### Abstract

Suppose that script C = {c_{ij} : i, 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 denned to be c(T) = S _{(i,j)?T} c_{ij}, where (i, j) denotes the directed edge from i to j in the tree T. Let T_{n} denote the 'optimal' tree, i.e. c(T_{n})=min{c(T) : T 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 c_{11}, c_{12},. . . , c_{in} which guarantee the existence of sequences {a_{n}}, {b_{n}}, and {d_{n}} such that b^{-1}_{n}(c(T_{n})-a_{n}) ? N(0, 1) in distribution, d^{-1}_{n} c(T_{n}) ? 1 in probability, and d^{-1}_{n}E(c(T_{n})) ? 1 as n ? 8, 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.

Original language | English |
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Pages (from-to) | 315-335 |

Number of pages | 21 |

Journal | Combinatorics, Probability and Computing |

Volume | 6 |

Issue number | 3 |

Publication status | Published - 1997 |

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*Combinatorics, Probability and Computing*, vol. 6, no. 3, pp. 315-335.

**Limit Laws for the Optimal Directed Tree with Random Costs.** / Hansen, Jennie C.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Limit Laws for the Optimal Directed Tree with Random Costs

AU - Hansen, Jennie C.

PY - 1997

Y1 - 1997

N2 - Suppose that script C = {cij : i, 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 denned to be c(T) = S (i,j)?T cij, 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) : T 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-1nE(c(Tn)) ? 1 as n ? 8, 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.

AB - Suppose that script C = {cij : i, 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 denned to be c(T) = S (i,j)?T cij, 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) : T 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-1nE(c(Tn)) ? 1 as n ? 8, 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.

UR - http://www.scopus.com/inward/record.url?scp=0031515486&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 315

EP - 335

JO - Combinatorics, Probability and Computing

JF - Combinatorics, Probability and Computing

SN - 0963-5483

IS - 3

ER -