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

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


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)CrossRefGoogle Scholar
  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)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Kelley Jr., J.E.: Critical-path planning and scheduling: Mathematical basis. Operations Research 10, 912–915 (1962)CrossRefGoogle Scholar
  10. 10.
    Kulkarni, V.G., Adlakha, V.G.: Markov and Markov-Regenerative PERT Networks. Operations Research 34, 769–781 (1986)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Martin, J.J.: Distribution of the time through a directed, acyclic network. Operations Research 13, 46–66 (1965)MATHCrossRefMathSciNetGoogle Scholar
  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)CrossRefGoogle Scholar
  13. 13.
    Thomas Jr., G.B.: Thomas’ Calculus International Edition, pp. 965–1066. Pearson Education, London (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ei Ando
    • 1
  • Hirotaka Ono
    • 1
    • 2
  • Kunihiko Sadakane
    • 1
  • Masafumi Yamashita
    • 1
    • 2
  1. 1.Department of Computer Science and Communication Engineering,Graduate School of Information Science and Electrical EngineeringKyushu University 
  2. 2.Institute of Systems, Information Technologies and Nanotechnologies 

Personalised recommendations