Workshop on Analytic Algorithmics and Combinatorics最新文献

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Protection Number of Recursive Trees 递归树保护数
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975505.5
Z. Golebiewski, Mateusz Klimczak
{"title":"Protection Number of Recursive Trees","authors":"Z. Golebiewski, Mateusz Klimczak","doi":"10.1137/1.9781611975505.5","DOIUrl":"https://doi.org/10.1137/1.9781611975505.5","url":null,"abstract":"","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122818122","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}
引用次数: 1
Ranked Schröder Trees 排名Schröder树
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975505.2
O. Bodini, Antoine Genitrini, M. Naima
{"title":"Ranked Schröder Trees","authors":"O. Bodini, Antoine Genitrini, M. Naima","doi":"10.1137/1.9781611975505.2","DOIUrl":"https://doi.org/10.1137/1.9781611975505.2","url":null,"abstract":"In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schröder tree model is not adapted to take into account the chronology between the branching nodes. In particular, it does not answer the question: how many different phylogenetic stories lead to the creation of n species and what is the average time to get there? In this paper, we enrich this model in two distinct ways in order to obtain two ranked tree models for phylogenetics, i.e. models coding chronology. For that purpose, we first develop a model of (strongly) increasing Schröder trees, symbolically described in the classical context of increasing labeling. Then we introduce a generalization for the labeling with some unusual order constraint in Analytic Combinatorics (namely the weakly increasing trees). Although these models are direct extensions of the Schröder tree model, it appears that they are also in one-to-one correspondence with several classical combinatorial objects. Through the paper, we present these links, exhibit some parameters in typical large trees and conclude the studies with efficient uniform samplers.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126093121","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}
引用次数: 6
QuickSort: Improved right-tail asymptotics for the limiting distribution, and large deviations (Extended Abstract) 快速排序:极限分布和大偏差的改进右尾渐近(扩展摘要)
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975505.9
J. A. Fill, Wei-Chun Hung
{"title":"QuickSort: Improved right-tail asymptotics for the limiting distribution, and large deviations (Extended Abstract)","authors":"J. A. Fill, Wei-Chun Hung","doi":"10.1137/1.9781611975505.9","DOIUrl":"https://doi.org/10.1137/1.9781611975505.9","url":null,"abstract":"We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $log[1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(log x)^2$; the corresponding order for the Janson (2015) bound is the lead order, $x log x$. \u0000Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129085456","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}
引用次数: 3
Subcritical random hypergraphs, high-order components, and hypertrees 亚临界随机超图、高阶分量和超树
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2018-10-18 DOI: 10.1137/1.9781611975505.12
Oliver Cooley, Wenjie Fang, N. Giudice, Mihyun Kang
{"title":"Subcritical random hypergraphs, high-order components, and hypertrees","authors":"Oliver Cooley, Wenjie Fang, N. Giudice, Mihyun Kang","doi":"10.1137/1.9781611975505.12","DOIUrl":"https://doi.org/10.1137/1.9781611975505.12","url":null,"abstract":"One of the central topics in the theory of random graphs deals with the phase transition in the order of the largest components. In the binomial random graph $mathcal{G}(n,p)$, the threshold for t...","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114195034","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}
引用次数: 8
Esthetic Numbers and Lifting Restrictions on the Analysis of Summatory Functions of Regular Sequences 正则序列求和函数分析的审美数与提升限制
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2018-08-02 DOI: 10.1137/1.9781611975505.3
C. Heuberger, Daniel Krenn
{"title":"Esthetic Numbers and Lifting Restrictions on the Analysis of Summatory Functions of Regular Sequences","authors":"C. Heuberger, Daniel Krenn","doi":"10.1137/1.9781611975505.3","DOIUrl":"https://doi.org/10.1137/1.9781611975505.3","url":null,"abstract":"When asymptotically analysing the summatory function of a $q$-regular sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of matrices of the linear representation of the sequence determine the \"shape\" (in particular the growth) of the asymptotic formula. Existing general results for determining the precise behavior (including the Fourier coefficients of the appearing fluctuations) have previously been restricted by a technical condition on these eigenvalues. \u0000The aim of this work is to lift these restrictions by providing a insightful proof based on generating functions for the main pseudo Tauberian theorem for all cases simultaneously. (This theorem is the key ingredient for overcoming convergence problems in Mellin--Perron summation in the asymptotic analysis.) \u0000One example is discussed in more detail: A precise asymptotic formula for the amount of esthetic numbers in the first~$N$ natural numbers is presented. Prior to this only the asymptotic amount of these numbers with a given digit-length was known.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134179274","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}
引用次数: 1
Reducing Simply Generated Trees by Iterative Leaf Cutting 通过迭代切叶减少简单生成的树
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2018-08-01 DOI: 10.1137/1.9781611975505.4
Benjamin Hackl, C. Heuberger, S. Wagner
{"title":"Reducing Simply Generated Trees by Iterative Leaf Cutting","authors":"Benjamin Hackl, C. Heuberger, S. Wagner","doi":"10.1137/1.9781611975505.4","DOIUrl":"https://doi.org/10.1137/1.9781611975505.4","url":null,"abstract":"We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using an additive tree parameter model combined with a recursive characterization. \u0000Our results include asymptotic formulas for mean and variance of this quantity as well as a central limit theorem.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132744838","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}
引用次数: 1
Asymptotic Enumeration of Graph Classes with Many Components 多分量图类的渐近枚举
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2018-01-14 DOI: 10.1137/1.9781611975062.12
K. Panagiotou, Leon Ramzews
{"title":"Asymptotic Enumeration of Graph Classes with Many Components","authors":"K. Panagiotou, Leon Ramzews","doi":"10.1137/1.9781611975062.12","DOIUrl":"https://doi.org/10.1137/1.9781611975062.12","url":null,"abstract":"We consider graph classes $mathcal G$ in which every graph has components in a class $mathcal{C}$ of connected graphs. We provide a framework for the asymptotic study of $lvertmathcal{G}_{n,N}rvert$, the number of graphs in $mathcal{G}$ with $n$ vertices and $N:=lfloorlambda nrfloor$ components, where $lambdain(0,1)$. Assuming that the number of graphs with $n$ vertices in $mathcal{C}$ satisfies begin{align*} lvert mathcal{C}_nrvertsim b n^{-(1+alpha)}rho^{-n}n!, quad nto infty end{align*} for some $b,rho>0$ and $alpha>1$ -- a property commonly encountered in graph enumeration -- we show that begin{align*} lvertmathcal{G}_{n,N}rvertsim c(lambda) n^{f(lambda)} (log n)^{g(lambda)} rho^{-n}h(lambda)^{N}frac{n!}{N!}, quad nto infty end{align*} for explicitly given $c(lambda),f(lambda),g(lambda)$ and $h(lambda)$. These functions are piecewise continuous with a discontinuity at a critical value $lambda^{*}$, which we also determine. The central idea in our approach is to sample objects of $cal G$ randomly by so-called Boltzmann generators in order to translate enumerative problems to the analysis of iid random variables. By that we are able to exploit local limit theorems and large deviation results well-known from probability theory to prove our claims. The main results are formulated for generic combinatorial classes satisfying the SET-construction.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121248333","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}
引用次数: 1
Arithmetic Progression Hypergraphs: Examining the Second Moment Method 等差数列超图:检验二阶矩法
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2017-12-01 DOI: 10.1137/1.9781611975505.14
M. Mitzenmacher
{"title":"Arithmetic Progression Hypergraphs: Examining the Second Moment Method","authors":"M. Mitzenmacher","doi":"10.1137/1.9781611975505.14","DOIUrl":"https://doi.org/10.1137/1.9781611975505.14","url":null,"abstract":"In many data structure settings, it has been shown that using \"double hashing\" in place of standard hashing, by which we mean choosing multiple hash values according to an arithmetic progression instead of choosing each hash value independently, has asymptotically negligible difference in performance. We attempt to extend these ideas beyond data structure settings by considering how threshold arguments based on second moment methods can be generalized to \"arithmetic progression\" versions of problems. With this motivation, we define a novel \"quasi-random\" hypergraph model, random arithmetic progression (AP) hypergraphs, which is based on edges that form arithmetic progressions and unifies many previous problems. Our main result is to show that second moment arguments for 3-NAE-SAT and 2-coloring of 3-regular hypergraphs extend to the double hashing setting. We leave several open problems related to these quasi-random hypergraphs and the thresholds of associated problem variations.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126625881","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}
引用次数: 0
Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs 具有素数节点的分裂分解树:仙人掌图的枚举和随机生成
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2017-11-29 DOI: 10.1137/1.9781611975062.13
Maryam Bahrani, Jérémie O. Lumbroso
{"title":"Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs","authors":"Maryam Bahrani, Jérémie O. Lumbroso","doi":"10.1137/1.9781611975062.13","DOIUrl":"https://doi.org/10.1137/1.9781611975062.13","url":null,"abstract":"In this paper, we build on recent results by Chauve et al. (2014) and Bahrani and Lumbroso (2017), which combined the split-decomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs---in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper, can be thought of as trees with their edges replaced by cycles (of arbitrary lengths). Their split-decomposition trees contain prime nodes that are cycles, making them ideal to study. We derive a characterization for the split-decomposition trees of cactus graphs, produce a general template of symbolic grammars for cactus graphs, and implement random generation for these graphs, building on work by Iriza (2015).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131301222","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}
引用次数: 4
The complexity of the Multiple Pattern Matching Problem for random strings 随机字符串多模式匹配问题的复杂性
Workshop on Analytic Algorithmics and Combinatorics Pub Date : 2017-06-15 DOI: 10.1137/1.9781611975062.5
Frédérique Bassino, Tsinjo Rakotoarimalala, A. Sportiello
{"title":"The complexity of the Multiple Pattern Matching Problem for random strings","authors":"Frédérique Bassino, Tsinjo Rakotoarimalala, A. Sportiello","doi":"10.1137/1.9781611975062.5","DOIUrl":"https://doi.org/10.1137/1.9781611975062.5","url":null,"abstract":"We generalise a multiple string pattern matching algorithm, recently proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size $s$. If $r_m$ is the number of words of length $m$ in the dictionary, and $phi(r) = max_m ln(s, m, r_m)/m$, the complexity rate for the string characters to be read by this algorithm is at most $kappa_{{}_textrm{UB}}, phi(r)$ for some constant $kappa_{{}_textrm{UB}}$. On the other side, we generalise the classical lower bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern, to deal with arbitrary dictionaries, and determine it to be at least $kappa_{{}_textrm{LB}}, phi(r)$. This proves the optimality of the algorithm, improving and correcting previous claims.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"9 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121007764","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}
引用次数: 1
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