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

  • Klaus Simon
  • Davide Crippa
  • Fabian Collenberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 726)


In the usual Gn, 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 qi/(1-qi) 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Klaus Simon
    • 1
  • Davide Crippa
    • 1
  • Fabian Collenberg
    • 1
  1. 1.Institut für Theoretische Informatik ETH-ZentrumZürich

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