Jennie C. Hansen

Order statistics for decomposable combinatorial structures

In this paper we consider the component structure of decomposable combinatorial objects, both labeled and unlabeled, from a probabilistic point of view. In both cases we show that when the generating function for the components of a structure is a logarithmic function, then the joint distribution of the normalized order statistics of the component sizes of a random object of size n coverges to the Poisson–Dirichlet distribution on the simplex ∇{{xi}: Σ xi = 1 x1 ⩾ x2 ⩾ … ⩾ 0}. This result complements recent results obtained by Flajolet and Soria on the total number of components in a random combinatorial structure.

A functional central limit theorem for the Ewens sampling formula.

For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n . To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D [0, 1] converge to Wiener measure. This result complements Kingmans frequency limit theorem [10] for the Ewens partition structure.

Distributed Largest-First Algorithm for Graph Coloring

In the paper we present a distributed probabilistic algorithm for coloring the vertices of a graph. Since this algorithm resembles a largest-first strategy, we call it the distributed LF (DLF) algorithm. The coloring obtained by DLF is optimal or near optimal for numerous classes of graphs e.g. complete k-partite, caterpillars, crowns, bipartite wheels. We also show that DLF runs in O(Δ2 log n) rounds for an arbitrary graph, where n is the number of vertices and Δ denotes the largest vertex degree.

Local properties of random mappings with exchangeable in-degrees

In this paper we investigate the ‘local’ properties of a random mapping model, T n D̂, which maps the set {1, 2, …, n} into itself. The random mapping T n D̂ , which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂ 1, …, D̂ n which satisfy In the random digraph, G n D̂ , which represents the mapping T n D̂ , the in-degree sequence for the vertices is given by the variables D̂ 1, D̂ 2, …, D̂ n , and, in some sense, G n D̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G n D̂ - for example, the numbers of predecessors and successors of v in G n D̂ . We show that the distribution of several variables associated with the local structure of G n D̂ can be expressed in terms of expectations of simple functions of D̂ 1, D̂ 2, …, D̂ n . We also consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

SUBADDITIVE ERGODIC THEOREMS FOR RANDOM SETS IN INFINITE DIMENSIONS

We prove pointwise and mean versions of the subadditive ergodic theorem for superstationary families of compact, convex random subsets of a real Banach space, extending previously known results that were obtained in finite dimensions or with additional hypotheses on the random sets. We also show how the techniques can be used to obtain the strong law of large numbers for pairwise independent random sets, as well as results in the weak topology.

Factorization in Fq[ x] and Brownian Motion

We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n→∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial. This research was partially supported by NSF grant DMS90 099074. 1

Limit Laws for the Optimal Directed Tree with Random Costs

Suppose that C={cijri, j≥1} is a collection of i.i.d. nonnegative continuous random variables and suppose T is a rooted, directed tree on vertices labelled 1,2,...,n. Then the ‘cost’ of T is defined to be c(T)=∑ (i,j)∈Tcij, where (i, j) denotes the directed edge from i to j in the tree T. Let Tn denote the ‘optimal’ tree, i.e. c(Tn) =min{c(T)rT is a directed, rooted tree in with n vertices}. We establish general conditions on the asymptotic behaviour of the moments of the order statistics of the variables c11, c12, ..., cin which guarantee the existence of sequences {an}, {bn}, and {dn} such that b−1n (c(Tn)−an) →N(0, 1) in distribution, d−1n c(Tn)→1 in probability, and d−1n E(c(Tn))→1 as n→∞, and we explicitly determine these sequences. The proofs of the main results rely upon the properties of general random mappings of the set {1, 2, ..., n} into itself. Our results complement and extend those obtained by McDiarmid [9] for optimal branchings in a complete directed graph.

Covering random points in a unit disk

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, …, X n }, where X 1, X 2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .

Near-optimal bounded-degree spanning trees

Random costsC(i, j) are assigned to the arcs of a complete directed graph onn labeled vertices. Given the cost matrixCn =(C(i, j)), letT*k =T*k (Cn ) be the spanning tree that has minimum cost among spanning trees with in-degree less than or equal tok. Since it is NP-hard to findT*k , we instead consider an efficient algorithm that finds a near-optimal spanning treeTka. If the edge costs are independent, with a common exponential(I) distribution, then, asn → ∞,