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**The transitive closure of a random digraph.**
*(English)*
Zbl 0712.68076

Summary: In a random n-vertex digraph, each arc is present with probability p, independently of the presence or absence of other arcs. We investigate the structure of the strong components of a random digraph and present an algorithm for the construction of the transitive closure of a random digraph. We show that, when n is large and np is equal to a constant c greater than 1, it is very likely that all but one of the strong components are very small, and that the unique large strong component contains about \(\Theta^ 2n\) vertices, where \(\Theta\) is the unique root in [0,1] of the equation \(1-x-e^{-cx}=0\). Nearly all the vertices outside the large strong component lie in strong components of size 1. Provided that the expected degree of a vertex is bounded away from 1, our transitive closure algorithm runs in expected time O(n). For all choices of n and p, the expected execution time of the algorithm is O(w(n) (n log n)\({}^{4/3})\), where w(n) is an arbitrary nondecreasing unbounded function. To circumvent the fact that the size of the transitive closure may be \(\Omega (n^ 2)\) the algorithm presents the transitive closure in the compact form (A\(\times B)\cup C\), where A and B are sets of vertices, and C is a set of arcs.

### MSC:

68R10 | Graph theory (including graph drawing) in computer science |

05C80 | Random graphs (graph-theoretic aspects) |

68Q25 | Analysis of algorithms and problem complexity |

05C20 | Directed graphs (digraphs), tournaments |

05C85 | Graph algorithms (graph-theoretic aspects) |

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\textit{R. M. Karp}, Random Struct. Algorithms 1, No. 1, 73--93 (1990; Zbl 0712.68076)

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### References:

[1] | Palásti, Studia Sci. Math. Hungar. 1 pp 205– (1966) |

[2] | ”The phase transition in the evolution of random graphs,” manuscript (1988). |

[3] | Fenner, Combinatorica 2 pp 347– (1988) |

[4] | Schnorr, SIAM J. Comput. 7 pp 127– (1978) |

[5] | Random Graphs, Academic, New York, 1985. |

[6] | Nagaev, Theory of Probability and its Applications pp 98– (1970) |

[7] | ”Probabilistic Construction of deterministic algorithms: approximating packing integer programs,” Proceedings of the 27th Annual IEEE Symp. on Foundations of Computer Science, 1986, pp. 10–18. |

[8] | The Theory of Branching Processes, Springer, New York, 1963. |

[9] | and , Branching Processes, Springer-Verlag, New York, 1972. |

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