Pub Date : 2012-01-16DOI: 10.1137/1.9781611973020.10
A. Helmi, A. Panholzer
The hiring problem is a recent research problem, which has been introduced and studied first by Broder et al. [2] in 2008. It belongs to the category of on-line decision making under uncertainty. In such kind of research, the input is a sequence of instances and a decision must be taken for each instance depending on the instances seen so far while no information on the future is available. The hiring problem can be considered as a natural extension of the well-known secretary problem [4], where the employer is now looking for many candidates rather than only one (as it is the case for the secretary problem). Here the goal is to design some hiring strategy to meet the demands of the employer, which essentially are to obtain a good quality staff at a reasonable hiring rate, which is a main difference to the secretary problem, where an optimization policy, namely the demand of hiring the best candidate, occurs. Broder et al. introduced two so-called "Lake Wobegon strategies", namely "hiring above the current mean" and "hiring above the current median", applied for a continuous probabilistic model for the sequence of scores of the candidates. Archibald and Martinez [1] have reformulated the problem for a discrete model that considers the relative ranks amongst candidates as it is the case in the secretary problem. For this model in [1] the authors studied two general strategies, namely "hiring above the m-th best candidate", and "hiring above the median" (and other quantiles). In this work we give a detailed study of the "hiring above the median" strategy under this discrete model for the input sequence of scores of the candidates. This strategy processes the sequence of candidates as follows: hire the first interviewed candidate, and then any coming candidate is hired if and only if his rank is better than the rank of the median of the already hired staff, and discarded otherwise. Compared to the previous work of [1] we use a somewhat different recursive approach for a study of the "hiring above the median" strategy leading to rather explicit results. The key ingredients are to take into account the score of the median (the so-called threshold candidate) of the hired staff during the hiring process as well as to distinguish between two cases according to the parity of the size of the hiring set. Considering the transition probabilities during the hiring process yields, for fundamental hiring quantities, a system of linear recurrences that can be translated into a system of partial differential equations for the corresponding generating functions. In order to solve the PDEs appearing it turned out to be crucial to find suitable normalization factors of the studied recursive sequences, such that one of the corresponding generating functions itself reduces to a first order linear PDE. The exact solutions obtained for the differential equations yield to a rather detailed description of the exact probability distributions together with limiting
{"title":"Analysis of \"Hiring Above the Median\": A \"Lake Wobegon\" Strategy for The Hiring Problem","authors":"A. Helmi, A. Panholzer","doi":"10.1137/1.9781611973020.10","DOIUrl":"https://doi.org/10.1137/1.9781611973020.10","url":null,"abstract":"The hiring problem is a recent research problem, which has been introduced and studied first by Broder et al. [2] in 2008. It belongs to the category of on-line decision making under uncertainty. In such kind of research, the input is a sequence of instances and a decision must be taken for each instance depending on the instances seen so far while no information on the future is available. The hiring problem can be considered as a natural extension of the well-known secretary problem [4], where the employer is now looking for many candidates rather than only one (as it is the case for the secretary problem). Here the goal is to design some hiring strategy to meet the demands of the employer, which essentially are to obtain a good quality staff at a reasonable hiring rate, which is a main difference to the secretary problem, where an optimization policy, namely the demand of hiring the best candidate, occurs. Broder et al. introduced two so-called \"Lake Wobegon strategies\", namely \"hiring above the current mean\" and \"hiring above the current median\", applied for a continuous probabilistic model for the sequence of scores of the candidates. \u0000 \u0000Archibald and Martinez [1] have reformulated the problem for a discrete model that considers the relative ranks amongst candidates as it is the case in the secretary problem. For this model in [1] the authors studied two general strategies, namely \"hiring above the m-th best candidate\", and \"hiring above the median\" (and other quantiles). In this work we give a detailed study of the \"hiring above the median\" strategy under this discrete model for the input sequence of scores of the candidates. This strategy processes the sequence of candidates as follows: hire the first interviewed candidate, and then any coming candidate is hired if and only if his rank is better than the rank of the median of the already hired staff, and discarded otherwise. \u0000 \u0000Compared to the previous work of [1] we use a somewhat different recursive approach for a study of the \"hiring above the median\" strategy leading to rather explicit results. The key ingredients are to take into account the score of the median (the so-called threshold candidate) of the hired staff during the hiring process as well as to distinguish between two cases according to the parity of the size of the hiring set. Considering the transition probabilities during the hiring process yields, for fundamental hiring quantities, a system of linear recurrences that can be translated into a system of partial differential equations for the corresponding generating functions. In order to solve the PDEs appearing it turned out to be crucial to find suitable normalization factors of the studied recursive sequences, such that one of the corresponding generating functions itself reduces to a first order linear PDE. \u0000 \u0000The exact solutions obtained for the differential equations yield to a rather detailed description of the exact probability distributions together with limiting","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132695068","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 : 2012-01-16DOI: 10.1137/1.9781611973020.5
J. Munro
We will address several issues, old and new, in the computational complexity of comparison based problems, focusing on the long term development of techniques. Most notably, we will look at the comparison minimization problems of placing a set of n values into an arbitrary given partial order, and of completing the sort from this stage. The parallel development of the algorithmics and of graph entropy will be discussed in this context.
{"title":"The Complexity of Partial Orders","authors":"J. Munro","doi":"10.1137/1.9781611973020.5","DOIUrl":"https://doi.org/10.1137/1.9781611973020.5","url":null,"abstract":"We will address several issues, old and new, in the computational complexity of comparison based problems, focusing on the long term development of techniques. Most notably, we will look at the comparison minimization problems of placing a set of n values into an arbitrary given partial order, and of completing the sort from this stage. The parallel development of the algorithmics and of graph entropy will be discussed in this context.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115155460","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 : 2012-01-16DOI: 10.1137/1.9781611973020.12
Jérémie O. Lumbroso, O. Bodini
In 2001, Duchon, Flajolet, Louchard and Schaeffer introduced Boltzmann samplers, a radically novel way to efficiently generate huge random combinatorial objects without any preprocessing; the insight was that the probabilities can be obtained directly by evaluating the generating functions of combinatorials classes. Over the following decade, a vast array of papers has increased the formal expressiveness of these random samplers. Our paper introduces a new kind of sampler which generates multiplicative combinatorial structures, which enumerated by Dirichlet generating functions. Such classes, which are significantly harder to analyze than their additive counterparts, are at the intersection of combinatorics and analytic number theory. Indeed, one example we fully discuss is that of ordered factorizations. While we recycle many of the concepts of Boltzmann random sampling, our samplers no longer obey a Boltzmann distribution; we thus have coined a new name for them: Dirichlet samplers. These are very efficient as they can generate objects of size n in O((log n)2) worst-case time complexity. By providing a means by which to generate very large random multiplicative objects, our Dirichlet samplers can facilitate the investigation of these interesting, yet little studied structures. We also hope to illustrate some of our general ideas regarding the future direction for random sampling.
{"title":"Dirichlet Random Samplers for Multiplicative Combinatorial Structures","authors":"Jérémie O. Lumbroso, O. Bodini","doi":"10.1137/1.9781611973020.12","DOIUrl":"https://doi.org/10.1137/1.9781611973020.12","url":null,"abstract":"In 2001, Duchon, Flajolet, Louchard and Schaeffer introduced Boltzmann samplers, a radically novel way to efficiently generate huge random combinatorial objects without any preprocessing; the insight was that the probabilities can be obtained directly by evaluating the generating functions of combinatorials classes. Over the following decade, a vast array of papers has increased the formal expressiveness of these random samplers. \u0000 \u0000Our paper introduces a new kind of sampler which generates multiplicative combinatorial structures, which enumerated by Dirichlet generating functions. Such classes, which are significantly harder to analyze than their additive counterparts, are at the intersection of combinatorics and analytic number theory. Indeed, one example we fully discuss is that of ordered factorizations. While we recycle many of the concepts of Boltzmann random sampling, our samplers no longer obey a Boltzmann distribution; we thus have coined a new name for them: Dirichlet samplers. These are very efficient as they can generate objects of size n in O((log n)2) worst-case time complexity. \u0000 \u0000By providing a means by which to generate very large random multiplicative objects, our Dirichlet samplers can facilitate the investigation of these interesting, yet little studied structures. We also hope to illustrate some of our general ideas regarding the future direction for random sampling.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132184443","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 : 2011-11-22DOI: 10.1137/1.9781611973020.6
Michael J. Bannister, D. Eppstein
We describe a variant of the Bellman--Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.
{"title":"Randomized Speedup of the Bellman-Ford Algorithm","authors":"Michael J. Bannister, D. Eppstein","doi":"10.1137/1.9781611973020.6","DOIUrl":"https://doi.org/10.1137/1.9781611973020.6","url":null,"abstract":"We describe a variant of the Bellman--Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126464849","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 : 2011-07-15DOI: 10.1137/1.9781611973020.11
E. Y. Jin, M. Nebel
Regarding so-called hairpin-loops as the building blocks of a RNA secondary structure, the order (as introduced by Waterman as a parameter on graphs in 1978) provides information on the (balanced) nesting-depth of hairpin-loops and thus on the overall complexity of the structure. Subsequent to Waterman's seminal work, Zucker et al. and Clote introduced a more realistic combinatorial model for RNA secondary structures, the so-called saturated secondary structures. Compared to the traditional model ofWaterman, unpaired nucleotides (vertices) which are in favorable position for a pairing do not exist, i.e. no base pair (edge) can be added without violating at least one restriction for the graphs. That way, one major shortcoming of the traditional model has been cleared. However, the resulting model gets much more challenging from a mathematical point of view. As a consequence, the current state of knowledge concerning saturated structures is limited to (1) their asymptotic number, (2) the expected number of base pairs, (3) the asymptotic normal density of states [4]. Nothing is known about the nature of the branching topology of saturated structures -- a topic that the current paper completely solves. In this paper we show how it is possible to attack saturated structures and especially how to analyze their order. This is of special interest since in the past it has been proven to be one of the most demanding parameters to address (for the traditional model it has been an open problem for more than 20 years to find asymptotic results for the number of structures of given order and similar). We show the expected order of RNA saturated secondary structures of size n is log4 n (1 + O (1/log2n)), if we select the saturated secondary structure uniformly at random. Furthermore, the order of saturated secondary structures is sharply concentrated around its mean. As a consequence saturated structures and structures in the traditional model behave the same with respect to the expected order. Thus we may conclude that the traditional model has already drawn the right picture and conclusions inferred from it with respect to the order (the overall shape) of a structure remain valid even if enforcing saturation (at least in expectation).
将所谓的发夹环作为RNA二级结构的构建块，顺序(由沃特曼在1978年作为图上的参数引入)提供了关于发夹环的(平衡的)嵌套深度的信息，从而提供了结构的总体复杂性。在Waterman的开创性工作之后，Zucker等人和Clote引入了一种更现实的RNA二级结构组合模型，即所谓的饱和二级结构。与传统的waterman模型相比，不存在对配对有利的未配对的核苷酸(顶点)，即没有碱基对(边)可以在不违反图的至少一个限制的情况下添加。这样，传统模式的一个主要缺点就被消除了。然而，从数学的角度来看，得到的模型更具挑战性。因此，目前关于饱和结构的知识仅限于(1)它们的渐近数，(2)期望的碱基对数，(3)状态的渐近正态密度[4]。我们对饱和结构的分支拓扑的性质一无所知——这篇论文完全解决了这个问题。在本文中，我们展示了如何攻击饱和结构，特别是如何分析它们的顺序。这是特别有趣的，因为在过去，它已被证明是最需要解决的参数之一(对于传统模型，20多年来一直是一个开放的问题，以找到给定顺序和类似结构的数量的渐近结果)。我们表明，如果我们均匀随机选择饱和二级结构，则大小为n的RNA饱和二级结构的期望阶数为log4n (1 + O (1/log2n))。此外，饱和二级结构的顺序明显集中在其平均值附近。因此，饱和结构和传统模型中的结构对于期望阶的行为是相同的。因此，我们可以得出结论，传统模型已经绘制了正确的图像，并且即使强制饱和(至少在预期中)，从它推断出的关于结构的顺序(整体形状)的结论仍然有效。
{"title":"The Expected Order of Saturated RNA Secondary Structures","authors":"E. Y. Jin, M. Nebel","doi":"10.1137/1.9781611973020.11","DOIUrl":"https://doi.org/10.1137/1.9781611973020.11","url":null,"abstract":"Regarding so-called hairpin-loops as the building blocks of a RNA secondary structure, the order (as introduced by Waterman as a parameter on graphs in 1978) provides information on the (balanced) nesting-depth of hairpin-loops and thus on the overall complexity of the structure. \u0000 \u0000Subsequent to Waterman's seminal work, Zucker et al. and Clote introduced a more realistic combinatorial model for RNA secondary structures, the so-called saturated secondary structures. Compared to the traditional model ofWaterman, unpaired nucleotides (vertices) which are in favorable position for a pairing do not exist, i.e. no base pair (edge) can be added without violating at least one restriction for the graphs. That way, one major shortcoming of the traditional model has been cleared. However, the resulting model gets much more challenging from a mathematical point of view. As a consequence, the current state of knowledge concerning saturated structures is limited to (1) their asymptotic number, (2) the expected number of base pairs, (3) the asymptotic normal density of states [4]. Nothing is known about the nature of the branching topology of saturated structures -- a topic that the current paper completely solves. \u0000 \u0000In this paper we show how it is possible to attack saturated structures and especially how to analyze their order. This is of special interest since in the past it has been proven to be one of the most demanding parameters to address (for the traditional model it has been an open problem for more than 20 years to find asymptotic results for the number of structures of given order and similar). We show the expected order of RNA saturated secondary structures of size n is log4 n (1 + O (1/log2n)), if we select the saturated secondary structure uniformly at random. Furthermore, the order of saturated secondary structures is sharply concentrated around its mean. As a consequence saturated structures and structures in the traditional model behave the same with respect to the expected order. Thus we may conclude that the traditional model has already drawn the right picture and conclusions inferred from it with respect to the order (the overall shape) of a structure remain valid even if enforcing saturation (at least in expectation).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130300614","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}
Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. We prove that the Walksat algorithm from Papadimitriou (FOCS 1991)/Schoning (FOCS 1999) finds a satisfying assignment of Φ in polynomial time w.h.p. if m/n ≤ ρ · 2k/k for a certain constant ρ > 0. This is an improvement by a factor of Θ(k) over the best previous analysis of Walksat from Coja-Oghlan, Feige, Frieze, Krivelevich, Vilenchik (SODA 2009).
{"title":"Analyzing Walksat on Random Formulas","authors":"A. Coja-Oghlan, A. Frieze","doi":"10.1137/12090191X","DOIUrl":"https://doi.org/10.1137/12090191X","url":null,"abstract":"Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. We prove that the Walksat algorithm from Papadimitriou (FOCS 1991)/Schoning (FOCS 1999) finds a satisfying assignment of Φ in polynomial time w.h.p. if m/n ≤ ρ · 2k/k for a certain constant ρ > 0. This is an improvement by a factor of Θ(k) over the best previous analysis of Walksat from Coja-Oghlan, Feige, Frieze, Krivelevich, Vilenchik (SODA 2009).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125848120","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 : 2011-01-22DOI: 10.1137/1.9781611973013.15
S. Miracle, Dana Randall, A. Streib
Colloids are mixtures of two different types of molecules. The model has a hard-core constraint forcing all the molecules to occupy non-overlapping positions, but there are no additional interactions between molecules; all non-overlapping arrangements are equally likely. It is believed that colloids undergo a phase transition whereby at low density the two types of molecules will be uniformly interspersed, while at high density large clusters will form and the two types of molecules will effectively separate. While local algorithms are not believed to work at or beyond the critical point, an algorithm due to Dress and Krauth [3] offers an alternative approach to sampling potentially beyond the critical point where clusters begin to form. We study the DK algorithm on a colloid model consisting of long bars and small diamonds on the periodic lattice Z2n. We show that if we restrict the model to allow at most one bar in each column of the lattice region, then local algorithms are slow, but the DK algorithm is provably efficient (if the bars are long enough). However, we show that when we allow any number of bars per column, the DK algorithm also requires exponential time to reach equilibrium.
{"title":"Cluster Algorithms for Discrete Models of Colloids with Bars","authors":"S. Miracle, Dana Randall, A. Streib","doi":"10.1137/1.9781611973013.15","DOIUrl":"https://doi.org/10.1137/1.9781611973013.15","url":null,"abstract":"Colloids are mixtures of two different types of molecules. The model has a hard-core constraint forcing all the molecules to occupy non-overlapping positions, but there are no additional interactions between molecules; all non-overlapping arrangements are equally likely. It is believed that colloids undergo a phase transition whereby at low density the two types of molecules will be uniformly interspersed, while at high density large clusters will form and the two types of molecules will effectively separate. While local algorithms are not believed to work at or beyond the critical point, an algorithm due to Dress and Krauth [3] offers an alternative approach to sampling potentially beyond the critical point where clusters begin to form. We study the DK algorithm on a colloid model consisting of long bars and small diamonds on the periodic lattice Z2n. We show that if we restrict the model to allow at most one bar in each column of the lattice region, then local algorithms are slow, but the DK algorithm is provably efficient (if the bars are long enough). However, we show that when we allow any number of bars per column, the DK algorithm also requires exponential time to reach equilibrium.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122239904","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 : 2011-01-22DOI: 10.1137/1.9781611973013.3
O. Bodini, Danièle Gardy, Bernhard Gittenberger
We aim at the asymptotic enumeration of lambda-terms of a given size where the order of nesting of abstractions is bounded whereas the size is tending to infinity. This is done by means of a generating function approach and singularity analysis. The generating functions appear to be composed of nested square roots which exhibit unexpected phenomena. We derive the asymptotic number of such lambda-terms and it turns out that the order depends on the bound of the height. Furthermore, we present some observations when generating such lambda randomly and explain why powerful tools for random generation, such as Boltzmann samplers, face serious difficulties in generating lambda-terms.
{"title":"Lambda-terms of Bounded Unary Height","authors":"O. Bodini, Danièle Gardy, Bernhard Gittenberger","doi":"10.1137/1.9781611973013.3","DOIUrl":"https://doi.org/10.1137/1.9781611973013.3","url":null,"abstract":"We aim at the asymptotic enumeration of lambda-terms of a given size where the order of nesting of abstractions is bounded whereas the size is tending to infinity. This is done by means of a generating function approach and singularity analysis. The generating functions appear to be composed of nested square roots which exhibit unexpected phenomena. We derive the asymptotic number of such lambda-terms and it turns out that the order depends on the bound of the height. Furthermore, we present some observations when generating such lambda randomly and explain why powerful tools for random generation, such as Boltzmann samplers, face serious difficulties in generating lambda-terms.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116662074","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 : 2011-01-22DOI: 10.1137/1.9781611973013.13
Sanaa Kholfi, H. Mahmoud
We study a class of tenable irreducible nondegenerate zero-balanced Polya urn schemes. We give a full characterization of the class by sufficient and necessary conditions. Only forms with a certain cyclic structure in their replacement matrix are admissible. The scheme has a steady state into proportions governed by the principal left eigenvector of the average replacement matrix. We study the gradual change for any such urn containing n → ∞ balls from the initial condition to the steady state. We look at the status of the urn after jn draws. We identify three phases of jn: The growing sublinear, the linear, and the superlinear. In the growing sublinear phase the number of balls of different colors has an asymptotic joint multivariate normal distribution, with mean and covariance structure that are influenced by the initial conditions. In the linear phase a different multivariate normal distribution kicks in, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. These Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory.
{"title":"The Class of Tenable Zero-balanced Polya Urn Schemes: Characterization and Gaussian Phases","authors":"Sanaa Kholfi, H. Mahmoud","doi":"10.1137/1.9781611973013.13","DOIUrl":"https://doi.org/10.1137/1.9781611973013.13","url":null,"abstract":"We study a class of tenable irreducible nondegenerate zero-balanced Polya urn schemes. We give a full characterization of the class by sufficient and necessary conditions. Only forms with a certain cyclic structure in their replacement matrix are admissible. The scheme has a steady state into proportions governed by the principal left eigenvector of the average replacement matrix. We study the gradual change for any such urn containing n → ∞ balls from the initial condition to the steady state. We look at the status of the urn after jn draws. We identify three phases of jn: The growing sublinear, the linear, and the superlinear. In the growing sublinear phase the number of balls of different colors has an asymptotic joint multivariate normal distribution, with mean and covariance structure that are influenced by the initial conditions. In the linear phase a different multivariate normal distribution kicks in, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. These Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130954344","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 : 2011-01-22DOI: 10.1137/1.9781611973013.6
R. Sedgewick
Before the advent of computing, a standard science and mathematics curriculum emerged in secondary education, supported and expanded by first-year college courses, that serves as the technical basis for every student in science and engineering. For whatever reason, we have seen two unfortunate developments at the dawn of the new millenium. First, many computer science students have somehow been exempted from having to know basic precepts of science (instead, they learn about developing large programs and about theoretical issues). Second, many students in science and engineering have somehow been exempted from being exposed to basic precepts of computer science and algorithmics (instead, they learn a few specific programming tools). In both cases, students are being shortchanged.
{"title":"Algorithms for the Masses","authors":"R. Sedgewick","doi":"10.1137/1.9781611973013.6","DOIUrl":"https://doi.org/10.1137/1.9781611973013.6","url":null,"abstract":"Before the advent of computing, a standard science and mathematics curriculum emerged in secondary education, supported and expanded by first-year college courses, that serves as the technical basis for every student in science and engineering. For whatever reason, we have seen two unfortunate developments at the dawn of the new millenium. First, many computer science students have somehow been exempted from having to know basic precepts of science (instead, they learn about developing large programs and about theoretical issues). Second, many students in science and engineering have somehow been exempted from being exposed to basic precepts of computer science and algorithmics (instead, they learn a few specific programming tools). In both cases, students are being shortchanged.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124009660","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：