Limit Laws for the Optimal Directed Tree with Random Costs

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Abstract

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.

Original languageEnglish
Pages (from-to)315-335
Number of pages21
JournalCombinatorics, Probability and Computing
Volume6
Issue number3
Publication statusPublished - 1997

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