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.


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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