Abstract: We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman and colleagues (Random Struct Algor 22 (2003), 33-42; Random Struct Algor 24 (2004), 105-117). Here we obtain a sharp threshold for the appearance of a fixed subgraph and for certain Ramsey properties. We also consider a related model of random k-SAT formulas, where an instance is obtained by adding random k-clauses to a fixed formula with a given number of clauses, and derive tight bounds for the non-satisfiability of the thus-obtained random formula. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: We analyze Markov chains for generating a random k-coloring of a random graph Gn,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k-coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: We present a (non-standard) probabilistic analysis of dynamic data structures whose sizes are considered dynamic random walks. The basic operations (insertion, deletion, positive and negative queries, batched insertion, lazy deletion, etc.) are time-dependent random variables. This model is a (small) step toward the analysis of these structures when the distribution of the set of histories is not uniform. As an illustration, we focus on list structures (linear lists, priority queues, and dictionaries) but the technique is applicable as well to more advanced data structures. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. L ...

Abstract: In this article we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on the recent results of Nagle, Schacht, and the authors, we give here solutions to these problems.In particular, we prove the following: Let F be a k-uniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(nt) copies of F. Then one can delete o(nk) edges of H to make it F-free.Similar results were recently obtained by W. T. Gowers. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Probabilistic cellular automata form a very large and general class of stochastic processes. These automata exhibit a wide range of complex behavior and are of interest in a number of fields of study, including mathematical physics, percolation theory, computer science, and neurobiology. Very little has been proved about these models, even in simple cases, so it is common to compare the models to mean field models. It is normally assumed that mean field models are essentially trivial. However, we show here that even the mean field models can exhibit surprising behavior. We prove some rigorous results on mean field models, including the existence of a surrogate for the “energy” in certain non-reversible models. We also briefly discuss some differences that occur between the mean field and lattice models. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: On input a random 3-CNF formula of clauses-to-variables ratio r3 applies repeatedly the following simple heuristic: Set to True a literal that appears in the maximum number of clauses, irrespective of their size and the number of occurrences of the negation of the literal (ties are broken randomly; 1-clauses when they appear get priority) We prove that for r3 < 342 this heuristic succeeds with probability asymptotically bounded away from zero Previously, heuristics of increasing sophistication were shown to succeed for r3 < 326 We improve up to r3 < 352 by further exploiting the degree of the negation of the evaluated to True literal © 2005 Wiley Periodicals, Inc Random Struct Alg, 2006

Abstract: Let Gn = lGmin(n, M)rM≥0 denote a min-degree random multigraph process in which Gmin(n, M + 1) is obtained from Gmin(n, M) by connecting a randomly chosen vertex of a minimum degree with another vertex of the multigraph. We study the probability that the random multigraph Gmin(n, M) is connected. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: We consider the topological characteristics of orientable surfaces generated by randomly gluing n triangles together. Our results are most conveniently expressed in terms of a parameter h = n / 2 + χ, where χ is the Euler characteristic of the surface. Simulations and results for similar models suggest that Ex [h] = log(3n) + γ + o(1) and Var [h] = log(3n) + γ - π2 / 6 + o(1). We prove that Ex [h] = log n + O(1) and Var [h] = O(log n). We also derive results concerning a number of other topological invariants and combinatorial characteristics of these random surfaces. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC0, that is, a pseudorandom generator that maps n-bit strings to m-bit strings such that every bit of the output depends on a constant number k of bits of the seed.They show that for k = 3, if m ≥ 4n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias.They leave the question open for k ≥ 4. In fact, they ask whether every NC0 generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC0 generator can sample an e-biased space with negligible e?We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2-Ω(n/c4). For large values of k, we construct generators that map n bits to $n^{\Omega(\sqrt{k})}$ bits such that every XOR of outputs has bias $2^{-{n^{{1 \over 2\sqrt k}}}}$.We also present a polynomial-time distinguisher for k = 4,m ≥ 24n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2kn⌈k/2⌉).Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2-Ω(n) and which maps n bits to Ω(n2) bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: The minimum bisection problem is to partition the vertices of a graph into two classes of equal size so as to minimize the number of crossing edges. Computing a minimum bisection is NP-hard in the worst case. In this paper we study a spectral heuristic for bisecting random graphs Gn(p,p′) with a planted bisection obtained as follows: partition n vertices into two classes of equal size randomly, and then insert edges inside the two classes with probability p′ and edges crossing the partition with probability p independently. If $n(p'-p)\geq c_0\sqrt{np'\ln(np')}$, where c0 is a suitable constant, then with probability 1 - o(1) the heuristic finds a minimum bisection of Gn(p,p′) along with a certificate of optimality. Furthermore, we show that the structure of the set of all minimum bisections of Gn(p,p′) undergoes a phase transition as $n(p'-p)=\Theta(\sqrt{np'\ln n})$. The spectral heuristic solves instances in the subcritical, the critical, and the supercritical phases of the phase transition optimally with probability 1 - o(1). These results extend previous work of Boppana [Proc. 28th FOCS (1987) 280–285]. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006An extended abstract version of this paper appeared in Proc. 16th SODA (2005), pp. 850–859.

Abstract: We study three families of labeled plane trees. In all these trees, the root is labeled 0 and the labels of two adjacent nodes differ by 0,1, or -1.One part of the paper is devoted to enumerative results. For each family, and for all j ∈ N, we obtain closed form expressions for the following three generating functions: the generating function of trees having no label larger than j; the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled j; and finally the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled at least, j. Strangely enough, all these series turn out to be algebraic, but we have no combinatorial intuition for this algebraicity.The other part of the paper is devoted to deriving limit laws from these enumerative results. In each of our families of trees, we endow the trees of size n with the uniform distribution and study the following random variables: Mn, the largest label occurring in a (random) tree; Xn(j), the number of nodes labeled j; and X +n (j), the number of nodes labeled j or more. We obtain limit laws for scaled versions of these random variables.Finally, we translate the above limit results into statements dealing with the integrated superBrownian excursion. In particular, we describe the law of the supremum of its support (thus recovering some earlier results obtained by Delmas) and the law of its distribution function at a given point. We also conjecture the law of its density (at a given point). © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47–56] that if the edge costs of the complete graph Kn are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to $\zeta(3)=\sum_{i=1}^{\infty}i^{-3}$. Here we consider the following stochastic two-stage version of this optimization problem. There are two sets of edge costs cM: E → R and cT: E → R, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs cM(e) and cT(e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price cM(e), or to wait until its Tuesday price cT(e) appears. The set of edges XM bought on Monday is then completed by the set of edges XT bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + o(1). We show that, in the case of two-stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant e > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ(3) - 1/2 + o(1). The threshold heuristic is shown to be sub-optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning out-arborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 - 1/e + o(1). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: For 0 0 let Gq(n,p) denote the random graph with vertex set [n]=l1,…,nr such that, for each graph G on [n] with e(G) edges and c(G) components, the probability that Gq(n,p)=G is proportional to $p^{e(G)}(1-p)^{({n \atop 2})-e(G)}q^{c(G)}$. The first systematic study of Gq(n,p) was undertaken by [Bollobas, Grimmett, and Janson (Probab Theory Relat Fields 104 (1996), 283–317)], who analyzed the phase transition phenomenon corresponding to the emergence of the giant component. In this paper we describe the structure of Gq(n,p) very close the critical threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Let ∑ = (V,E) be a finite, d-regular bipartite graph. For any λ > 0 let πλ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ|I| (πλ is the hard-core measure with activity λ on ∑). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is πλ. We show that when λ is large enough (as a function of d and the expansion of subsets of single-parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d-dimensional hypercube l0,1rd we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: We investigate the degree sequences of scale-free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non-rigorous arguments of Dorogovtsev, Mendes, and Samukhin (Phys Rev Lett 85 (2000), 4633). We also consider a generalization of the model with more randomization, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localized eigenfunctions of the adjacency matrix can be found. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a, and b be vertices. An earlier paper (J. Van den Berg and J. Kahn, (Ann Probab 29 (2001), 123–126) proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events ”there is an open path from s to a” and “there is an open path from s to b” are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-) associated and that it is conditionally negatively correlated with the open cluster of t.We also present analogues of some of our results for (a) random-cluster measures and (b) directed percolation and contact processes and observe that the latter lead to improvements of some of the results in a paper of Belitsky et al. (Stoch Proc Appl 67 (1997), 213–225). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Let c be a constant and (e1,f1),(e2,f2),…,(ecn,fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2, then whp every graph which contains at least one edge from each ordered pair (ei,fi) has a component of size Ω(n), and, if c < c2, then whp there is a graph containing at least one edge from each pair that has no component with more than O(n1-e vertices, where e is a constant that depends on c2 - c. The constant c2 is roughly 0.97677. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: We see that the entropy method yields strong concentration results for general self-bounding functions of independent random variables. These give an improvement of a concentration result of Talagrand much used in discrete mathematics. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Using the cavity equations of Mzard, Parisi, and Zecchina [Science 297 (2002), 812]; Mzard and Zecchina, [Phys Rev E 66 (2002), 056126] we derive the various threshold values for the number of clau...

Abstract: We consider the MAX k-CUT problem on random graphs Gn,p. First, we bound the probable weight of a MAX k-CUT using probabilistic counting arguments and by analyzing a simple greedy heuristic. Then, we give an algorithm that approximates MAX k-CUT in expected polynomial time, with approximation ratio 1 + O((np)-1/2). Our main technical tool is a new bound on the probable value of Frieze and Jerrum's semidefinite programming (SDP)-relaxation of MAX k-CUT on random graphs. To obtain this bound, we show that the value of the SDP is tightly concentrated. As a further application of our bound on the probable value of the SDP, we obtain an algorithm for approximating the chromatic number of Gn,p, 1/n ≤ p ≤ 0.99, within a factor of O((np)1/2) in polynomial expected time, thereby answering a question of Krivelevich and Vu. We give similar algorithms for random regular graphs. The techniques for studying the SDP apply to a variety of SDP relaxations of further NP-hard problems on random structures and may therefore be of independent interest. For instance, to bound the SDP we estimate the eigenvalues of random graphs with given degree sequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006An extended abstract version of this paper appeared in Proc. ICALP 2003, Springer LNCS 2719, pp. 200–211.

Abstract: Limit theorems (including a Berry-Esseen bound) are derived for the number of comparisons taken by the Boyer-Moore algorithm for finding the occurrences of a given pattern in a random text. Previously, only special variants of this algorithm have been analyzed. We also propose a means of computing the limiting constants for the mean and the variance. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006.

Abstract: We show that the operations of permuting columns and rows separately and independently mix a square matrix in constant time. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: Let n be a positive integer and λ > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set Vn, in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let $\lambda = c \sqrt {\ln n/n}$ and let X be the number of isolated points in G(λ, n). We prove that almost always X ∼ n1-c2 when 0 ≤ c 2.26164 …, the diameter of G(λ, n) is bounded by (4 + o(1))/λ; and we modify this construction to yield a function c(δ) > 0 such that the diameter is at most 2(1 + δ + o(1))/λ when c > c(δ). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: This paper proposesapolynomialtime perfect (exact)sampling algorithm for2 xn contingencytables.Our algorithm isa LasVegas type randomized algorithm and the expected running time is boundedby 0(n3InN) where n isthenumber of columnsandN is the total sum of whole entries ina table.The algorithm is based on monotone coupling from thepast(monotone CFTP) algorithm and new Markov chain for sampling two-rowed contingency tables uniformly. We employed thepathcoupling methodandshowed the mixing rate of our chain. Our result indicatesthatuniform generation of two-rowed contingency tables is easier than the corresponding counting problem,since the counting problem is known to be #P-complete.

Abstract: Abstract We present a novel analysis of a random sampling approach for four clustering problems in metric spaces: k‐median , k‐means , min‐sum k‐clustering , and balanced k‐median . For all these problems, we consider the following simple sampling scheme: select a small sample set of input points uniformly at random and then run some approximation algorithm on this sample set to compute an approximation of the best possible clustering of this set. Our main technical contribution is a significantly strengthened analysis of the approximation guarantee by this scheme for the clustering problems. The main motivation behind our analyses was to design sublinear‐time algorithms for clustering problems. Our second contribution is the development of new approximation algorithms for the aforementioned clustering problems. Using our random sampling approach, we obtain for these problems the first time approximation algorithms that have running time independent of the input size, and depending on k and the diameter of the metric space only. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Abstract: The k–server problem is one of the most important and well-studied problems in the area of on–line computation. Its importance stems from the fact that it models many practical problems like multi-level memory paging encountered in operating systems, weighted caching used in the management of web caches, head motion planning of multi-headed disks, and robot motion planning. In this paper, we investigate its randomized version for which Θ(log k)–competitiveness is conjectured and yet hardly any