首页 > 最新文献

Workshop on Analytic Algorithmics and Combinatorics最新文献

英文 中文
On the Number of Hamilton Cycles in Bounded Degree Graphs 关于有界度图中哈密顿环的个数
Pub Date : 2008-01-19 DOI: 10.1137/1.9781611972986.8
Heidi Gebauer
The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs -- deriving thereby extremal bounds. We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276n), improving on Eppstein's previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein's in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle C and then proceed around C, succesively producing partial Hamilton cycles. Our approach can also be used to show that the number of Hamilton cycles of a 4-regular n-vertex graph is at most O(18n/5) ≤ O(1.783n), which improves a previous bound by Sharir and Welzl. This result is complemented by a lower bound of 48n/8 ≥ 1.622n. Then we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time √3n · poly(n) = O(1.733n), improving a previous result of Eppstein. This algorithm can be modified to compute the number of Hamilton cycles in the same time bound and to enumerate all Hamilton cycles in time (√3n +hc(G))·poly(n) with hc(G) denoting the number of Hamilton cycles of the given graph G. So our upper bound of O(1.783n) for the number of Hamilton cycles serves also as a time bound for enumeration. Using similar techniques as in the 3-regular case we establish upper bounds for the number of Hamilton cycles in 5-regular graphs and in graphs of average degree 3, 4, and 5.
本文的主要贡献是在有界度图中枚举Hamilton环的一种新方法——由此推导出极值界。我们描述了一种算法,该算法在O(1.276n)时间内枚举给定的3-正则n顶点图的所有Hamilton环,改进了Eppstein的前界。由此得到的3正则n顶点图中Hamilton环最大数目的新上界O(1.276n)接近已知的下界Ω(1.259n)。我们的方法与Eppstein的不同之处在于,他在每一步中都考虑一个新的图并对其进行修改,而我们(在一开始)固定一个汉密尔顿环C,然后围绕C继续,依次产生部分汉密尔顿环。我们的方法还可以用来证明一个4-正则n顶点图的Hamilton环的个数不超过O(18n/5)≤O(1.783n),这改进了Sharir和Welzl之前的一个界。这个结果被48n/8≥1.622n的下界所补充。然后,我们提出了一种算法,该算法在√3n·poly(n) = O(1.733n)的时间内找到给定4正则图的最小权汉密尔顿环,改进了先前的Eppstein结果。该算法可以修改为计算同一时间界内的Hamilton圈数,并在时间(√3n +hc(G))·poly(n)内枚举所有Hamilton圈,其中hc(G)表示给定图G的Hamilton圈数,因此我们给出的Hamilton圈数的上界O(1.783n)也可以作为枚举的时间界。使用与3正则情况类似的技术,我们建立了5正则图和平均次数为3、4和5的图中Hamilton环数的上界。
{"title":"On the Number of Hamilton Cycles in Bounded Degree Graphs","authors":"Heidi Gebauer","doi":"10.1137/1.9781611972986.8","DOIUrl":"https://doi.org/10.1137/1.9781611972986.8","url":null,"abstract":"The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs -- deriving thereby extremal bounds. \u0000 \u0000We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276n), improving on Eppstein's previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein's in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle C and then proceed around C, succesively producing partial Hamilton cycles. \u0000 \u0000Our approach can also be used to show that the number of Hamilton cycles of a 4-regular n-vertex graph is at most O(18n/5) ≤ O(1.783n), which improves a previous bound by Sharir and Welzl. This result is complemented by a lower bound of 48n/8 ≥ 1.622n. \u0000 \u0000Then we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time √3n · poly(n) = O(1.733n), improving a previous result of Eppstein. This algorithm can be modified to compute the number of Hamilton cycles in the same time bound and to enumerate all Hamilton cycles in time (√3n +hc(G))·poly(n) with hc(G) denoting the number of Hamilton cycles of the given graph G. So our upper bound of O(1.783n) for the number of Hamilton cycles serves also as a time bound for enumeration. \u0000 \u0000Using similar techniques as in the 3-regular case we establish upper bounds for the number of Hamilton cycles in 5-regular graphs and in graphs of average degree 3, 4, and 5.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126146191","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: 26
Generating Random Derangements 生成随机排列
Pub Date : 2008-01-19 DOI: 10.1137/1.9781611972986.7
C. Martínez, A. Panholzer, H. Prodinger
In this short note, we propose a simple and efficient algorithm to generaterandom derangements, that is, permutations without fixed points. We discuss the algorithm correctness and its performance and compare it to other alternatives. We find that the algorithm has expected linear complexity, works in-place with little additional auxiliary memory and qualitatively behaves like the well-known Fisher-Yates shuffle for random permutations or Sattolo's algorithm for random cyclic permutations.
在这篇短文中,我们提出了一个简单而有效的算法来生成随机排列,即没有不动点的排列。讨论了算法的正确性和性能,并与其他替代算法进行了比较。我们发现该算法具有预期的线性复杂性,在很少额外辅助内存的情况下工作,并且定性地表现为随机排列的著名的Fisher-Yates shuffle或随机循环排列的Sattolo算法。
{"title":"Generating Random Derangements","authors":"C. Martínez, A. Panholzer, H. Prodinger","doi":"10.1137/1.9781611972986.7","DOIUrl":"https://doi.org/10.1137/1.9781611972986.7","url":null,"abstract":"In this short note, we propose a simple and efficient algorithm to generaterandom derangements, that is, permutations without fixed points. We discuss the algorithm correctness and its performance and compare it to other alternatives. We find that the algorithm has expected linear complexity, works in-place with little additional auxiliary memory and qualitatively behaves like the well-known Fisher-Yates shuffle for random permutations or Sattolo's algorithm for random cyclic permutations.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127404371","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: 22
Augmented Graph Models for Small-World Analysis with Geographic Factors 具有地理因素的小世界分析增广图模型
Pub Date : 2008-01-19 DOI: 10.1137/1.9781611972986.5
V. Nguyen, C. Martel
Small-world properties, such as small-diameter and clustering, and the power-law property are widely recognized as common features of large-scale real-world networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting. These two are the distance-bias tendency (links tend to connect to closer nodes) and the property of bounded growth in localities. However, existing formal models for real-world complex networks usually don't fully consider these geographical factors. We describe a flexible approach using a standard augmented graph model (e.g. Watt and Strogatz's [33], and Kleinberg's [20] models) and present important initial results on a refined model where we focus on the small-diameter characteristic and the above two geographical factors. We start with a general model where an arbitrary initial node-weighted graph H is augmented with additional random links specified by a generic 'distribution rule' τ and the weights of nodes in H. We consider a refined setting where the initial graph H is associated with a growth-bounded metric, and τ has a distance-bias characteristic, specified by parameters as follows. The base graph H has neighborhood growth bounded from both below and above, specified by parameters β1, β2 > 0. (These parameters can be thought of as the dimension of the graph, e.g. β1 = 2 and β1 = 3 for a graph modeling a setting with nodes in both 2D and 3D settings.) That is 2β1 ≤ Nu(2r)/Nu(r) ≤ 2β2 where Nu(r) is the number of nodes v within metric distance r from u: d(u, v) ≤ r. When we add random links using distribution τ, this distribution is specified by parameter α > 0 such that the probability that a link from u goes to v ≠ u is ∝ 1/dα(u,v). We show which parameters produce a small-diameter graph and how the diameter changes depending on the relationship between the distance-bias parameter α and the two bounded growth parameters β1, β2 > 0. In particular, for most connected base graphs, the diameter of our augmented graph is logarithmic if α ≤ β1, and poly-log if β2 ≤ α 2β2. These results also suggest promising implications for applications in designing routing algorithms.
小世界性质,如小直径和聚类,以及幂律性质被广泛认为是大规模现实世界网络的共同特征。最近的研究还注意到两个重要的地理因素发挥了重要作用,特别是在与互联网相关的环境中。这两个是距离偏倚倾向(链接倾向于连接到更近的节点)和局部有界增长的性质。然而,现实世界复杂网络的现有正式模型通常没有充分考虑这些地理因素。我们使用标准增宽图模型(例如Watt和Strogatz的[33]和Kleinberg的[20]模型)描述了一种灵活的方法,并在一个细化模型上提出了重要的初步结果,其中我们专注于小直径特征和上述两个地理因素。我们从一个一般模型开始,其中任意初始节点加权图H被一个通用的“分布规则”τ和H中节点的权重指定的附加随机链接所增强。我们考虑了一个改进的设置,其中初始图H与一个增长有界度量相关联,并且τ具有距离偏差特征,由参数指定如下。基图H具有上下有界的邻域增长,由参数β1, β2 >指定。(这些参数可以被认为是图的维度,例如β1 = 2和β1 = 3对于一个在2D和3D设置中都有节点的图建模设置。)即2β1≤Nu(2r)/Nu(r)≤2β2,其中Nu(r)是距离u的度量距离r内的节点数v: d(u, v)≤r。当我们使用分布τ添加随机链接时,该分布由参数α > 0指定,使得从u到v的链接≠u的概率为∝1/dα(u,v)。我们展示了哪些参数产生小直径图,以及直径如何根据距离偏置参数α与两个有界生长参数β1, β2 > 0之间的关系而变化。特别地,对于大多数连通基图,当α≤β1时,增广图的直径是对数的,当β2≤α 2β2时,增广图的直径是多对数的。这些结果也为设计路由算法的应用提供了有希望的启示。
{"title":"Augmented Graph Models for Small-World Analysis with Geographic Factors","authors":"V. Nguyen, C. Martel","doi":"10.1137/1.9781611972986.5","DOIUrl":"https://doi.org/10.1137/1.9781611972986.5","url":null,"abstract":"Small-world properties, such as small-diameter and clustering, and the power-law property are widely recognized as common features of large-scale real-world networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting. These two are the distance-bias tendency (links tend to connect to closer nodes) and the property of bounded growth in localities. However, existing formal models for real-world complex networks usually don't fully consider these geographical factors. \u0000 \u0000We describe a flexible approach using a standard augmented graph model (e.g. Watt and Strogatz's [33], and Kleinberg's [20] models) and present important initial results on a refined model where we focus on the small-diameter characteristic and the above two geographical factors. We start with a general model where an arbitrary initial node-weighted graph H is augmented with additional random links specified by a generic 'distribution rule' τ and the weights of nodes in H. We consider a refined setting where the initial graph H is associated with a growth-bounded metric, and τ has a distance-bias characteristic, specified by parameters as follows. The base graph H has neighborhood growth bounded from both below and above, specified by parameters β1, β2 > 0. (These parameters can be thought of as the dimension of the graph, e.g. β1 = 2 and β1 = 3 for a graph modeling a setting with nodes in both 2D and 3D settings.) That is 2β1 ≤ Nu(2r)/Nu(r) ≤ 2β2 where Nu(r) is the number of nodes v within metric distance r from u: d(u, v) ≤ r. When we add random links using distribution τ, this distribution is specified by parameter α > 0 such that the probability that a link from u goes to v ≠ u is ∝ 1/dα(u,v). We show which parameters produce a small-diameter graph and how the diameter changes depending on the relationship between the distance-bias parameter α and the two bounded growth parameters β1, β2 > 0. In particular, for most connected base graphs, the diameter of our augmented graph is logarithmic if α ≤ β1, and poly-log if β2 ≤ α 2β2. These results also suggest promising implications for applications in designing routing algorithms.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133057068","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
Exact Analysis of the Recurrence Relations Generalized from the Tower of Hanoi 由河内塔推广的递推关系的精确分析
Pub Date : 2007-11-30 DOI: 10.1137/1.9781611972986.6
A. Matsuura
In this paper, we analyze the recurrence relations generalized from the Tower of Hanoi problem of the form T(n, α, β) = min1≤t≤n{α T(n − t, α, β)+β S(t, 3)}, where S(t, 3) = 2t − 1 is the optimal solution for the 3-peg Tower of Hanoi problem. It is shown that when α and β are natural numbers and α ≥ 2, the sequence of differences of T(n, α, β)'s, i.e., T(n, α, β) − T(n − 1, α, β), consists of numbers of the form β2iαj (i, j ≥ 0) lined in the increasing order.
本文分析了由河内塔问题推广而来的T(n, α, β) = min1≤T≤n{α T(n−T, α, β)+β S(T, 3)}的递推关系,其中S(T, 3) = 2t−1是河内塔问题的最优解。结果表明,当α和β为自然数且α≥2时,T(n, α, β)的差值序列即T(n, α, β)−T(n−1,α, β)由β2i - αj (i, j≥0)形式的数按递增顺序排列。
{"title":"Exact Analysis of the Recurrence Relations Generalized from the Tower of Hanoi","authors":"A. Matsuura","doi":"10.1137/1.9781611972986.6","DOIUrl":"https://doi.org/10.1137/1.9781611972986.6","url":null,"abstract":"In this paper, we analyze the recurrence relations generalized from the Tower of Hanoi problem of the form T(n, α, β) = min1≤t≤n{α T(n − t, α, β)+β S(t, 3)}, where S(t, 3) = 2t − 1 is the optimal solution for the 3-peg Tower of Hanoi problem. It is shown that when α and β are natural numbers and α ≥ 2, the sequence of differences of T(n, α, β)'s, i.e., T(n, α, β) − T(n − 1, α, β), consists of numbers of the form β2iαj (i, j ≥ 0) lined in the increasing order.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122177848","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: 7
Bloom Maps 布鲁姆地图
Pub Date : 2007-10-17 DOI: 10.1137/1.9781611972986.4
David Talbot, J. Talbot
We consider the problem of succinctly encoding a static map to support approximate queries. We derive upper and lower bounds on the space requirements in terms of the error rate and the entropy of the distribution of values over keys: our bounds differ by a small constant factor. For the upper bound we introduce a novel data structure, the Bloom map, generalising the Bloom filter to this problem. The lower bound follows from an information theoretic argument.
我们考虑简洁地编码静态映射以支持近似查询的问题。我们根据错误率和键上值分布的熵推导出空间需求的上界和下界:我们的上界有一个小的常数因子的差异。对于上界,我们引入了一种新的数据结构,Bloom映射,将Bloom滤波器推广到这个问题。下界是由一个信息论的论点推导出来的。
{"title":"Bloom Maps","authors":"David Talbot, J. Talbot","doi":"10.1137/1.9781611972986.4","DOIUrl":"https://doi.org/10.1137/1.9781611972986.4","url":null,"abstract":"We consider the problem of succinctly encoding a static map to support approximate queries. We derive upper and lower bounds on the space requirements in terms of the error rate and the entropy of the distribution of values over keys: our bounds differ by a small constant factor. For the upper bound we introduce a novel data structure, the Bloom map, generalising the Bloom filter to this problem. The lower bound follows from an information theoretic argument.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128893184","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: 9
Analysis of Insertion Costs in Priority Trees 优先级树的插入代价分析
Pub Date : 2007-01-06 DOI: 10.1137/1.9781611972979.2
Markus Kuba, A. Panholzer
Priority trees are a data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . ., n are equally likely to construct a priority tree of size n we give a detailed average-case analysis of insertion cost measures: we study the recursion depth and the number of key comparisons when inserting an element into a random size-n priority tree. For inserting a random element we obtain exact and asymptotic results for the expectation and the variance and can further show a central limit law of the parameters studied and for inserting an element with specified priority we can show exact and asymptotic results for the expectation of these quantities.
优先级树是用于优先级队列管理的数据结构。在数字1,…,n的所有排列都同样有可能构建大小为n的优先树的模型下,我们给出了插入成本度量的详细平均情况分析:我们研究了在随机大小为n的优先树中插入元素时的递归深度和键比较次数。对于插入一个随机元素,我们得到了期望和方差的精确和渐近的结果,并进一步证明了所研究参数的中心极限规律;对于插入一个指定优先级的元素,我们可以得到这些量的期望的精确和渐近的结果。
{"title":"Analysis of Insertion Costs in Priority Trees","authors":"Markus Kuba, A. Panholzer","doi":"10.1137/1.9781611972979.2","DOIUrl":"https://doi.org/10.1137/1.9781611972979.2","url":null,"abstract":"Priority trees are a data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . ., n are equally likely to construct a priority tree of size n we give a detailed average-case analysis of insertion cost measures: we study the recursion depth and the number of key comparisons when inserting an element into a random size-n priority tree. For inserting a random element we obtain exact and asymptotic results for the expectation and the variance and can further show a central limit law of the parameters studied and for inserting an element with specified priority we can show exact and asymptotic results for the expectation of these quantities.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"41 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131865507","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: 2
The Average Profile of Suffix Trees 后缀树的平均轮廓
Pub Date : 2007-01-06 DOI: 10.1137/1.9781611972979.3
Mark Daniel Ward
The internal profile of a tree structure denotes the number of internal nodes found at a specific level of the tree. Similarly, the external profile denotes the number of leaves on a level. The profile is of great interest because of its intimate connection to many other parameters of trees. For instance, the depth, fill-up level, height, path length, shortest path, and size of trees can each be interpreted in terms of the profile. The current study is motivated by the work of Park et al. [22], which was a comprehensive study of the profile of tries constructed from independent strings (also, each string generated by a memoryless source). In the present paper, however, we consider suffix trees, which are constructed from suffixes of a common string. The dependency between suffixes demands a careful, intricate treatment of overlaps in words. We precisely analyze the average internal and external profiles of suffix trees generated by a memoryless source. We utilize combinatorics on words (in particular, autocorrelation, i.e., the degree to which a word overlaps with itself) generating functions, singularity analysis, and the Mellin transform. We make comparisons of the average profile of suffix trees to the average profile of tries constructed from independent strings. We emphasize that our methods are extensible to higher moments. The present report describes the first moment of both the internal and external profiles of suffix trees.
树结构的内部轮廓表示在树的特定级别上发现的内部节点的数量。类似地,外部配置文件表示一个级别上的叶子数量。由于它与树木的许多其他参数密切相关,因此剖面非常有趣。例如,树的深度、填充水平、高度、路径长度、最短路径和大小都可以根据剖面进行解释。目前的研究受到Park等人[22]工作的启发,该工作是对由独立字符串(也由无记忆源生成的每个字符串)构建的尝试概况的全面研究。然而,在本文中,我们考虑后缀树,它是由一个公共字符串的后缀构成的。后缀之间的依赖关系要求对单词中的重叠部分进行仔细、复杂的处理。我们精确地分析了由无内存源生成的后缀树的平均内部和外部轮廓。我们在单词上使用组合学(特别是自相关,即单词与自身重叠的程度)生成函数,奇点分析和Mellin变换。我们将后缀树的平均轮廓与由独立字符串构造的尝试的平均轮廓进行了比较。我们强调我们的方法可以扩展到更高的矩。本报告描述了后缀树的内部和外部轮廓的第一个时刻。
{"title":"The Average Profile of Suffix Trees","authors":"Mark Daniel Ward","doi":"10.1137/1.9781611972979.3","DOIUrl":"https://doi.org/10.1137/1.9781611972979.3","url":null,"abstract":"The internal profile of a tree structure denotes the number of internal nodes found at a specific level of the tree. Similarly, the external profile denotes the number of leaves on a level. The profile is of great interest because of its intimate connection to many other parameters of trees. For instance, the depth, fill-up level, height, path length, shortest path, and size of trees can each be interpreted in terms of the profile. \u0000 \u0000The current study is motivated by the work of Park et al. [22], which was a comprehensive study of the profile of tries constructed from independent strings (also, each string generated by a memoryless source). In the present paper, however, we consider suffix trees, which are constructed from suffixes of a common string. The dependency between suffixes demands a careful, intricate treatment of overlaps in words. \u0000 \u0000We precisely analyze the average internal and external profiles of suffix trees generated by a memoryless source. We utilize combinatorics on words (in particular, autocorrelation, i.e., the degree to which a word overlaps with itself) generating functions, singularity analysis, and the Mellin transform. We make comparisons of the average profile of suffix trees to the average profile of tries constructed from independent strings. We emphasize that our methods are extensible to higher moments. The present report describes the first moment of both the internal and external profiles of suffix trees.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130817424","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
Fast Sorting and Pattern-avoiding Permutations 快速排序和避免模式排列
Pub Date : 2007-01-06 DOI: 10.1137/1.9781611972979.1
David Arthur
We say a permutation π "avoids" a pattern σ if no length |σ| subsequence of π is ordered in precisely the same way as σ. For example, π avoids (1, 2, 3) if it contains no increasing subsequence of length three. It was recently shown by Marcus and Tardos that the number of permutations of length n avoiding any fixed pattern is at most exponential in n. This suggests the possibility that if π is known a priori to avoid a fixed pattern, it may be possible to sort π in as little as linear time. Fully resolving this possibility seems very challenging, but in this paper, we demonstrate a large class of patterns σ for which σ-avoiding permutations can be sorted in O(n log log log n) time.
如果π的子序列与σ的顺序完全相同,那么我们说一个排列π“避免”了一个模式σ。例如,π避免(1,2,3),如果它不包含长度为3的递增子序列。Marcus和Tardos最近表明,长度为n的排列的数量最多是n的指数。这表明,如果π是先验的,可以避免固定的模式,那么在线性时间内就可以对π进行排序。完全解决这种可能性似乎非常具有挑战性,但在本文中,我们展示了一大类模式σ,其中σ-避免排列可以在O(n log log log n)时间内排序。
{"title":"Fast Sorting and Pattern-avoiding Permutations","authors":"David Arthur","doi":"10.1137/1.9781611972979.1","DOIUrl":"https://doi.org/10.1137/1.9781611972979.1","url":null,"abstract":"We say a permutation π \"avoids\" a pattern σ if no length |σ| subsequence of π is ordered in precisely the same way as σ. For example, π avoids (1, 2, 3) if it contains no increasing subsequence of length three. It was recently shown by Marcus and Tardos that the number of permutations of length n avoiding any fixed pattern is at most exponential in n. This suggests the possibility that if π is known a priori to avoid a fixed pattern, it may be possible to sort π in as little as linear time. Fully resolving this possibility seems very challenging, but in this paper, we demonstrate a large class of patterns σ for which σ-avoiding permutations can be sorted in O(n log log log n) time.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124503632","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: 9
Estimating the Number of Active Flows in a Data Stream over a Sliding Window 估算滑动窗口上数据流中活动流的数量
Pub Date : 2007-01-06 DOI: 10.1137/1.9781611972979.9
Éric Fusy, F. Giroire
A new algorithm is introduced to estimate the number of distinct flows (or connections) in a data stream. The algorithm maintains an accurate estimate of the number of distinct flows over a sliding window. It is simple to implement, parallelizes optimally, and has a very good trade-off between auxiliary memory and accuracy of the estimate: a relative accuracy of order 1/√m requires essentially a memory of order mln(n/m) words, where n is an upper bound on the number of flows to be seen over the sliding window. For instance, a memory of only 64kB is sufficient to maintain an estimate with accuracy of order 4 percents for a stream with several million flows. The algorithm has been validated both by simulations and experimentations on real traffic. It proves very efficient to monitor traffic and detect attacks.
引入了一种新的算法来估计数据流中不同流(或连接)的数量。该算法保持对滑动窗口上不同流的数量的准确估计。它很容易实现,最优地并行化,并且在辅助内存和估计精度之间有很好的权衡:1/√m阶的相对精度本质上需要mln(n/m)个单词的内存,其中n是在滑动窗口上看到的流数量的上限。例如,对于具有数百万流的流,仅64kB的内存就足以维持大约4%的估计精度。通过仿真和实际交通实验验证了该算法的有效性。它被证明是非常有效的监控流量和检测攻击。
{"title":"Estimating the Number of Active Flows in a Data Stream over a Sliding Window","authors":"Éric Fusy, F. Giroire","doi":"10.1137/1.9781611972979.9","DOIUrl":"https://doi.org/10.1137/1.9781611972979.9","url":null,"abstract":"A new algorithm is introduced to estimate the number of distinct flows (or connections) in a data stream. The algorithm maintains an accurate estimate of the number of distinct flows over a sliding window. It is simple to implement, parallelizes optimally, and has a very good trade-off between auxiliary memory and accuracy of the estimate: a relative accuracy of order 1/√m requires essentially a memory of order mln(n/m) words, where n is an upper bound on the number of flows to be seen over the sliding window. For instance, a memory of only 64kB is sufficient to maintain an estimate with accuracy of order 4 percents for a stream with several million flows. The algorithm has been validated both by simulations and experimentations on real traffic. It proves very efficient to monitor traffic and detect attacks.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125339633","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: 50
On the Average Cost of Insertions on Random Relaxed K-d Trees 随机松弛K-d树上插入的平均代价
Pub Date : 2007-01-06 DOI: 10.1137/1.9781611972979.4
Amalia Duch Brown, C. Martínez
In this work we refine the average case analysis of randomized insertions and deletions in random relaxed K-d trees, first given by Broutin et al. in [3]. The analysis is based in the analysis of the split and join algorithms, which recursively call one another and are the basis of the randomized update operations under consideration. For K = 2 the average cost of insertions and deletions is Θ(log n). For K > 2, this average cost is Θ(np(K)-1), for some p(K) > 1. This immediately follows from the analysis of the expected cost sn of splitting a tree of size n, which is the same as the expected cost mn of joining a pair of trees with total size n. These costs are, for K = 2, sn = mn = Θ(n) and, for K > 2, sn = mn = Ω(np(K)). In this abstract we find a closed form for the value of the exponent p(K), as well as the constant factor multiplying the main order term in sn.
在这项工作中,我们改进了随机松弛K-d树中随机插入和删除的平均情况分析,该分析首先由Broutin等人在[3]中给出。该分析基于对分割和连接算法的分析,这两种算法递归地相互调用,并且是所考虑的随机更新操作的基础。对于K = 2,插入和删除的平均代价是Θ(log n)。对于K > 2,这个平均代价是Θ(np(K)-1),对于某些p(K) > 1。这可以从分割大小为n的树的期望成本sn的分析中得出,这与连接总大小为n的树的期望成本mn相同。这些成本是,对于K = 2, sn = mn = Θ(n),对于K > 2, sn = mn = Ω(np(K))。在这个摘要中,我们找到了指数p(K)值的封闭形式,以及常数因子乘以sn中的主阶项。
{"title":"On the Average Cost of Insertions on Random Relaxed K-d Trees","authors":"Amalia Duch Brown, C. Martínez","doi":"10.1137/1.9781611972979.4","DOIUrl":"https://doi.org/10.1137/1.9781611972979.4","url":null,"abstract":"In this work we refine the average case analysis of randomized insertions and deletions in random relaxed K-d trees, first given by Broutin et al. in [3]. The analysis is based in the analysis of the split and join algorithms, which recursively call one another and are the basis of the randomized update operations under consideration. \u0000 \u0000For K = 2 the average cost of insertions and deletions is Θ(log n). For K > 2, this average cost is Θ(np(K)-1), for some p(K) > 1. This immediately follows from the analysis of the expected cost sn of splitting a tree of size n, which is the same as the expected cost mn of joining a pair of trees with total size n. These costs are, for K = 2, sn = mn = Θ(n) and, for K > 2, sn = mn = Ω(np(K)). In this abstract we find a closed form for the value of the exponent p(K), as well as the constant factor multiplying the main order term in sn.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121682160","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
期刊
Workshop on Analytic Algorithmics and Combinatorics
全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1