Approximating the longest path length of a stochastic DAG by a normal distribution in linear time

This paper presents a linear time algorithm for approximating, in the sense below, the longest path length of a given directed acyclic graph (DAG), where each edge length is given as a normally distributed random variable. Let F (x) be the distribution function of the longest path length of the DAG. Our algorithm computes the mean and the variance of a normal distribution whose distribution function over(F, ̃) (x) satisfies over(F, ̃) (x) ≤ F (x) as long as F (x) ≥ a, given a constant a (1 / 2 ≤ a < 1). In other words, it computes an upper bound 1 - over(F, ̃) (x) on the tail probability 1 - F (x), provided x ≥ F^{- 1} (a). To evaluate the accuracy of the approximation of F (x) by over(F, ̃) (x), we first conduct two experiments using a standard benchmark set ITC'99 of logical circuits, since a typical application of the algorithm is the delay analysis of logical circuits. We also perform a worst case analysis to derive an upper bound on the difference over(F, ̃)^{- 1} (a) - F^{- 1} (a).