On the Distribution of the Transitive Closure in a Random Acyclic Digraph

In the usual Gn;p{model of a random acyclic digraph let n (1) be the size of the reeexive, transitive closure of node 1, a source node; then the distribution of n (1) is given by where q = 1 ? p. Our analysis points out some surprising relations between this distribution and known functions of the number theory. In particular we nd for the expectation of n (1): lim n!1 n ? E(n (1)) = L(q) where L(q) = P 1 i=1 q i =(1 ? q i) is the so{called Lambert Series, which corresponds to the generating function of the divisor{function. These results allow us to improve the expected running time for the computation of the transitive closure in a random acyclic digraph and in particular we can ameliorate in some cases the analysis of the Goral c kovv a{Koubek Algorithm.