Pub Date : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2022.16
Christoffer Olsson, S. Wagner
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.
{"title":"Automorphisms of Random Trees","authors":"Christoffer Olsson, S. Wagner","doi":"10.4230/LIPIcs.AofA.2022.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.16","url":null,"abstract":"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.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122603655","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2020.3
Frédérique Bassino, Tsinjo Rakotoarimalala, A. Sportiello
15 We describe a multiple string pattern matching algorithm which is well-suited for approximate 16 search, and dictionaries composed of words of different lengths. We prove that this algorithm has 17 optimal complexity rate up to a multiplicative constant, for arbitrary dictionaries. This extends to 18 arbitrary dictionaries the classical results of Yao [SIAM J. Comput. 8, 1979] and Chang and Marr 19 [Proc. CPM94, 1994]. 2
我们描述了一种多字符串模式匹配算法，它非常适合于近似16搜索，以及由不同长度的单词组成的字典。我们证明了该算法对于任意字典具有17个最优的复杂度，且复杂度可达一个乘法常数。姚[SIAM J. Comput. 8, 1979]和Chang and Marr 19 [Proc. CPM94, 1994]的经典结果扩展到18个任意词典。2
{"title":"The Complexity of the Approximate Multiple Pattern Matching Problem for Random Strings","authors":"Frédérique Bassino, Tsinjo Rakotoarimalala, A. Sportiello","doi":"10.4230/LIPIcs.AofA.2020.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.3","url":null,"abstract":"15 We describe a multiple string pattern matching algorithm which is well-suited for approximate 16 search, and dictionaries composed of words of different lengths. We prove that this algorithm has 17 optimal complexity rate up to a multiplicative constant, for arbitrary dictionaries. This extends to 18 arbitrary dictionaries the classical results of Yao [SIAM J. Comput. 8, 1979] and Chang and Marr 19 [Proc. CPM94, 1994]. 2","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121403950","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2022.3
Gabriel Berzunza Ojeda, Cecilia Holmgren
We study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree tn 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 tn 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. 2012 ACM Subject Classification Mathematics of computing → Probabilistic algorithms
{"title":"Fragmentation Processes Derived from Conditioned Stable Galton-Watson Trees","authors":"Gabriel Berzunza Ojeda, Cecilia Holmgren","doi":"10.4230/LIPIcs.AofA.2022.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.3","url":null,"abstract":"We study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree tn 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 tn 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. 2012 ACM Subject Classification Mathematics of computing → Probabilistic algorithms","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127469977","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2020.12
Ilse Fischer, Matjaž Konvalinka
Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but a bijective proof for any of these equivalences has been elusive for almost 40 years. In this extended abstract, we provide a sketch of the first bijective proof of the enumeration formula for alternating sign matrices, and of the fact that alternating sign matrices are equinumerous with descending plane partitions. The bijections are based on the operator formula for the number of monotone triangles due to the first author. The starting point for these constructions were known “computational” proofs, but the combinatorial point of view led to several drastic modifications and simplifications. We also provide computer code where all of our constructions have been implemented. 2012 ACM Subject Classification Mathematics of computing → Combinatoric problems; Mathematics of computing → Combinatorial algorithms; Mathematics of computing → Enumeration
{"title":"The First Bijective Proof of the Alternating Sign Matrix Theorem Theorem","authors":"Ilse Fischer, Matjaž Konvalinka","doi":"10.4230/LIPIcs.AofA.2020.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.12","url":null,"abstract":"Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but a bijective proof for any of these equivalences has been elusive for almost 40 years. In this extended abstract, we provide a sketch of the first bijective proof of the enumeration formula for alternating sign matrices, and of the fact that alternating sign matrices are equinumerous with descending plane partitions. The bijections are based on the operator formula for the number of monotone triangles due to the first author. The starting point for these constructions were known “computational” proofs, but the combinatorial point of view led to several drastic modifications and simplifications. We also provide computer code where all of our constructions have been implemented. 2012 ACM Subject Classification Mathematics of computing → Combinatoric problems; Mathematics of computing → Combinatorial algorithms; Mathematics of computing → Enumeration","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":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126131623","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2018.32
D. Ralaivaosaona, J. B. Ravelomanana, S. Wagner
Tanglegrams are structures consisting of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the two trees. We say that a tanglegram is planar if it can be drawn in the plane without crossings. Using a blend of combinatorial and analytic techniques, we determine an asymptotic formula for the number of planar tanglegrams with n leaves on each side.
{"title":"Counting Planar Tanglegrams","authors":"D. Ralaivaosaona, J. B. Ravelomanana, S. Wagner","doi":"10.4230/LIPIcs.AofA.2018.32","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2018.32","url":null,"abstract":"Tanglegrams are structures consisting of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the two trees. We say that a tanglegram is planar if it can be drawn in the plane without crossings. Using a blend of combinatorial and analytic techniques, we determine an asymptotic formula for the number of planar tanglegrams with n leaves on each side.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128635223","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2022.17
D. Ralaivaosaona
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.
对于正整数n，实数p∈(0,1)，由条件为无环的二项随机有向图模型D (n, p)，即禁止有向环，得到一个随机有向无环有向图D ac (n, p)。在二项随机有向图模型D (n, p)中，每个可能的有向边(不包括环路)以p的概率独立出现。源和汇是有向无环图最自然的特征之一。我们研究了当p为λ/n形式时，D ac (n, p)中源数的分布，其中λ是一个固定的正常数。由于对称性，吸收的数量将与源的数量具有相同的分布。我们的主要动机是了解当我们通过临界点p = 1 /n时分布是如何变化的。由于我们处于稀疏状态，包含孤立顶点的数量也是有意义的。在有向图中，孤立顶点可以看作既是源又是汇的顶点。当p = λ/n时，我们证明了这些参数的渐近正态性。我们的方法是基于对Gessel著作中多元生成函数的分析。
{"title":"The Number of Sources and Isolated Vertices in Random Directed Acyclic Graphs","authors":"D. Ralaivaosaona","doi":"10.4230/LIPIcs.AofA.2022.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.17","url":null,"abstract":"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.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130823121","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2018.24
D. Gillman, Dana Randall
Many physical models undergo phase transitions as some parameter of the system is varied. This phenomenon has bearing on the convergence times for local Markov chains walking among the configurations of the physical system. One of the most basic examples of this phenomenon is the ferromagnetic Ising model on an n× n square lattice region Λ with mixed boundary conditions. For this spin system, if we fix the spins on the top and bottom sides of the square to be + and the left and right sides to be −, a standard Peierls argument based on energy shows that below some critical temperature tc, any local Markov chainM requires time exponential in n to mix. Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under 90 degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This class includes the ferromagnetic Ising model with mixed boundary conditions described above, where the frustrated faces are on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of Λ and yields weaker results when they are on the boundary of Λ but not near the corners. We show that there is a universal temperature T below which M will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing. 2012 ACM Subject Classification Theory of computation→ Random walks and Markov chains
{"title":"Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices","authors":"D. Gillman, Dana Randall","doi":"10.4230/LIPIcs.AofA.2018.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2018.24","url":null,"abstract":"Many physical models undergo phase transitions as some parameter of the system is varied. This phenomenon has bearing on the convergence times for local Markov chains walking among the configurations of the physical system. One of the most basic examples of this phenomenon is the ferromagnetic Ising model on an n× n square lattice region Λ with mixed boundary conditions. For this spin system, if we fix the spins on the top and bottom sides of the square to be + and the left and right sides to be −, a standard Peierls argument based on energy shows that below some critical temperature tc, any local Markov chainM requires time exponential in n to mix. Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under 90 degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This class includes the ferromagnetic Ising model with mixed boundary conditions described above, where the frustrated faces are on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of Λ and yields weaker results when they are on the boundary of Λ but not near the corners. We show that there is a universal temperature T below which M will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing. 2012 ACM Subject Classification Theory of computation→ Random walks and Markov chains","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"61 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126019900","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2020.6
Gabriel Berzunza Ojeda, Cecilia Holmgren
We consider the model of random trees introduced by Devroye [13], the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. The approach we develop may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we have also studied the case of complete d-regular trees. 2012 ACM Subject Classification Mathematics of computing → Probabilistic algorithms
{"title":"Largest Clusters for Supercritical Percolation on Split Trees","authors":"Gabriel Berzunza Ojeda, Cecilia Holmgren","doi":"10.4230/LIPIcs.AofA.2020.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.6","url":null,"abstract":"We consider the model of random trees introduced by Devroye [13], the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. The approach we develop may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we have also studied the case of complete d-regular trees. 2012 ACM Subject Classification Mathematics of computing → Probabilistic algorithms","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115790536","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2018.19
M. Drmota, Guanglong Yu
The purpose of this paper is to provide a central limit theorem for the number of occurrences of double triangles in random planar maps. This is the first result of this kind that goes beyond face counts of given valency. The method is based on generating functions, an involved combinatorial decomposition scheme that leads to a system of catalytic functional equations and an analytic extension of the Quadratic Method to systems of equations.
{"title":"The Number of Double Triangles in Random Planar Maps","authors":"M. Drmota, Guanglong Yu","doi":"10.4230/LIPIcs.AofA.2018.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2018.19","url":null,"abstract":"The purpose of this paper is to provide a central limit theorem for the number of occurrences of double triangles in random planar maps. This is the first result of this kind that goes beyond face counts of given valency. The method is based on generating functions, an involved combinatorial decomposition scheme that leads to a system of catalytic functional equations and an analytic extension of the Quadratic Method to systems of equations.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122742748","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 : 1900-01-01DOI: 10.4230/LIPIcs.AofA.2022.1
A. Akhavi, F. Paccaut, B. Vallée
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.
{"title":"Building Sources of Zero Entropy: Rescaling and Inserting Delays (Invited Talk)","authors":"A. Akhavi, F. Paccaut, B. Vallée","doi":"10.4230/LIPIcs.AofA.2022.1","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2022.1","url":null,"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.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124734046","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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