Limit Laws for the Optimal Directed Tree with Random Costs

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 c₁₁, c₁₂,. . . , 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_nE(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.