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).
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics