Computing the Exact Distribution Function of the Stochastic Longest Path Length in a DAG

• Ei Ando
• Hirotaka Ono
• Masafumi Yamashita
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)

Abstract

Consider the longest path problem for directed acyclic graphs (DAGs), where a mutually independent random variable is associated with each of the edges as its edge length. Given a DAG G and any distributions that the random variables obey, let F MAX(x) be the distribution function of the longest path length. We first represent F MAX(x) by a repeated integral that involves n − 1 integrals, where n is the order of G. We next present an algorithm to symbolically execute the repeated integral, provided that the random variables obey the standard exponential distribution. Although there can be Ω(2 n ) paths in G, its running time is bounded by a polynomial in n, provided that k, the cardinality of the maximum anti-chain of the incidence graph of G, is bounded by a constant. We finally propose an algorithm that takes x and ε> 0 as inputs and approximates the value of repeated integral of x, assuming that the edge length distributions satisfy the following three natural conditions: (1) The length of each edge (v i ,v j ) ∈ E is non-negative, (2) the Taylor series of its distribution function F ij (x) converges to F ij (x), and (3) there is a constant σ that satisfies $$\sigma^p \le \left|\left(\frac{d}{dx}\right)^p F_{ij}(x)\right|$$ for any non-negative integer p. It runs in polynomial time in n, and its error is bounded by ε, when x, ε, σ and k can be regarded as constants.

Keywords

Polynomial Time Edge Length Directed Acyclic Graph Independent Random Variable Longe Path
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

References

1. 1.
Ando, E., Nakata, T., Yamashita, M.: Approximating the longest path length of a stochastic DAG by a normal distribution in linear time. Journal of Discrete Algorithms (2009), doi:10.1016/j.jda.2009.01.001Google Scholar
2. 2.
Ando, E., Ono, H., Sadakane, K., Yamashita, M.: A Generic Algorithm for Approximately Solving Stochastic Graph Optimization Problems (submitted for publication)Google Scholar
3. 3.
Ando, E., Yamashita, M., Nakata, T., Matsunaga, Y.: The Statistical Longest Path Problem and Its Application to Delay Analysis of Logical Circuits. In: Proc. TAU, pp. 134–139 (2002)Google Scholar
4. 4.
Ball, M.O., Colbourn, C.J., Proban, J.S.: Network Reliability. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds.) Handbooks in Operations Research and Management Science. Network Models, vol. 7, pp. 673–762. Elsevier Science B. V., Amsterdam (1995)Google Scholar
5. 5.
Berkelaar, M.: Statistical delay calculation, a linear time method. In: Proceedings of the International Workshop on Timing Analysis (TAU 1997), pp. 15–24 (1997)Google Scholar
6. 6.
Clark, C.E.: The PERT model for the distribution of an activity time. Operations Research 10, 405–406 (1962)
7. 7.
Hashimoto, M., Onodera, H.: A performance optimization method by gate sizing using statistical static timing analysis. IEICE Trans. Fundamentals E83-A(12), 2558–2568 (2000)Google Scholar
8. 8.
Hagstrom, J.N.: Computational Complexity of PERT Problems. Networks 18, 139–147 (1988)
9. 9.
Kelley Jr., J.E.: Critical-path planning and scheduling: Mathematical basis. Operations Research 10, 912–915 (1962)
10. 10.
Kulkarni, V.G., Adlakha, V.G.: Markov and Markov-Regenerative PERT Networks. Operations Research 34, 769–781 (1986)
11. 11.
Martin, J.J.: Distribution of the time through a directed, acyclic network. Operations Research 13, 46–66 (1965)
12. 12.
Nikolova, E.: Stochastic Shortest Paths Via Quasi-convex Maximization. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 552–563. Springer, Heidelberg (2006)
13. 13.
Thomas Jr., G.B.: Thomas’ Calculus International Edition, pp. 965–1066. Pearson Education, London (2005)Google Scholar

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

• Ei Ando
• 1
• Hirotaka Ono
• 1
• 2 