Random Struct. Algorithms最新文献

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On the distribution of rank of a random matrix over a finite field 有限域上随机矩阵的秩分布

Random Struct. Algorithms

Pub Date : 2000-10-10 DOI: 10.1002/1098-2418(200010/12)17:3/4%3C197::AID-RSA2%3E3.0.CO;2-K

C. Cooper

Let M = (mij) be a random n × n matrix over GF(t) in which each matrix entry mij is independently and identically distributed, with Pr(mij = 0) = 1 − p(n) and Pr(mij = r) = p(n)/(t − 1), r 6= 0. If we choose t ≥ 3, and condition on M having no zero rows or columns, then the probability that M is non-singular tends to ct ∼ ∏∞ j=1(1 − t−j) provided p ≥ (log n + d)/n, where d → −∞ slowly.

设M = (mij)是GF(t)上的一个随机n × n矩阵，其中每个矩阵项mij独立地同分布，Pr(mij = 0) = 1 - p(n)， Pr(mij = r) = p(n)/(t - 1)， r 6= 0。若取t≥3，且条件M不存在零行零列，则当p≥(log n + d)/n时，M非奇异的概率趋于ct ~∏∞j=1(1−t−j)，其中d缓慢→−∞。

引用次数: 89

On a random graph with immigrating vertices: Emergence of the giant component 在顶点迁移的随机图上:巨分量的出现

Random Struct. Algorithms

Pub Date : 2000-09-01 DOI: 10.1002/1098-2418(200009)17:2%3C79::AID-RSA1%3E3.0.CO;2-W

D. Aldous, B. Pittel

Author(s): Aldous, DJ; Pittel, B | Abstract: A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with O(n2/3) vertices is shown to be the same as in the Erdos-Renyi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t = π/2, rather than t = 1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum-of-squares and sum-of-cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky-type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812-854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley a Sons, Inc. Random Struct. Alg., 17, 79-102, 2000.

作者:Aldous, DJ;摘要研究了一个随机演化图，其顶点的迁移速率为n，每条可能边的出现速率为1/n。O(n2/3)个顶点的巨型构件出现的详细图与Erdos-Renyi图过程相同，开始时顶点数固定为n。一个主要的区别是现在的转变发生在t = π/2的时间，而不是t = 1。这个证明有三个要素。亚临界阶段最大组分的大小是通过与某一多类型分支过程的比较确定的。有了这个界限，通过鞅方法显示了分量大小的平方和和和的增长，与斯摩鲁乔斯基型方程的解密切相关。该近似允许我们将Aldous [brown短途，临界随机图和乘法聚聚，Ann Probab 25(1997)， 812-854]的结果应用于乘法聚聚中巨分量的出现，即非均匀随机图过程。©2000 John Wiley a Sons, Inc随机结构。Alg。科学通报，17,79-102,2000。

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引用次数: 42

Bounding the unsatisfiability threshold of random 3-SAT 随机3-SAT不满意阈值的边界

Random Struct. Algorithms

Pub Date : 2000-09-01 DOI: 10.1002/1098-2418(200009)17:2%3C103::AID-RSA2%3E3.0.CO;2-P

S. Janson, Y. Stamatiou, M. Vamvakari

The satisfiability threshold conjecture states that fur a randomly generated formula of m clauses of exactly k literals over n variables, the probability that it is satisfiable, as n tends to infinity, changes abruptly from I to 0, as the ratio I = m/n is

可满足阈值猜想表明，对于随机生成的m个分句公式，其中恰好有k个字面量/n个变量，当n趋于无穷时，该公式可满足的概率从I到0突然变化，因为比值I = m/n为

引用次数: 83

Permutations with roots 带根排列

Random Struct. Algorithms

Pub Date : 2000-09-01 DOI: 10.1002/1098-2418(200009)17:2%3C157::AID-RSA4%3E3.0.CO;2-2

M. Bóna, A. McLennan, D. White

We prove that the probability p2(n) that a random permutation of length n has a square root is monotonically nonincreasing in n. More generally, we prove that the probability pr(n) that a random permutation of length n has an rth root, r prime, is monotonically nonincreasing in n. We also show for all r≥2 that pr(n)0 as n∞. While doing this, we combinatorially prove that pr(n)=pr(n+1) for r prime and for all n not congruent to −1 mod r, and we construct several bijections for sets of permutations defined by modular class restrictions on the cycle lengths. We also include a simple probabilistic proof that, for r≥2, pr(n)0 as n∞. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 157–167, 2000

我们证明了长度为n的随机排列有平方根的概率p2(n)在n上是单调不增加的。更一般地，我们证明了长度为n的随机排列有第r根r '的概率pr(n)在n上是单调不增加的。我们还证明了对于所有r≥2,pr(n)0是n∞。在此过程中，我们组合证明了pr(n)=pr(n+1)对于r '和对于所有n不等于- 1 mod r，并且我们构造了几个由模类限制在循环长度上定义的置换集的双射。我们还包括一个简单的概率证明，当r≥2时，pr(n)0为n∞。©2000 John Wiley & Sons, Inc随机结构。Alg。中文信息学报，17 (7):157-167,2000

引用次数: 15

Asymptotics of the list-chromatic index for multigraphs 多重图的表色指数的渐近性

Random Struct. Algorithms

Pub Date : 2000-09-01 DOI: 10.1002/1098-2418(200009)17:2%3C117::AID-RSA3%3E3.0.CO;2-9

J. Kahn

The list-chromatic index, χl′(G) of a multigraph G is the least t such that if S(A) is a set of size t for each A∈E≔E(G), then there exists a proper coloring σ of G with σ(A)∈S(A) for each A∈E. The list-chromatic index is bounded below by the ordinary chromatic index, χ′(G), which in turn is at least the fractional chromatic index, χ′*(G). In previous work we showed that the chromatic and fractional chromatic indices are asymptotically the same; here we extend this to the list-chromatic index: χl′(G)∼χ′*(G) as χl′(G)∞. The proof uses sampling from “hard-core” distributions on the set of matchings of a multigraph to go from fractional to list colorings. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 117–156, 2000

多重图G的表色指标χ 1′(G)最小t，使得对于每个a∈E，如果S(a)是一个大小为t的集合，则对于每个a∈E，对于σ(a)∈S(a)， G存在一个适当的着色σ。表色指数以普通色指数χ ' (G)为界，而普通色指数又至少是分数色指数χ ' *(G)。在以前的工作中，我们证明了色指标和分数色指标是渐近相同的;这里我们将其扩展到表色指数:χ 1 ' (G) ~ χ ' *(G) = χ 1 ' (G)∞。证明使用从多图的匹配集上的“硬核”分布中抽样，从分数到列表着色。©2000 John Wiley & Sons, Inc随机结构。Alg。中文信息学报，17 (7):117-156,2000

引用次数: 51

Construction of expanders and superconcentrators using Kolmogorov complexity 利用柯尔莫哥洛夫复杂度构造膨胀机和超浓缩机

Random Struct. Algorithms

Pub Date : 2000-08-01 DOI: 10.1002/1098-2418(200008)17:1%3C64::AID-RSA5%3E3.0.CO;2-3

U. Schöning

We show the existence of various versions of expander graphs using Kolmogorov complexity. This method seems superior to the usual probabilistic construction. It turns out that the best known bounds on the size of expanders and superconcentrators can be attained based on this method. In the case of (acyclic) superconcentrators we attain a density of about 34 edges/vertices. Furthermore, related graph properties are reviewed, like magnification, edge-magnification, and isolation, and we develop bounds based on the Kolmogorov approach. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 64–77, 2000

我们用Kolmogorov复杂度证明了各种版本的展开图的存在性。这种方法似乎优于通常的概率构造。结果表明，用这种方法可以得到膨胀剂和超浓缩剂的最佳尺寸界限。在(无环)超聚光器的情况下，我们获得了大约34个边/顶点的密度。此外，回顾了相关的图属性，如放大，边缘放大和隔离，并基于Kolmogorov方法开发了界。©2000 John Wiley & Sons, Inc随机结构。Alg。中文信息学报，17:64-77,2000

引用次数: 6

Complete minors in pseudorandom graphs 伪随机图中的完全小调

Random Struct. Algorithms

Pub Date : 2000-08-01 DOI: 10.1002/1098-2418(200008)17:1%3C26::AID-RSA3%3E3.0.CO;2-7

A. Thomason

引用次数: 8

New bounds on nearly perfect matchings in hypergraphs: Higher codegrees do help 超图中近乎完美匹配的新边界:更高的余度确实有帮助

Random Struct. Algorithms

Pub Date : 2000-08-01 DOI: 10.1002/1098-2418(200008)17:1%3C29::AID-RSA4%3E3.0.CO;2-W

V. Vu

Let H be a (k+1)-uniform, D-regular hypergraph on n vertices and let (H) be the minimum number of vertices left uncovered by a matching in H. Cj(H), the j-codegree of H, is the maximum number of edges sharing a set of j vertices in common. We prove a general upper bound on (H), based on the codegree sequence C2 (H), C3 (H),…. Our bound improves and generalizes many results on the topic, including those of Grable, Alon, Kim, and Spencer, and Kostochka and Rodl. It also leads to a substantial improvement in several applications. The key ingredient of the proof is the so-called polynomial technique, which is a new and useful tool to prove concentration results for functions with large Lipschitz coefficient. This technique is of independent interest. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 29–63, 2000

设H为n个顶点上的(k+1)一致的d正则超图，设(H)为H中匹配剩余未覆盖的最小顶点数。Cj(H)， H的j余度，是一组共有j个顶点的边的最大数目。基于余度序列C2 (H)， C3 (H)，....证明了(H)上的一般上界我们的界改进和推广了许多关于这个主题的结果，包括Grable, Alon, Kim, and Spencer, Kostochka and Rodl的结果。它还在几个应用程序中带来了实质性的改进。证明的关键是所谓的多项式技术，它是证明具有大利普希茨系数的函数的集中结果的一种新的有用的工具。这项技术具有独立的意义。©2000 John Wiley & Sons, Inc随机结构。Alg。， 17: 29-63, 2000

引用次数: 40

Sharp thresholds for certain Ramsey properties of random graphs 随机图的某些拉姆齐性质的尖锐阈值

Random Struct. Algorithms

Pub Date : 2000-08-01 DOI: 10.1002/1098-2418(200008)17:1%3C1::AID-RSA1%3E3.0.CO;2-4

E. Friedgut, M. Krivelevich

引用次数: 36

Degrees and choice numbers 度和选择数

Random Struct. Algorithms

Pub Date : 2000-07-01 DOI: 10.1002/1098-2418(200007)16:4%3C364::AID-RSA5%3E3.0.CO;2-0

N. Alon

The choice number ch(G) of a graph G = (V,E) is the minimum number k such that for every assignment of a list S(v) of at least k colors to each vertex v ∈ V , there is a proper vertex coloring of G assigning to each vertex v a color from its list S(v). We prove that if the minimum degree of G is d, then its choice number is at least ( 1 2 − o(1)) log2 d, where the o(1)-term tends to zero as d tends to infinity. This is tight up to a constant factor of 2 + o(1), improves an estimate established in [1], and settles a problem raised in [2].

图G = (V,E)的选择数ch(G)是k的最小值，使得对于每一个至少有k种颜色的列表S(V)分配给每个顶点V∈V, G存在一个适当的顶点着色，从它的列表S(V)中分配给每个顶点V一种颜色。证明了如果G的最小度为d，则其选择数至少为(1 2−o(1)) log2d，其中当d趋于无穷时，o(1)项趋于零。这接近于2 + 0(1)的常数因子，改进了[1]中建立的估计，并解决了[2]中提出的问题。

引用次数: 102

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