Pub Date : 2015-07-29DOI: 10.1137/1.9781611974324.11
Noëla Müller, Ralph Neininger
A cyclic urn is an urn model for balls of types $0,ldots,m-1$ where in each draw the ball drawn, say of type $j$, is returned to the urn together with a new ball of type $j+1 mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2le mle 6$. For $mge 7$ the normalized composition vector does not converge. However, there is an almost sure approximation by a periodic random vector. In this paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $mge 7$. However, they are of maximal dimension $m-1$ only when $6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are supported by a two-dimensional subspace.
循环钢球缸是一种用于$0,ldots,m-1$型钢球的钢球缸模型，在每次抽出的钢球中，抽出的钢球(例如$j$型)与一个新的$j+1 mod m$型钢球一起返回到钢球缸中。这个案例$m=2$就是著名的弗里德曼骨灰盒。在归一化之后，已知复合向量，即在$n$步之后每种类型球的数量的向量对于$2le mle 6$是渐近正态的。对于$mge 7$，归一化复合向量不收敛。然而，有一个周期随机向量的近似几乎是肯定的。本文对该周期随机向量的渐近波动进行了辨识。我们证明这些波动对于所有$mge 7$都是渐近正态的。然而，只有当$6$不除$m$时，它们才是最大维数$m-1$。对于$m$是$6$的倍数，波动由二维子空间支持。
{"title":"The CLT Analogue for Cyclic Urns","authors":"Noëla Müller, Ralph Neininger","doi":"10.1137/1.9781611974324.11","DOIUrl":"https://doi.org/10.1137/1.9781611974324.11","url":null,"abstract":"A cyclic urn is an urn model for balls of types $0,ldots,m-1$ where in each draw the ball drawn, say of type $j$, is returned to the urn together with a new ball of type $j+1 mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2le mle 6$. For $mge 7$ the normalized composition vector does not converge. However, there is an almost sure approximation by a periodic random vector. In this paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $mge 7$. However, they are of maximal dimension $m-1$ only when $6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are supported by a two-dimensional subspace.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122553322","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 : 2015-07-28DOI: 10.1137/1.9781611974324.10
Matús Mihalák, P. Uznański, Pencho Yordanov
We study the problem of enumerating all rooted directed spanning trees (arborescences) of a directed graph (digraph) $G=(V,E)$ of $n$ vertices. An arborescence $A$ consisting of edges $e_1,ldots,e_{n-1}$ can be represented as a monomial $e_1cdot e_2 cdots e_{n-1}$ in variables $e in E$. All arborescences $mathsf{arb}(G)$ of a digraph then define the Kirchhoff polynomial $sum_{A in mathsf{arb}(G)} prod_{ein A} e$. We show how to compute a compact representation of the Kirchhoff polynomial -- its prime factorization, and how it relates to combinatorial properties of digraphs such as strong connectivity and vertex domination. In particular, we provide digraph decomposition rules that correspond to factorization steps of the polynomial, and also give necessary and sufficient primality conditions of the resulting factors expressed by connectivity properties of the corresponding decomposed components. Thereby, we obtain a linear time algorithm for decomposing a digraph into components corresponding to factors of the initial polynomial, and a guarantee that no finer factorization is possible. The decomposition serves as a starting point for a recursive deletion-contraction algorithm, and also as a preprocessing phase for iterative enumeration algorithms. Both approaches produce a compressed output and retain some structural properties in the resulting polynomial. This proves advantageous in practical applications such as calculating steady states on digraphs governed by Laplacian dynamics, or computing the greatest common divisor of Kirchhoff polynomials. Finally, we initiate the study of a class of digraphs which allow for a practical enumeration of arborescences. Using our decomposition rules we observe that various digraphs from real-world applications fall into this class or are structurally similar to it.
研究了具有$n$个顶点的有向图$G=(V,E)$的所有有根有向生成树(树形)的枚举问题。由边组成的乔木$A$$e_1,ldots,e_{n-1}$可以表示为变量$e in E$中的单项$e_1cdot e_2 cdots e_{n-1}$。有向图的所有树形$mathsf{arb}(G)$定义Kirchhoff多项式$sum_{A in mathsf{arb}(G)} prod_{ein A} e$。我们展示了如何计算Kirchhoff多项式的紧凑表示——它的质因数分解，以及它如何与有向图的组合性质(如强连通性和顶点支配)相关。特别地，我们给出了与多项式分解步骤相对应的有向图分解规则，并给出了由相应分解分量的连通性表示的结果因子的充分必要素数条件。由此，我们得到了将有向图分解为与初始多项式的因子相对应的分量的线性时间算法，并保证不可能进行更精细的分解。分解是递归删除-收缩算法的起点，也是迭代枚举算法的预处理阶段。这两种方法都会产生压缩的输出，并在产生的多项式中保留一些结构属性。这在实际应用中被证明是有利的，例如计算由拉普拉斯动力学控制的有向图上的稳态，或计算基尔霍夫多项式的最大公约数。最后，我们开始研究一类有向图，它允许树杈的实际枚举。使用我们的分解规则，我们观察到来自实际应用程序的各种有向图都属于这一类，或者在结构上与之相似。
{"title":"Prime Factorization of the Kirchhoff Polynomial: Compact Enumeration of Arborescences","authors":"Matús Mihalák, P. Uznański, Pencho Yordanov","doi":"10.1137/1.9781611974324.10","DOIUrl":"https://doi.org/10.1137/1.9781611974324.10","url":null,"abstract":"We study the problem of enumerating all rooted directed spanning trees (arborescences) of a directed graph (digraph) $G=(V,E)$ of $n$ vertices. An arborescence $A$ consisting of edges $e_1,ldots,e_{n-1}$ can be represented as a monomial $e_1cdot e_2 cdots e_{n-1}$ in variables $e in E$. All arborescences $mathsf{arb}(G)$ of a digraph then define the Kirchhoff polynomial $sum_{A in mathsf{arb}(G)} prod_{ein A} e$. We show how to compute a compact representation of the Kirchhoff polynomial -- its prime factorization, and how it relates to combinatorial properties of digraphs such as strong connectivity and vertex domination. In particular, we provide digraph decomposition rules that correspond to factorization steps of the polynomial, and also give necessary and sufficient primality conditions of the resulting factors expressed by connectivity properties of the corresponding decomposed components. Thereby, we obtain a linear time algorithm for decomposing a digraph into components corresponding to factors of the initial polynomial, and a guarantee that no finer factorization is possible. The decomposition serves as a starting point for a recursive deletion-contraction algorithm, and also as a preprocessing phase for iterative enumeration algorithms. Both approaches produce a compressed output and retain some structural properties in the resulting polynomial. This proves advantageous in practical applications such as calculating steady states on digraphs governed by Laplacian dynamics, or computing the greatest common divisor of Kirchhoff polynomials. Finally, we initiate the study of a class of digraphs which allow for a practical enumeration of arborescences. Using our decomposition rules we observe that various digraphs from real-world applications fall into this class or are structurally similar to it.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129937439","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 : 2015-06-09DOI: 10.1137/1.9781611974324.4
Élie de Panafieu, Lander Ramos
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs with a given minimum degree, and establishes new ones, such as the enumeration of Euler graphs, i.e. where all vertices have an even degree. Those results are derived using analytic combinatorics.
{"title":"Graphs with degree constraints","authors":"Élie de Panafieu, Lander Ramos","doi":"10.1137/1.9781611974324.4","DOIUrl":"https://doi.org/10.1137/1.9781611974324.4","url":null,"abstract":"Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs with a given minimum degree, and establishes new ones, such as the enumeration of Euler graphs, i.e. where all vertices have an even degree. Those results are derived using analytic combinatorics.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128608203","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 : 2015-01-04DOI: 10.1137/1.9781611973761.12
Marcin Kardas, Mirosław Kutyłowski, Jakub Lemiesz
We introduce and analyze a distributed cardinality estimation algorithm for a network consisted of not synchronized nodes. Our solution can be regarded as a generalization of the classic approximate counting algorithm based on the balls and bins model and is connected to the well studied process of covering the circle with random arcs. Although the algorithm is presented in the context of a radio network, the basic idea is applicable to any system in which many uncoordinated nodes communicate over a shared medium. In the paper we prove the correctness of the algorithm and by the methods of complex analysis we carefully examine the accuracy and precision of the estimator we have proposed. We also show that the construction of the proposed algorithm is a backbone for simple distributed summation.
{"title":"On Distributed Cardinality Estimation: Random Arcs Recycled","authors":"Marcin Kardas, Mirosław Kutyłowski, Jakub Lemiesz","doi":"10.1137/1.9781611973761.12","DOIUrl":"https://doi.org/10.1137/1.9781611973761.12","url":null,"abstract":"We introduce and analyze a distributed cardinality estimation algorithm for a network consisted of not synchronized nodes. Our solution can be regarded as a generalization of the classic approximate counting algorithm based on the balls and bins model and is connected to the well studied process of covering the circle with random arcs. Although the algorithm is presented in the context of a radio network, the basic idea is applicable to any system in which many uncoordinated nodes communicate over a shared medium. In the paper we prove the correctness of the algorithm and by the methods of complex analysis we carefully examine the accuracy and precision of the estimator we have proposed. We also show that the construction of the proposed algorithm is a backbone for simple distributed summation.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132358678","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 : 2015-01-04DOI: 10.1137/1.9781611973761.6
O. Bodini, Antoine Genitrini
Increasing trees have been extensively studied, since it is a simple model for many natural phenomena. Our paper focuses on sub-families of increasing trees. We measure the number of connected components obtained after having removed the nodes whose labels are smaller than a given value. This measure of cut-length allows, for example, to analyse in average an algorithm for tree-labelling. It is noticeable that we give exact formulae for the distribution of trees according to their size and cut-lengths. Our approach is based on a construction using grafting processes.
{"title":"Cuts in Increasing Trees","authors":"O. Bodini, Antoine Genitrini","doi":"10.1137/1.9781611973761.6","DOIUrl":"https://doi.org/10.1137/1.9781611973761.6","url":null,"abstract":"Increasing trees have been extensively studied, since it is a simple model for many natural phenomena. Our paper focuses on sub-families of increasing trees. We measure the number of connected components obtained after having removed the nodes whose labels are smaller than a given value. This measure of cut-length allows, for example, to analyse in average an algorithm for tree-labelling. It is noticeable that we give exact formulae for the distribution of trees according to their size and cut-lengths. Our approach is based on a construction using grafting processes.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"141 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114344981","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 : 2015-01-04DOI: 10.1137/1.9781611973761.7
D. Ralaivaosaona, S. Wagner
A fringe subtree of a rooted tree is a subtree that consists of a node and all its descendants. In this paper, we are particularly interested in the number of fringe subtrees that occur repeatedly in a random rooted tree. Specifically, we show that the average number of fringe subtrees that occur at least r times is of asymptotic order n/(log n)3/2 for every r ≥ 2 (with small periodic fluctuations in the main term) if a tree is taken uniformly at random from a simply generated family. Moreover, we also prove a strong concentration result for a related parameter: the size of the smallest tree that does not occur as a fringe subtree is with high probability equal to one of at most two different values. The main proof ingredients are singularity analysis, bootstrapping and the first and second moment methods.
{"title":"Repeated fringe subtrees in random rooted trees","authors":"D. Ralaivaosaona, S. Wagner","doi":"10.1137/1.9781611973761.7","DOIUrl":"https://doi.org/10.1137/1.9781611973761.7","url":null,"abstract":"A fringe subtree of a rooted tree is a subtree that consists of a node and all its descendants. In this paper, we are particularly interested in the number of fringe subtrees that occur repeatedly in a random rooted tree. Specifically, we show that the average number of fringe subtrees that occur at least r times is of asymptotic order n/(log n)3/2 for every r ≥ 2 (with small periodic fluctuations in the main term) if a tree is taken uniformly at random from a simply generated family. Moreover, we also prove a strong concentration result for a related parameter: the size of the smallest tree that does not occur as a fringe subtree is with high probability equal to one of at most two different values. The main proof ingredients are singularity analysis, bootstrapping and the first and second moment methods.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123431325","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 : 2015-01-04DOI: 10.1137/1.9781611973761.5
Frédérique Bassino, A. Sportiello
Directed random walks in dimension two describe the diffusion dynamics of particles in a line. Through a well-known bijection, excursions, i.e. walks in the half-plane, describe families of "simply-generated" Galton--Watson trees. These random objects can be generated in linear time, through an algorithm due to Devroye, and crucially based on the fact that the steps of the walk form an exchangeable sequence of random variables. We consider here the random generation of a more general family of structures, in which the transition rates, instead of being fixed once and for all, evolve in time (but not in space). Thus, the steps are not exchangeable anymore. On one side, this generalises diffusion into time-dependent diffusion. On the other side, among other things, this allows to consider effects of excluded volume, for Galton--Watson trees arising from exploration processes on finite random graphs, both directed and undirected. In the directed version, a special case concerns partitions of N objects into M blocks (counted by Stirling numbers of the second kind), and rooted K-maps which are accessible from the root, which in turn are related to the uniform generation of random accessible deterministic complete automata. We present an algorithm, based on the block-decomposition of the problem, and a crucial procedure consisting of a generalised Devroye algorithm, for transition rates which are well-approximated by piecewise exponential functions. The achieved (bit-)complexity remains linear.
{"title":"Linear-time generation of inhomogeneous random directed walks","authors":"Frédérique Bassino, A. Sportiello","doi":"10.1137/1.9781611973761.5","DOIUrl":"https://doi.org/10.1137/1.9781611973761.5","url":null,"abstract":"Directed random walks in dimension two describe the diffusion dynamics of particles in a line. Through a well-known bijection, excursions, i.e. walks in the half-plane, describe families of \"simply-generated\" Galton--Watson trees. These random objects can be generated in linear time, through an algorithm due to Devroye, and crucially based on the fact that the steps of the walk form an exchangeable sequence of random variables. \u0000 \u0000We consider here the random generation of a more general family of structures, in which the transition rates, instead of being fixed once and for all, evolve in time (but not in space). Thus, the steps are not exchangeable anymore. \u0000 \u0000On one side, this generalises diffusion into time-dependent diffusion. On the other side, among other things, this allows to consider effects of excluded volume, for Galton--Watson trees arising from exploration processes on finite random graphs, both directed and undirected. In the directed version, a special case concerns partitions of N objects into M blocks (counted by Stirling numbers of the second kind), and rooted K-maps which are accessible from the root, which in turn are related to the uniform generation of random accessible deterministic complete automata. \u0000 \u0000We present an algorithm, based on the block-decomposition of the problem, and a crucial procedure consisting of a generalised Devroye algorithm, for transition rates which are well-approximated by piecewise exponential functions. The achieved (bit-)complexity remains linear.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131080579","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 : 2015-01-04DOI: 10.1137/1.9781611973761.9
P. Jacquet, A. Magner
Graph tries are a generalization of classical digital trees: instead of being built from strings, a G-trie is built from label functions on the graph G. In this work, we determine leading order asymptotics for the variance of the size of a G-trie built on a memoryless source on a uniform alphabet distribution, where G is a member of a large class of infinite, M-regular directed, acyclic graphs with M > 1 fixed. In particular, this covers the cases of trees and grids. We find that, in such tries, the variance is of order Θ(nρ'), for some ρ' depending on G which is minimized when G is a tree. We also give an explicit expression for ρ' in the case where G is a grid, with M = 2.
{"title":"Variance of Size in Regular Graph Tries","authors":"P. Jacquet, A. Magner","doi":"10.1137/1.9781611973761.9","DOIUrl":"https://doi.org/10.1137/1.9781611973761.9","url":null,"abstract":"Graph tries are a generalization of classical digital trees: instead of being built from strings, a G-trie is built from label functions on the graph G. In this work, we determine leading order asymptotics for the variance of the size of a G-trie built on a memoryless source on a uniform alphabet distribution, where G is a member of a large class of infinite, M-regular directed, acyclic graphs with M > 1 fixed. In particular, this covers the cases of trees and grids. We find that, in such tries, the variance is of order Θ(nρ'), for some ρ' depending on G which is minimized when G is a tree. We also give an explicit expression for ρ' in the case where G is a grid, with M = 2.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121811818","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 : 2014-08-13DOI: 10.1137/1.9781611973761.3
Marcos A. Kiwi, D. Mitsche
Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for $alpha> tfrac{1}{2}$, $Cinmathbb{R}$, $ninmathbb{N}$, set $R=2ln n+C$ and build the graph $G=(V,E)$ with $|V|=n$ as follows: For each $vin V$, generate i.i.d. polar coordinates $(r_{v},theta_{v})$ using the joint density function $f(r,theta)$, with $theta_{v}$ chosen uniformly from $[0,2pi)$ and $r_{v}$ with density $f(r)=frac{alphasinh(alpha r)}{cosh(alpha R)-1}$ for $0leq r< R$. Then, join two vertices by an edge, if their hyperbolic distance is at most $R$. We prove that in the range $tfrac{1}{2} < alpha < 1$ a.a.s. for any two vertices of the same component, their graph distance is $O(log^{C_0+1+o(1)}n)$, where $C_0=2/(tfrac{1}{2}-frac{3}{4}alpha+tfrac{alpha^2}{4})$, thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size $O(log^{2C_0+1+o(1)}n)$, thus answering a question of Bode, Fountoulakis and M"{u}ller [BFM13]. We also show that a.a.s. there exist isolated components forming a path of length $Omega(log n)$, thus yielding a lower bound on the size of the second largest component.
{"title":"A Bound for the Diameter of Random Hyperbolic Graphs","authors":"Marcos A. Kiwi, D. Mitsche","doi":"10.1137/1.9781611973761.3","DOIUrl":"https://doi.org/10.1137/1.9781611973761.3","url":null,"abstract":"Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for $alpha> tfrac{1}{2}$, $Cinmathbb{R}$, $ninmathbb{N}$, set $R=2ln n+C$ and build the graph $G=(V,E)$ with $|V|=n$ as follows: For each $vin V$, generate i.i.d. polar coordinates $(r_{v},theta_{v})$ using the joint density function $f(r,theta)$, with $theta_{v}$ chosen uniformly from $[0,2pi)$ and $r_{v}$ with density $f(r)=frac{alphasinh(alpha r)}{cosh(alpha R)-1}$ for $0leq r< R$. Then, join two vertices by an edge, if their hyperbolic distance is at most $R$. We prove that in the range $tfrac{1}{2} < alpha < 1$ a.a.s. for any two vertices of the same component, their graph distance is $O(log^{C_0+1+o(1)}n)$, where $C_0=2/(tfrac{1}{2}-frac{3}{4}alpha+tfrac{alpha^2}{4})$, thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size $O(log^{2C_0+1+o(1)}n)$, thus answering a question of Bode, Fountoulakis and M\"{u}ller [BFM13]. We also show that a.a.s. there exist isolated components forming a path of length $Omega(log n)$, thus yielding a lower bound on the size of the second largest component.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127971031","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 : 2014-07-21DOI: 10.1137/1.9781611973761.2
Ioannis Giotis, L. Kirousis, Kostas I. Psaromiligkos, D. Thilikos
The algorithm for Lovasz Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve reconstructing the history of the algorithm from the witness tree. We apply our technique to improve the best known upper bound to acyclic chromatic index. Specifically we show that a graph with maximum degree Δ has an acyclic proper edge coloring with at most ⌈3.74(Δ − 1)⌉ + 1 colors, whereas the previously known best bound was 4(Δ − 1). The same technique is also applied to improve corresponding bounds for graphs with bounded girth. An interesting aspect of this application is that the probability of the "undesirable" events do not have a uniform upper bound, i.e. it constitutes a case of the asymmetric Lovasz Local Lemma.
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