Pub Date : 2019-01-01DOI: 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}
Pub Date : 2019-01-01DOI: 10.1137/1.9781611975505.2
O. Bodini, Antoine Genitrini, M. Naima
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.
{"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}
Pub Date : 2019-01-01DOI: 10.1137/1.9781611975505.9
J. A. Fill, Wei-Chun Hung
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, $x log x$. Using 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).
我们对Svante Janson(2015)在极限快速排序分布函数$F$的右尾部以及Fill and Hung(2018)在相应密度$F$的右尾部以及每阶$F$的绝对导数的右尾部所产生的渐近对数上界进行了实质性的改进。例如，我们建立了$log[1 - F(x)]$的上界，该上界通过$(log x)^2$的阶项匹配Knessl和Szpankowski(1999)的猜想渐近性;Janson(2015)绑定的相应顺序是先导顺序，$x log x$。使用F$上的精炼渐近界，我们得到了快速排序所需的比较次数分布的右尾大偏差(LD)结果，该结果大大提高了McDiarmid和Hayward(1996)的双边LD结果。
{"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}
Pub Date : 2018-10-18DOI: 10.1137/1.9781611975505.12
Oliver Cooley, Wenjie Fang, N. Giudice, Mihyun Kang
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...
{"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}
Pub Date : 2018-08-02DOI: 10.1137/1.9781611975505.3
C. Heuberger, Daniel Krenn
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. The 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.) One 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.
{"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}
Pub Date : 2018-08-01DOI: 10.1137/1.9781611975505.4
Benjamin Hackl, C. Heuberger, S. Wagner
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. Our results include asymptotic formulas for mean and variance of this quantity as well as a central limit theorem.
{"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}
Pub Date : 2018-01-14DOI: 10.1137/1.9781611975062.12
K. Panagiotou, Leon Ramzews
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.
{"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}
Pub Date : 2017-12-01DOI: 10.1137/1.9781611975505.14
M. Mitzenmacher
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.
{"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}
Pub Date : 2017-11-29DOI: 10.1137/1.9781611975062.13
Maryam Bahrani, Jérémie O. Lumbroso
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).
{"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}
Pub Date : 2017-06-15DOI: 10.1137/1.9781611975062.5
Frédérique Bassino, Tsinjo Rakotoarimalala, A. Sportiello
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.
我们推广了Fredriksson和Grabowski最近提出的一种多字符串模式匹配算法[J]。Discr。[Alg. 7, 2009]，用于处理大小为$s$的字母表上的任意字典。如果$r_m$是字典中长度为$m$的单词数，$phi(r) = max_m ln(s, m, r_m)/m$，则对于某个常数$kappa_{{}_textrm{UB}}$，该算法读取字符串字符的复杂度最多为$kappa_{{}_textrm{UB}}, phi(r)$。另一方面，我们推广了Yao的经典下界[SIAM J. Comput. 8, 1979]，用于处理任意字典的单一模式问题，并确定其至少为$kappa_{{}_textrm{LB}}, phi(r)$。这证明了算法的最优性，改进和纠正了以前的说法。
{"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}
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