On the distribution of the transitive closure in a random acyclic digraph

Abstract

In the usual G _{n, p}-model of a random acyclic digraph let γ ^*_n (1) be the size of the reflexive, transitive closure of node 1, a source node; then the distribution of γ ^*_n (1) is given by

$$\forall 1 \leqslant h \leqslant n: Pr\left( {\gamma _n^ * \left( 1 \right) = h} \right) = q^{n - h} \prod\limits_{i = 1}^{h - 1} {\left( {1 - q^{n - i} } \right)} ,$$

where q=1−p. Our analysis points out some surprising relations between this distribution and known functions of the number theory. In particular we find for the expectation of γ ^*_n (1):

$$\mathop {lim}\limits_{n \to \infty } n - E\left( {\gamma _n^ * \left( 1 \right)} \right) = L\left( q \right)$$

where L(q)=∑ ^∞_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 Goralčíková-Koubek Algorithm.

Research supported by the Swiss National Science Foundation, Grant 21-34115.92.