Longest path distance in random circuits

Nicolas Broutin (1) , Omar Fawzi (2)

1 RAP - Networks, Algorithms and Probabilities
2 School of Computer Science [Montréal]

We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random DAG. This completes the study of natural distances in random DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also obtain large deviation bounds for the minimum of a branching random walk with constant branching, which can be seen as a simplified version of our main result.

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Nicolas Broutin, Omar Fawzi. Longest path distance in random circuits. Combinatorics, Probability and Computing, 2012, 21 (6), pp.856-881. ⟨10.1017/S0963548312000260⟩. ⟨hal-00773368⟩