A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $\delta^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $\delta > 0$), ii) the tree branches are given by curves which are regular in the sense of H\"older continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.

We give a precise average-case analysis of Ben-Or's algorithm for testing the irreducibility of polynomials over finite fields. First, we study the probability that a random polynomial of degree n over 𝔽q contains no irreducible factors of degree less than m, 1≤m≤n. The results are given in terms of the Buchstab function. Then, we compute the expectation and variance of the smallest degree among the irreducible factors of a random polynomial. The analysis of Ben-Or's algorithm readily follows from this expectation. We also compute the probability of a polynomial being irreducible when it has no irreducible factors of degree less than m, 1≤m≤n. This probability is useful in the analysis of some algorithms for factoring polynomials over finite fields. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 439–456, 1998

In the paper we study the asymptotic behavior of the number of trees with n vertices and diameter k=k(n), where (n−k)/n→a as n→∞ for some constant a<1. We use this result to determine the limit distribution of the diameter of the random graph G(n, p) in the subcritical phase. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 485–500, 1998

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 $n^\delta (\delta>0)$ 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 w 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 $p \ge n^{-{\alpha (k)}+\epsilon}}$ Where $\alpha(k) = \frac{(2k)}{((k-1)(k+2))}$ and \epsilon is a positive constant.

The size of ordered binary decision diagrams (OBDDs) strongly depends on the chosen variable ordering. It is an obvious heuristic to use symmetric variable orderings, i.e., variable orderings where symmetric variables are arranged adjacently. In order to evaluate this heuristic, methods for estimating the OBDD size for random partially symmetric functions are presented. Characterizations of cases where, with high probability, only symmetric variable orderings and, with high probability, only nonsymmetric variable orderings lead to minimum OBDD size are obtained. For this analysis estimates for the number of different blocks of random Boolean matrices are used. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 49–70, 1998

We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables. Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random variables. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 1–16, 1998

We prove that for every r and d≥2 there is a C such that for most choices of d permutations π1, π2,…,πd of Sn, the following holds: for any two r-tuples of distinct elements in {1,…,n}, there is a product of less than C log n of the πis which map the first r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to a general context of which random Cayley graph quotients of Sn are good expanders. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 335–350, 1998

For a pair of integers 1≤γ<r, the γ-chromatic number of an r-uniform hypergraph H=(V, E) is the minimal k, for which there exists a partition of V into subsets T1,…,Tk such that |e∩Ti|≤γ for every e∈E. In this paper we determine the asymptotic behavior of the γ-chromatic number of the random r-uniform hypergraph Hr(n, p) for all possible values of γ and for all values of p down to p=Θ(n−r+1). © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 381–403, 1998

This paper contributes to the study of nonlinear dynamical systems from a computational perspective. These systems are inherently more powerful than their linear counterparts (such as Markov chains), which have had a wide impact in computer science, and they seem likely to play an increasing role in the future. However, there are as yet no general techniques available for handling the computational aspects of discrete nonlinear systems, and even the simplest examples seem very hard to analyze. We focus in this paper on a class of quadratic systems that are widely used as a model in population genetics and also in genetic algorithms. These systems describe a process where random matings occur between parental chromosomes via a mechanism known as “crossover”: i.e., children inherit pieces of genetic material from different parents according to some random rule. Our results concern two fundamental quantitative properties of crossover systems: 1. We develop a general technique for computing the rate of convergence to equilibrium. We apply this technique to obtain tight bounds on the rate of convergence in several cases of biological and computational interest. In general, we prove that these systems are “rapidly mixing,” in the sense that the convergence time is very small in comparison with the size of the state space. 2. We show that, for crossover systems, the classical quadratic system is a good model for the behavior of finite populations of small size. This stands in sharp contrast to recent results of Arora et al. and Pudlák, who show that such a correspondence is unlikely to hold for general quadratic systems. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 313–334, 1998