Limit Laws for the Optimal Directed Tree with Random Costs

Suppose that 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 defined to be c(T ) = ∑ (i,j)∈T cij , where (i, j) is 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−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)) → 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.