Sparse pseudorandom distributions

The existence of sparse pseudorandom distributions is proved. These are probability distributions concentrated in a very small set of strings, yet it is infeasible for any polynomial-time algorithm to distinguish between truly random coins and coins selected according to these distributions. It is shown that such distributions can be generated by (nonpolynomial) probabilistic algorithms, while probabilistic polynomial-time algorithms cannot even approximate all the pseudorandom distributions. Moreover, we show the existence of evasive pseudorandom distributions which are not only sparse, but also have the property that no polynomial-time algorithm may find an element in their support, except for a negligible probability. All these results are proved independently of any intractability assumption.     

An approximate analysis of a cyclic server queue with limited service and reservations

In this paper, we examine a queueing problem motivated by the pipeline polling protocol in satellite communications. The model is an extension of the cyclic queueing system withM-limited service. In this service mechanism, each queue, after receiving service on cyclej, makes a reservation for its service requirement in cyclej + 1. The main contribution to queueing theory is that we propose an approximation for the queue length and sojourn-time distributions for this discipline. Most approximate studies on cyclic queues, which have been considered before, examine the means only. Our method is an iterative one, which we prove to be convergent by using stochastic dominance arguments. We examine the performance of our algorithm by comparing it to simulations and show that the results are very good.     

On solving sparse band systems with three algorithms of the ABS family

In this paper, we examine three algorithms in the ABS family and consider their storage requirements on sparse band systems. It is shown that, when using the implicit Cholesky algorithm on a band matrix with band width 2q+1, onlyq additional vectors are required. Indeed, for any matrix with upper band widthq, onlyq additional vectors are needed. More generally, ifa _kj 0,j>k, then thejth row ofH _i is effectively nonzero ifj>i>k. The arithmetic operations involved in solving a band matrix by this method are dominated by (1/2)n ² q. Special results are obtained forq-band tridiagonal matrices and cyclic band matrices.The implicit Cholesky algorithm may require pivoting if the matrixA does not possess positive-definite principal minors, so two further algorithms were considered that do not require this property. When using the implicit QR algorithm, a matrix with band widthq needs at most 2q additional vectors. Similar results forq-band tridiagonal matrices and cyclic band matrices are obtained.For the symmetric Huang algorithm, a matrix with band widthq requiresq–1 additional vectors. The storage required forq-band tridiagonal matrices and cyclic band matrices are again analyzed.This work was undertaken during the visit of Dr. J. Abaffy to Hatfield Polytechnic, sponsored by SERC Grant No. GR/E-07760.     

Randomized Algorithms over Finite Fields for the Exact Parity Base Problem

We present three randomized pseudo-polynomial algorithms for the problem of finding a base of specified value in a weighted represented matroid subject to parity conditions. These algorithms, the first two being an improved version of those presented by P. M. Camerini et al. (1992, J. Algorithms13, 258–273) use fast arithmetic working over a finite field chosen at random among a set of appropriate fields. We show that the choice of a best algorithm among those presented depends on a conjecture related to the best value of the so-called Linnik constant concerning the distribution of prime numbers in arithmetic progressions. This conjecture, which we call the C-conjecture, is a strengthened version of a conjecture formulated in 1934 by S. Chowla. If the C-conjecture is true, the choice of a best algorithm is simple, since the last algorithm exhibits the best performance, either when the performance is measured in arithmetic operations, or when it is measured in bit operations and mild assumptions hold. If the C-conjecture is false we are still able to identify a best algorithm, but in this case the choice is between the first two algorithms and depends on the asymptotic growth of m with respect to those of U and n, where 2n, 2m, U are the rank, the number of elements, and the maximum weight assigned to the elements of the matroid, respectively.     

The erlangization method for Markovian fluid flows

For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.     

Approximation Algorithms for Precedence-Constrained Scheduling Problems on Parallel Machines that Run at Different Speeds

We present new approximation algorithms for the problem of scheduling precedence-constrained jobs on parallel machines that are uniformly related. That is, there arenjobs andmmachines; each jobjrequiresp_junits of processing, and is to be processed on one machine without interruption; if it is assigned to machinei, which runs at a given speeds_i, it takesp_j/s_itime units. There also is a partial order on the jobs, wherej kimplies that jobkmay not start processing until jobjhas been completed. We consider two objective functions:C_max = max_j C_j, whereC_jdenotes the completion time of jobj, and ∑_jw_jC_j, wherew_jis a weight that is given for each jobj. For the first objective, the best previously known result is an -approximation algorithm, which was shown by Jaffe more than 15 years ago. We give anO(log m)-approximation algorithm. We also show how to extend this result to obtain anO(log m)-approximation algorithm for the second objective, albeit with a somewhat larger constant. These results also extend to settings in which each jobjhas a release dater_jbefore which the job may not begin processing. In addition, we obtain stronger performance guarantees if there are a limited number of distinct speeds. Our results are based on a new linear programming-based technique for estimating the speed at which each job should be run, and a variant of the list scheduling algorithm of Graham that can exploit this additional information.     

Distributions of Random Partitions and Their Applications

Assume that a random sample of size m is selected from a population containing a countable number of classes (subpopulations) of elements (individuals). A partition of the set of sample elements into (unordered) subsets, with each subset containing the elements that belong to same class, induces a random partition of the sample size m, with part sizes {Z ₁,Z ₂,...,Z _N } being positive integer-valued random variables. Alternatively, if N _j is the number of different classes that are represented in the sample by j elements, for j=1,2,...,m, then (N ₁,N ₂,...,N _m ) represents the same random partition. The joint and the marginal distributions of (N ₁,N ₂,...,N _m ), as well as the distribution of are of particular interest in statistical inference. From the inference point of view, it is desirable that all the information about the population is contained in (N ₁,N ₂,...,N _m ). This requires that no physical, genetical or other kind of significance is attached to the actual labels of the population classes. In the present paper, combinatorial, probabilistic and compound sampling models are reviewed. Also, sampling models with population classes of random weights (proportions), and in particular the Ewens and Pitman sampling models, on which many publications are devoted, are extensively presented.      

The size of the giant high‐order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any and .     

Walks on the slit plane

 In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line . We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable is algebraic of degree 8. Moreover, for any point (i, j), the length for walks of this type ending at (i, j) is also algebraic, of degree 2 or 4, and involves the famous Catalan numbers. Our method is based on the solution of a functional equation, established via a simple combinatorial argument. It actually works for more general models, in which walks take their steps in a finite subset of ℤ² satisfying two simple conditions. The corresponding are always algebraic. In the second part of the paper, we derive from our enumerative results a number of probabilistic corollaries. For instance, we can compute exactly the probability that an ordinary random walk starting from (i, j) hits for the first time the half-line at position (k, 0), for any triple (i, j, k). This generalizes a question raised by R. Kenyon, which was the starting point of this paper. Taking uniformly at random all n-step walks on the slit plane, we also compute the probability that they visit a given point (k, 0), and the average number of visits to this point. In other words, we quantify the transience of the walks. Finally, we derive an explicit limit law for the coordinates of their endpoint. Received: 22 December 2001 / Revised version: 19 February 2002 / Published online: 30 September 2002 Both authors were partially supported by the INRIA, via the cooperative research action Alcophys. Mathematics Subject Classification (2000): O5A15-60C05     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

New steps in the amazing world of sequences and schedules

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timep _i,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan. We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules. Furthermore, we investigate the problemO∥C _max on an operation set with spanning tree structure. Supported by Deutsche Forschungsgemeinschaft, Project ScheMA     

Efficient generation of random nonsingular matrices

We present an efficient algorithm for generating an n × n nonsingular matrix uniformly over a finite field. This algorithm is useful for several cryptographic and checking applications. Over GF[2] our algorithm runs in expected time M(n) + O(n²), where M(n) is the time needed to multiply two n × n matrices, and the expected number of random bits it uses is n² + 3. (Over other finite fields we use n² + O(1) random field elements on average.) This is more efficient than the standard method for solving this problem, both in terms of expected running time and the expected number of random bits used. The standard method is to generate random n × n matrices until we produce one with nonzero determinant. In contrast, our technique directly produces a random matrix guaranteed to have nonzero determinant. We also introduce efficient algorithms for related problems such as uniformly generating singular matrices or matrices with fixed determinant. © 1993 John Wiley & Sons, Inc.     

The Number of Unlabeled Orders on Fourteen Elements

Lacking an explicit formula for the numbers T ₀(n) of all order relations (equivalently: T ₀ topologies) on n elements, those numbers have been explored only up to n=13 (unlabeled posets) and n=15 (labeled posets), respectively.In a new approach, we used an orderly algorithm to (i) generate each unlabeled poset on up to 14 elements and (ii) collect enough information about the posets on 13 elements to be able to compute the number of labeled posets on 16 elements by means of a formula by Erné. Unlike other methods, our algorithm avoids isomorphism tests and can therefore be parallelized quite easily. The underlying principle of successively adding new elements to small objects is applicable to lattices and other kinds of order structures, too.     

The largest eigenvalue of small rank perturbations of Hermitian random matrices

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M _ij ,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix and its rank. This rank is also allowed to increase with N in some restricted way. An erratum to this article is available at .     

Disjoint skolem sequences and related disjoint structures

A Skolem sequence of order n is a sequence S = (s₁, s₂…, s²ⁿ) of 2n integers satisfying the following conditions: (1) for every k ∈ {1, 2,… n} there exist exactly two elements s_i,S_j such that S_i = S_j = k; (2) If s_i = s_j = k,i < j then j ? i = k. In this article we show the existence of disjoint Skolem, disjoint hooked Skolem, and disjoint near-Skolem sequences. Then we apply these concepts to the existence problems of disjoint cyclic Steiner and Mendelsohn triple systems and the existence of disjoint 1-covering designs. © 1993 John Wiley & Sons, Inc.     

Likelihood ratio tests for symmetry against one-sided alternatives

A random variableX is said to have a symmetric distribution (about 0) if and only ifX and –X are, identically distributed. By considering various types of partial orderings between the distributions ofX and –X, one obtains various notions of skewness or one-sided bias. In this paper we study likelihood ratio tests for testing the symmetry of a discrete distribution about zero against the alternatives, (i)X is stochastically greater than –X; and (ii) pr(X=j)pr(X=–j) for allj>0. In the process, we obtain maximum likelihood estimators of the distribution function under the above alternatives. The asymptotic null distributions of the test statistics have been obtained and are of the chi-bar square type. A simulation study was performed to compare the powers of these tests with other tests.     

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require n^δ (δ>0) colors even for bounded chromatic (k-colorable for fixed k) n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19 , 204–234 (1995)] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work as long as p≥n^−α(k)+ϵ where α(k)=(2k)/((k−1)(k+2)) and ϵ is a positive constant. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 125–158 (1998)     

A Survey of the Coupon Collector’s Problem with Random Sample Sizes

This paper surveys the coupon collector’s waiting time problem with random sample sizes and equally likely balls. Consider an urn containing m red balls. For each draw, a random number of balls are removed from the urn. The group of removed balls is painted white and returned to the urn. Several approaches to addressing this problem are discussed, including a Markov chain approach to compute the distribution and expected value of the number of draws required for the urn to contain j white balls given that it currently contains i white balls. As a special case, E[N], the expected number of draws until all the balls are white given that all are currently red is also obtained.      

Coherent random allocations, and the Ewens-Pitman formula

Assume that there is a random number K of positive integer random variables S₁, …, S_K that are conditionally independent given K and all have identical distributions. A random integer partition N = S₁ + S₂ + … + S_K arises, and we denote by P_N the conditional distribution of this partition for a fixed value of N. We prove that the distributions {P_N} _N=1 ^∞ form a partition structure in the sense of Kingman if and only if they are governed by the Ewens-Pitman Formula. The latter generalizes the celebrated Ewens sampling formula, which has numerous applications in pure and applied mathematics. The distributions of the random variables K and S_j belong to a family of integer distributions with two real parameters, which we call quasi-binomial. Hence every Ewens-Pitman distribution arises as a result of a two-stage random procedure based on this simple class of integer distributions. Bibliography: 25 titles. This paper is an edited and actualized version of the unpublished PDMI preprint 21/1995. Further development of the ideas of this work can be found in [21, 25]. A number of detected misprints was fixed without notice, the bibliography was extended beyond the original 19 references, and a few comments were added as footnotes. (Comments by Alexander Gnedin.) __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 127–145.