LIPIcs.AofA.2022.2
article
On the Independence Number of Random Trees via Tricolourations
Bellin, Etienne
1
https://orcid.org/0000-0002-8638-1929
Ecole Polytechnique, Palaiseau, France
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.2/LIPIcs.AofA.2022.2.pdf
Independence number
simply generated tree
Galton-Watson tree
tricolouration
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
3:1
3:14
10.4230/LIPIcs.AofA.2022.3
article
Fragmentation Processes Derived from Conditioned Stable Galton-Watson Trees
Berzunza Ojeda, Gabriel
1
https://orcid.org/0000-0002-7924-8775
Holmgren, Cecilia
2
https://orcid.org/0000-0003-0717-4671
Department of Mathematical Sciences, University of Liverpool, United Kingdom
Department of Mathematics, Uppsala University, Sweden
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.3/LIPIcs.AofA.2022.3.pdf
Additive coalescent
fragmentation
Galton-Watson trees
spectrally positive stable Lévy processes
stable Lévy tree
Prim’s algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
4:1
4:14
10.4230/LIPIcs.AofA.2022.4
article
A Modification of the Random Cutting Model
Burghart, Fabian
1
Department of Mathematics, Uppsala University, Sweden
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.4/LIPIcs.AofA.2022.4.pdf
Random cutting model
Random separation of graphs
Percolation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
5:1
5:13
10.4230/LIPIcs.AofA.2022.5
article
Enumeration of d-Combining Tree-Child Networks
Chang, Yu-Sheng
1
Fuchs, Michael
1
Liu, Hexuan
1
Wallner, Michael
2
Yu, Guan-Ru
3
Department of Mathematical Sciences, National Chengchi University, Taipei, 116, Taiwan
Institute of Discrete Mathematics and Geometry, TU Wien, Vienna, 1040, Austria
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, 824, Taiwan
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.5/LIPIcs.AofA.2022.5.pdf
Phylogenetic network
tree-child network
d-combining tree-child network
exact enumeration
asymptotic enumeration
reticulation node
limit law
stretched exponential
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
6:1
6:12
10.4230/LIPIcs.AofA.2022.6
article
Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps
Chapuy, Guillaume
1
https://orcid.org/0000-0001-7730-5417
Louf, Baptiste
2
https://orcid.org/0000-0002-7799-6948
Walsh, Harriet
3
1
https://orcid.org/0000-0002-4199-4975
Université Paris Cité, IRIF, CNRS, F-75013 Paris, France
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden
Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.6/LIPIcs.AofA.2022.6.pdf
Random partitions
limit shapes
transposition factorisations
map enumeration
Hurwitz numbers
RSK algorithm
giant components
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
7:1
7:15
10.4230/LIPIcs.AofA.2022.7
article
Universal Properties of Catalytic Variable Equations
Drmota, Michael
1
https://orcid.org/0000-0002-6876-6569
Hainzl, Eva-Maria
1
https://orcid.org/0000-0001-9431-5596
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.7/LIPIcs.AofA.2022.7.pdf
catalytic equation
singular expansion
univeral asymptotics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
8:1
8:16
10.4230/LIPIcs.AofA.2022.8
article
Partial Match Queries in Quad- K-d Trees
Duch, Amalia
1
https://orcid.org/0000-0003-4371-1286
Martínez, Conrado
1
https://orcid.org/0000-0003-1302-9067
Universitat Politècnica de Catalunya, Barcelona, Spain
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.8/LIPIcs.AofA.2022.8.pdf
Quadtree
Partial match queries
Associative queries
Multidimensional search
Analysis of algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
9:1
9:13
10.4230/LIPIcs.AofA.2022.9
article
Improved Error Bounds for the Number of Irreducible Polynomials and Self-Reciprocal Irreducible Monic Polynomials with Prescribed Coefficients over a Finite Field
Gao, Zhicheng
1
School of Mathematics and Statistics, Carleton University, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.9/LIPIcs.AofA.2022.9.pdf
finite fields
irreducible polynomials
prescribed coefficients
generating functions
Weil bounds
self-reciprocal
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
10:1
10:17
10.4230/LIPIcs.AofA.2022.10
article
Uncovering a Random Tree
Hackl, Benjamin
1
2
https://orcid.org/0000-0003-2998-9599
Panholzer, Alois
3
https://orcid.org/0000-0003-2813-3457
Wagner, Stephan
1
https://orcid.org/0000-0001-5533-2764
Uppsala University, Sweden
University of Klagenfurt, Austria
TU Wien, Austria
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.10/LIPIcs.AofA.2022.10.pdf
Labeled tree
uncover process
functional central limit theorem
limiting distribution
phase transition
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
11:1
11:15
10.4230/LIPIcs.AofA.2022.11
article
Depth-First Search Performance in a Random Digraph with Geometric Degree Distribution
Jacquet, Philippe
1
https://orcid.org/0000-0001-7919-1206
Janson, Svante
2
https://orcid.org/0000-0002-9680-2790
Inria Saclay Ile de France, France
Department of Mathematics, Uppsala University, Uppsala, Sweden
We present an analysis of the depth-first search algorithm in a random digraph model with geometric outdegree distribution. We give also some extensions to general outdegree distributions. This problem posed 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.11/LIPIcs.AofA.2022.11.pdf
Combinatorics
Depth-First Search
Random Digraphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
12:1
12:17
10.4230/LIPIcs.AofA.2022.12
article
Affirmative Sampling: Theory and Applications
Lumbroso, Jérémie
1
https://orcid.org/0000-0002-5563-687X
Martínez, Conrado
2
https://orcid.org/0000-0003-1302-9067
Department of Computer Science, Princeton University, NJ, USA
Department of Computer Science, Universitat Politècnica de Catalunya, Barcelona, Spain
Affirmative Sampling is a practical and efficient novel algorithm to obtain random samples of distinct elements from a data stream. Its most salient feature is that the size S of the sample will, on expectation, grow with the (unknown) number n of distinct elements in the data stream. As any distinct element has the same probability to be sampled, and the sample size is greater when the "diversity" (the number of distinct elements) is greater, the samples that Affirmative Sampling delivers are more representative than those produced by any scheme where the sample size is fixed a priori - hence its name. Our algorithm is straightforward to implement, and several implementations already exist.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.12/LIPIcs.AofA.2022.12.pdf
Data streams
Distinct sampling
Random sampling
Cardinality estimation
Analysis of algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
13:1
13:17
10.4230/LIPIcs.AofA.2022.13
article
Mean Field Analysis of an Incentive Algorithm for a Closed Stochastic Network
Moreno, Bianca Marin
1
Fricker, Christine
1
Mohamed, Hanene
2
Philippe, Amaury
3
Trépanier, Martin
3
INRIA Paris, DI-ENS, PSL Research University, Paris, France
MODAL'X, UMR CNRS 9023, UPL, Université Paris Nanterre, France
Mathematical and Industrial Engineering, Polytechnique Montreal, Montreal, Canada
The paper deals with a load-balancing algorithm for a closed stochastic network with two zones with different demands. The algorithm is motivated by an incentive algorithm for redistribution of cars in a large-scale car-sharing system. The service area is divided into two zones. When cars stay too long in the low-demand zone, users are encouraged to pick them up and return them in the high-demand zone. The zones are divided in cells called stations. The cars are the network customers. The mean-field limit solution of an ODE gives the large scale distribution of the station state in both clusters for this incentive policy in a discrete Markovian framework. An equilibrium point of this ODE is characterized via the invariant measure of a random walk in the quarter-plane. The proportion of empty and saturated stations measures how the system is balanced. Numerical experiments illustrate the impact of the incentive policy. Our study shows that the incentive policy helps when the high-demand zone observes a lack of cars but a saturation must be prevented especially when the high-demand zone is small.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.13/LIPIcs.AofA.2022.13.pdf
Large scale analysis
mean-field
car-sharing
incentive algorithm
stochastic network
cluster
load balancing
closed Jackson networks
product-form distribution
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
14:1
14:13
10.4230/LIPIcs.AofA.2022.14
article
On the Contraction Method with Reduced Independence Assumptions
Neininger, Ralph
1
Straub, Jasmin
1
Institute for Mathematics, Goethe University, 60054 Frankfurt a.M., Germany
Recursive sequences of laws of random variables (and random vectors) are considered where an independence assumption which is usually made within the setting of the contraction method is dropped. This restricts the study to sequences which after normalization lead to asymptotic normality. We provide a general univariate central limit theorem which can directly be applied to problems from the analysis of algorithms and random recursive structures without further knowledge of the contraction method. Also multivariate central limit theorems are shown and bounds on rates of convergence are provided. Examples include some previously shown central limit analogues as well as new applications on Fibonacci matchings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.14/LIPIcs.AofA.2022.14.pdf
Probabilistic Analysis of Algorithms
random Trees
weak Convergence
Probability Metrics
Contraction Method
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
15:1
15:16
10.4230/LIPIcs.AofA.2022.15
article
Polyharmonic Functions in the Quarter Plane
Nessmann, Andreas
1
2
https://orcid.org/0000-0001-7394-7519
Technische Universität Wien, Austria
Université de Tours, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.15/LIPIcs.AofA.2022.15.pdf
Polyharmonic functions
Functional equations
Lattice paths
Random walks
Brownian motion
Generating functions
Laplace transforms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
16:1
16:16
10.4230/LIPIcs.AofA.2022.16
article
Automorphisms of Random Trees
Olsson, Christoffer
1
https://orcid.org/0000-0002-0864-0497
Wagner, Stephan
1
https://orcid.org/0000-0001-5533-2764
Department of Mathematics, Uppsala University, Sweden
We study the size of the automorphism group of two different types of random trees: Galton-Watson trees and Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution. While the proof for Galton-Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of Pólya trees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.16/LIPIcs.AofA.2022.16.pdf
random tree
Galton-Watson tree
Pólya tree
automorphism group
central limit theorem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
17:1
17:14
10.4230/LIPIcs.AofA.2022.17
article
The Number of Sources and Isolated Vertices in Random Directed Acyclic Graphs
Ralaivaosaona, Dimbinaina
1
https://orcid.org/0000-0002-6350-5538
University of Stellenbosch, South Africa
For a positive integer n and a real number p ∈ (0,1), a random directed acyclic digraph 𝔻_{ac}(n,p) is obtained from the binomial random digraph model 𝔻(n,p) conditioned to be acyclic, i.e., directed cycles are forbidden. In the binomial random digraph model 𝔻(n,p), every possible directed edge (excluding loops) occurs independently with probability p. Sources and sinks are among the most natural characteristics of directed acyclic graphs. We investigate the distribution of the number of sources in 𝔻_{ac}(n,p) when p is of the form λ/n, where λ is a fixed positive constant. Because of symmetry, the number of sinks will have the same distribution as the number of sources. Our main motivation is to understand how this distribution changes as we pass through the critical point p = 1/n. Since we are in the sparse regime, it makes sense to include the number of isolated vertices as well. In a directed graph an isolated vertex can be regarded as a vertex that is both a source and a sink. We prove asymptotic normality for each of these parameters when p = λ/n. Our method is based on the analysis of a multivariate generating function from a work of Gessel.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.17/LIPIcs.AofA.2022.17.pdf
Directed acyclic graph
generating function
central limit theorem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
225
18:1
18:12
10.4230/LIPIcs.AofA.2022.18
article
Parking Functions, Multi-Shuffle, and Asymptotic Phenomena
Yin, Mei
1
https://orcid.org/0000-0002-3706-9357
Department of Mathematics, University of Denver, Denver, CO, USA
Given a positive integer-valued vector u = (u_1, … , u_m) with u_1 < ⋯ < u_m, a u-parking function of length m is a sequence π = (π_1, … , π_m) of positive integers whose non-decreasing rearrangement (λ_1, … , λ_m) satisfies λ_i ≤ u_i for all 1 ≤ i ≤ m. We introduce a combinatorial construction termed a parking function multi-shuffle to generic u-parking functions and obtain an explicit characterization of multiple parking coordinates. As an application, we derive various asymptotic probabilistic properties of a uniform u-parking function of length m when u_i = cm+ib. The asymptotic scenario in the generic situation c > 0 is in sharp contrast with that of the special situation c = 0.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.18/LIPIcs.AofA.2022.18.pdf
Parking function
Multi-shuffle
Asymptotic expansion
Abel’s multinomial theorem