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Jennie Hansen

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Adam Mickiewicz University in Poznań

Budapest University of Technology and Economics

Alois Panholzer

Ehsan Ebrahimzadeh

Florentin Smarandache

University of New Mexico

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Papers by Jennie Hansen

Structural transition in random mappings

The electronic journal of combinatorics

In this paper we characterise the structural transition in random mappings with in-degree restric... more In this paper we characterise the structural transition in random mappings with in-degree restrictions. Specifically, for integers 0≤r≤n, we consider a random mapping model T ^ n r from [n]={1,2,···,n} into [n] such that G ^ n r , the directed graph on n labelled vertices which represents the mapping T ^ n r , has r vertices that are constrained to have in-degree at most 1 and the remaining vertices have in-degree at most 2. When r=n, T ^ n r is a uniform random permutation and when r<n, we can view T ^ n r as a ’corrupted’ permutation. We investigate structural transition in G ^ n r as we vary the integer parameter r relative to the total number of vertices n. We obtain exact and asymptotic distributions for the number of cyclic vertices, the number of components, and the size of the typical component in G ^ n r , and we characterise the dependence of the limiting distributions of these variables on the relationship between the parameters n and r as n→∞. We show that the number ...

Herriot-Watt University

Probabilistic Analysis of Rule 2

Corr, Aug 31, 2004

Li and Wu proposed Rule 2, a localized approximation algorithm that attempts to find a small conn... more Li and Wu proposed Rule 2, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. Here we study the asymptotic performance of Rule 2 on random unit disk graphs formed from n random points in an s_n by s_n square region of the plane. If s_n is below the threshold for connectivity, then Rule 2 produces a dominating set whose expected size is O(n/(loglog n)^{3/2}). We conjecture that this bound is not optimal.

Order statistics for decomposable combinatorial structures

Random Structures & Algorithms, 1994

In this paper we consider the component structure of decomposable combinatorial objects, both lab... more

Covering Random Points in a Unit Ball

Choose random points X1,X2,X3,... independently from a uniform distribution in a unit ball in ,. ... more Choose random points X1,X2,X3,... independently from a uniform distribution in a unit ball in ,. Call Xn a dominator i distance( Xn,Xi) 1 for all i < n, i.e. the first n points are all contained in the unit ball that is centered at the n’th point Xn. We prove that, with probability one, only finitely many of the points are dominators. For the special case m = 2, we consider the unit disk graph Gn determined by n random points X1,X2,...,Xn in the unit disk. With asymptotic probability one, Gn has a connected dominating set consisting of just two points. keywords and phrases: stochastic geometry, dominating set, geometric graph, unit ball graph

A Functional Central Limit Theorem for the Ewens Sampling Formula

Journal of Applied Probability, 1990

Factorization in Fq[ x] and Brownian Motion

Combinatorics, Probability and Computing, 1993

We consider the set of polynomials of degree n over a finite field and put the uniform probabilit... more 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.

Limit Laws for the Optimal Directed Tree with Random Costs

Combinatorics, Probability and Computing, 1997

ABSTRACT Suppose that C = fcij : i;j 1g is a collection of i.i.d. nonneg- ative continuous random... more ABSTRACT Suppose that C = fcij : i;j 1g is a collection of i.i.d. nonneg- ative continuous random variables and suppose T is a rooted, directed tree on vertices labelled 1,2,. . . ,n. Then the &#39;cost&#39; of T is dened to be c(T )= P (i;j)2T cij ,w here (i;j) is denotes the directed edge from i to j in the tree T.L et T ndenote the &#39;optimal&#39; tree, i.e. c(Tn )=m inf c ( T ): Tis a directed, rooted tree in with n verticesg: We establish general con- ditions on the asymptotic behaviour of the moments of the order statistics of the variablesc11;c12;:::;cin which guarantee the existence of sequences fang;fbng; andfdng such that b 1 n (c(Tn) an)! N(0; 1) in distribution, d 1 n c(Tn)! 1 in probability, and d 1 n E(c(Tn)) ! 1a s n !1 ,a nd we explicitly determine these sequences. The proofs of the main results rely upon the properties of general random mappings of the setf1; 2;:::;ng into itself. Our results complement and extend those obtained by McDi- armid (9) for optimal branchings in a complete directed graph.

How random is the characteristic polynomial of a random matrix ?

Mathematical Proceedings of the Cambridge Philosophical Society, 1993

... How random is the characteristic polynomial of a random matrix? BY JENNIE C. HANSENf Actuaria... more

A Functional Central Limit Theorem for Random Mappings

The Annals of Probability, 1989

We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ ... more We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ and put a uniform probability measure on this set. Any such mapping can be represented as a directed graph on $n$ labelled vertices. We study the component structure of the associated graphs as $n \rightarrow \infty$. To each mapping we associate a step function on $\lbrack 0, 1 \rbrack$. Each jump in the function equals the number of connected components of a certain size in the graph which represents the map. We normalize these functions and show that the induced measures on $D\lbrack 0, 1 \rbrack$ converge to Wiener measure. This result complements another result by Aldous on random mappings.

Correction: A Functional Central Limit Theorem for Random Mappings

The Annals of Probability, 1991

Near-optimal bounded-degree spanning trees

Algorithmica, 2001

are assigned to the arcs of a complete directed graph on n labelled vertices. Given the cost matr... more are assigned to the arcs of a complete directed graph on n labelled vertices. Given the cost matrix C n = (C(i, j)), let T * k = T * k (C n ) be the spanning tree that has minimum cost among spanning trees with in-degree less than or equal to k. Since it is NP-hard to find T * k , we instead consider an efficient algorithm that finds a nearoptimal spanning tree T a k . If the edge costs are independent, with a common exponential(1) distribution, then as n → ∞ E(Cost(T a k )) = E(Cost(T * k )) + o(1). Upper and lower bounds for E(Cost(T * k )) are also obtained for k ≥ 2.

Covering random points in a unit disk

Advances in Applied Probability, 2008

ABSTRACT Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D i... more ABSTRACT 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 Vn = {X1, X2, ..., Xn}, where X1, X2, ... are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in Vn that cover all of Vn.

Subadditive ergodic theorems for random sets in infinite dimensions

by Paul Hulse and Jennie Hansen

Statistics & Probability Letters, 2000

ABSTRACT We prove pointwise and mean versions of the subadditive ergodic theorem for superstation... more ABSTRACT 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.

The Expected Size of the Rule k Dominating Set: II

Rule k (k=1,2,3,...) is a well known family of approximation algorithms that can be used to find ... more Rule k (k=1,2,3,...) is a well known family of approximation algorithms that can be used to find connected dominating sets in a graph. They were originally proposed by Dai, Li, and Wu as the basis for ecient routing methods for ad hoc wireless networks. In this paper we study the asymptotic performance of Rules 1 and 2 on random,unit

Distributed Largest-First Algorithm for Graph Coloring

by Jennie Hansen and Łukasz Kuszner

Lecture Notes in Computer Science, 2004

A random mapping with preferential attachment

by Jennie Hansen and Jerzy Jaworski

Random Structures and Algorithms, 2009

In this paper we investigate the asymptotic structure of a random mapping model with preferential... more In this paper we investigate the asymptotic structure of a random mapping model with preferential attachment, T ρ n , which maps the set {1, 2, ..., n} into itself. The model T ρ n was introduced in a companion paper and the asymptotic structure of the associated directed graph G ρ n which represents the action of T ρ n on the set {1, 2, ..., n} was investigated in [11] and [12] in the case when the attraction parameter ρ > 0 is fixed as n → ∞. In this paper we consider the asymptotic structure of G ρ n when the attraction parameter ρ ≡ ρ(n) is a function of n as n → ∞. We show that there are three distinct regimes during the evolution of G ρ n : (i) ρn → ∞ as n → ∞, (ii) ρn → β > 0 as n → ∞, and (iii) ρn → 0 as n → ∞. It turns out that the asymptotic structure of G ρ n is, in some cases, quite different from the asymptotic structure of well-known models such as the uniform random mapping model and models with an attracting center. In particular, in regime (ii) we obtain some interesting new limiting distributions which are related to the incomplete gamma function.

A cutting process for random mappings

by Jennie Hansen and Jerzy Jaworski

Random Structures and Algorithms, 2007

In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we ... more In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph G n on n labelled vertices, and we delete (if possible), uniformly and at random, m non-cyclic directed edges from G n . The maximal random digraph consisting of the uni-cyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by G m n . If the number of non-cyclic directed edges is less than m, then G m n consists of the cycles, including loops, of the initial mapping G n . We consider the component structure of the trimmed mapping G m n . In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of G m n as n, m → ∞. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases:

Large components of bipartite random mappings

by Jerzy Jaworski and Jennie Hansen

Random Structures and Algorithms, 2000

A bipartite random mapping T K,L of a finite set V = V 1 ∪ V 2 , |V 1 | = K and |V 2 | = L , into... more A bipartite random mapping T K,L of a finite set V = V 1 ∪ V 2 , |V 1 | = K and |V 2 | = L , into itself assigns independently to each i ∈ V 1 its unique image j ∈ V 2 with probability 1/L and to each i ∈ V 2 its unique image j ∈ V 1 with probability 1/K. We study the connected component structure of a random digraph G(T K,L ) , representing T K,L , as K → ∞ and L → ∞. We show that, no matter how K and L tend to infinity relative to each other, the joint distribution of the normalized order statistics for the component sizes converges in distribution to the Poisson-Dirichlet distribution on the

Random mappings with exchangeable in-degrees

by Jerzy Jaworski and Jennie Hansen

Random Structures and Algorithms, 2008

In this paper we introduce a new random mapping model, TD n , which maps the set {1, 2, ..., n} i... more In this paper we introduce a new random mapping model, TD n , which maps the set {1, 2, ..., n} into itself. The random mapping TD n is constructed using a collection of exchangeable random variableŝ D 1 , ....,D n which satisfy n i=1D i = n. In the random digraph, GD n , which represents the mapping TD n , the in-degree sequence for the vertices is given by the variablesD 1 ,D 2 , ...,D n , and, in some sense, GD n can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions ofD 1 ,D 2 , ...,D n . We also consider two special examples of TD n which correspond to random mappings with preferential and anti-preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above.

Structural transition in random mappings

The electronic journal of combinatorics

In this paper we characterise the structural transition in random mappings with in-degree restric... more In this paper we characterise the structural transition in random mappings with in-degree restrictions. Specifically, for integers 0≤r≤n, we consider a random mapping model T ^ n r from [n]={1,2,···,n} into [n] such that G ^ n r , the directed graph on n labelled vertices which represents the mapping T ^ n r , has r vertices that are constrained to have in-degree at most 1 and the remaining vertices have in-degree at most 2. When r=n, T ^ n r is a uniform random permutation and when r<n, we can view T ^ n r as a ’corrupted’ permutation. We investigate structural transition in G ^ n r as we vary the integer parameter r relative to the total number of vertices n. We obtain exact and asymptotic distributions for the number of cyclic vertices, the number of components, and the size of the typical component in G ^ n r , and we characterise the dependence of the limiting distributions of these variables on the relationship between the parameters n and r as n→∞. We show that the number ...

Herriot-Watt University

Probabilistic Analysis of Rule 2

Corr, Aug 31, 2004

Li and Wu proposed Rule 2, a localized approximation algorithm that attempts to find a small conn... more Li and Wu proposed Rule 2, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. Here we study the asymptotic performance of Rule 2 on random unit disk graphs formed from n random points in an s_n by s_n square region of the plane. If s_n is below the threshold for connectivity, then Rule 2 produces a dominating set whose expected size is O(n/(loglog n)^{3/2}). We conjecture that this bound is not optimal.

Order statistics for decomposable combinatorial structures

Random Structures & Algorithms, 1994

In this paper we consider the component structure of decomposable combinatorial objects, both lab... more

Covering Random Points in a Unit Ball

Choose random points X1,X2,X3,... independently from a uniform distribution in a unit ball in ,. ... more Choose random points X1,X2,X3,... independently from a uniform distribution in a unit ball in ,. Call Xn a dominator i distance( Xn,Xi) 1 for all i < n, i.e. the first n points are all contained in the unit ball that is centered at the n’th point Xn. We prove that, with probability one, only finitely many of the points are dominators. For the special case m = 2, we consider the unit disk graph Gn determined by n random points X1,X2,...,Xn in the unit disk. With asymptotic probability one, Gn has a connected dominating set consisting of just two points. keywords and phrases: stochastic geometry, dominating set, geometric graph, unit ball graph

A Functional Central Limit Theorem for the Ewens Sampling Formula

Journal of Applied Probability, 1990

Factorization in Fq[ x] and Brownian Motion

Combinatorics, Probability and Computing, 1993

We consider the set of polynomials of degree n over a finite field and put the uniform probabilit... more 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.

Limit Laws for the Optimal Directed Tree with Random Costs

Combinatorics, Probability and Computing, 1997

ABSTRACT Suppose that C = fcij : i;j 1g is a collection of i.i.d. nonneg- ative continuous random... more ABSTRACT Suppose that C = fcij : i;j 1g is a collection of i.i.d. nonneg- ative continuous random variables and suppose T is a rooted, directed tree on vertices labelled 1,2,. . . ,n. Then the &#39;cost&#39; of T is dened to be c(T )= P (i;j)2T cij ,w here (i;j) is denotes the directed edge from i to j in the tree T.L et T ndenote the &#39;optimal&#39; tree, i.e. c(Tn )=m inf c ( T ): Tis a directed, rooted tree in with n verticesg: We establish general con- ditions on the asymptotic behaviour of the moments of the order statistics of the variablesc11;c12;:::;cin which guarantee the existence of sequences fang;fbng; andfdng such that b 1 n (c(Tn) an)! N(0; 1) in distribution, d 1 n c(Tn)! 1 in probability, and d 1 n E(c(Tn)) ! 1a s n !1 ,a nd we explicitly determine these sequences. The proofs of the main results rely upon the properties of general random mappings of the setf1; 2;:::;ng into itself. Our results complement and extend those obtained by McDi- armid (9) for optimal branchings in a complete directed graph.

How random is the characteristic polynomial of a random matrix ?

Mathematical Proceedings of the Cambridge Philosophical Society, 1993

... How random is the characteristic polynomial of a random matrix? BY JENNIE C. HANSENf Actuaria... more

A Functional Central Limit Theorem for Random Mappings

The Annals of Probability, 1989

We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ ... more We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ and put a uniform probability measure on this set. Any such mapping can be represented as a directed graph on $n$ labelled vertices. We study the component structure of the associated graphs as $n \rightarrow \infty$. To each mapping we associate a step function on $\lbrack 0, 1 \rbrack$. Each jump in the function equals the number of connected components of a certain size in the graph which represents the map. We normalize these functions and show that the induced measures on $D\lbrack 0, 1 \rbrack$ converge to Wiener measure. This result complements another result by Aldous on random mappings.

Correction: A Functional Central Limit Theorem for Random Mappings

The Annals of Probability, 1991

Near-optimal bounded-degree spanning trees

Algorithmica, 2001

are assigned to the arcs of a complete directed graph on n labelled vertices. Given the cost matr... more are assigned to the arcs of a complete directed graph on n labelled vertices. Given the cost matrix C n = (C(i, j)), let T * k = T * k (C n ) be the spanning tree that has minimum cost among spanning trees with in-degree less than or equal to k. Since it is NP-hard to find T * k , we instead consider an efficient algorithm that finds a nearoptimal spanning tree T a k . If the edge costs are independent, with a common exponential(1) distribution, then as n → ∞ E(Cost(T a k )) = E(Cost(T * k )) + o(1). Upper and lower bounds for E(Cost(T * k )) are also obtained for k ≥ 2.

Covering random points in a unit disk

Advances in Applied Probability, 2008

ABSTRACT Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D i... more ABSTRACT 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 Vn = {X1, X2, ..., Xn}, where X1, X2, ... are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in Vn that cover all of Vn.

Subadditive ergodic theorems for random sets in infinite dimensions

by Paul Hulse and Jennie Hansen

Statistics & Probability Letters, 2000

ABSTRACT We prove pointwise and mean versions of the subadditive ergodic theorem for superstation... more ABSTRACT 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.

The Expected Size of the Rule k Dominating Set: II

Rule k (k=1,2,3,...) is a well known family of approximation algorithms that can be used to find ... more Rule k (k=1,2,3,...) is a well known family of approximation algorithms that can be used to find connected dominating sets in a graph. They were originally proposed by Dai, Li, and Wu as the basis for ecient routing methods for ad hoc wireless networks. In this paper we study the asymptotic performance of Rules 1 and 2 on random,unit

Distributed Largest-First Algorithm for Graph Coloring

by Jennie Hansen and Łukasz Kuszner

Lecture Notes in Computer Science, 2004

A random mapping with preferential attachment

by Jennie Hansen and Jerzy Jaworski

Random Structures and Algorithms, 2009

In this paper we investigate the asymptotic structure of a random mapping model with preferential... more In this paper we investigate the asymptotic structure of a random mapping model with preferential attachment, T ρ n , which maps the set {1, 2, ..., n} into itself. The model T ρ n was introduced in a companion paper and the asymptotic structure of the associated directed graph G ρ n which represents the action of T ρ n on the set {1, 2, ..., n} was investigated in [11] and [12] in the case when the attraction parameter ρ > 0 is fixed as n → ∞. In this paper we consider the asymptotic structure of G ρ n when the attraction parameter ρ ≡ ρ(n) is a function of n as n → ∞. We show that there are three distinct regimes during the evolution of G ρ n : (i) ρn → ∞ as n → ∞, (ii) ρn → β > 0 as n → ∞, and (iii) ρn → 0 as n → ∞. It turns out that the asymptotic structure of G ρ n is, in some cases, quite different from the asymptotic structure of well-known models such as the uniform random mapping model and models with an attracting center. In particular, in regime (ii) we obtain some interesting new limiting distributions which are related to the incomplete gamma function.

A cutting process for random mappings

by Jennie Hansen and Jerzy Jaworski

Random Structures and Algorithms, 2007

In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we ... more In this paper we consider a cutting process for random mappings. Specifically, for 0 < m < n, we consider the initial (uniform) random mapping digraph G n on n labelled vertices, and we delete (if possible), uniformly and at random, m non-cyclic directed edges from G n . The maximal random digraph consisting of the uni-cyclic components obtained after cutting the m edges is called the trimmed random mapping and is denoted by G m n . If the number of non-cyclic directed edges is less than m, then G m n consists of the cycles, including loops, of the initial mapping G n . We consider the component structure of the trimmed mapping G m n . In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component of G m n as n, m → ∞. This asymptotic distribution depends on the relationship between n and m and we show that there are three distinct cases:

Large components of bipartite random mappings

by Jerzy Jaworski and Jennie Hansen

Random Structures and Algorithms, 2000

A bipartite random mapping T K,L of a finite set V = V 1 ∪ V 2 , |V 1 | = K and |V 2 | = L , into... more A bipartite random mapping T K,L of a finite set V = V 1 ∪ V 2 , |V 1 | = K and |V 2 | = L , into itself assigns independently to each i ∈ V 1 its unique image j ∈ V 2 with probability 1/L and to each i ∈ V 2 its unique image j ∈ V 1 with probability 1/K. We study the connected component structure of a random digraph G(T K,L ) , representing T K,L , as K → ∞ and L → ∞. We show that, no matter how K and L tend to infinity relative to each other, the joint distribution of the normalized order statistics for the component sizes converges in distribution to the Poisson-Dirichlet distribution on the

Random mappings with exchangeable in-degrees

by Jerzy Jaworski and Jennie Hansen

Random Structures and Algorithms, 2008

In this paper we introduce a new random mapping model, TD n , which maps the set {1, 2, ..., n} i... more In this paper we introduce a new random mapping model, TD n , which maps the set {1, 2, ..., n} into itself. The random mapping TD n is constructed using a collection of exchangeable random variableŝ D 1 , ....,D n which satisfy n i=1D i = n. In the random digraph, GD n , which represents the mapping TD n , the in-degree sequence for the vertices is given by the variablesD 1 ,D 2 , ...,D n , and, in some sense, GD n can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions ofD 1 ,D 2 , ...,D n . We also consider two special examples of TD n which correspond to random mappings with preferential and anti-preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above.