Computing the Exact Distribution Function of the Stochastic Longest Path Length in a DAG
 Ei Ando,
 Hirotaka Ono,
 Kunihiko Sadakane,
 Masafumi Yamashita
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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 antichain 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 nonnegative, (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 nonnegative 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.
 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.2009.01.001
 Ando, E., Ono, H., Sadakane, K., Yamashita, M.: A Generic Algorithm for Approximately Solving Stochastic Graph Optimization Problems (submitted for publication)
 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)
 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. (1995) Handbooks in Operations Research and Management Science. Elsevier Science B. V., Amsterdam, pp. 673762
 Berkelaar, M.: Statistical delay calculation, a linear time method. In: Proceedings of the International Workshop on Timing Analysis (TAU 1997), pp. 15–24 (1997)
 Clark, C.E. (1962) The PERT model for the distribution of an activity time. Operations Research 10: pp. 405406 CrossRef
 Hashimoto, M., Onodera, H. (2000) A performance optimization method by gate sizing using statistical static timing analysis. IEICE Trans. Fundamentals E83A: pp. 25582568
 Hagstrom, J.N. (1988) Computational Complexity of PERT Problems. Networks 18: pp. 139147 CrossRef
 Kelley, J.E. (1962) Criticalpath planning and scheduling: Mathematical basis. Operations Research 10: pp. 912915 CrossRef
 Kulkarni, V.G., Adlakha, V.G. (1986) Markov and MarkovRegenerative PERT Networks. Operations Research 34: pp. 769781 CrossRef
 Martin, J.J. (1965) Distribution of the time through a directed, acyclic network. Operations Research 13: pp. 4666 CrossRef
 Nikolova, E. Stochastic Shortest Paths Via Quasiconvex Maximization. In: Azar, Y., Erlebach, T. eds. (2006) Algorithms – ESA 2006. Springer, Heidelberg, pp. 552563 CrossRef
 Thomas, G.B. (2005) Thomas’ Calculus International Edition. Pearson Education, London
 Title
 Computing the Exact Distribution Function of the Stochastic Longest Path Length in a DAG
 Book Title
 Theory and Applications of Models of Computation
 Book Subtitle
 6th Annual Conference, TAMC 2009, Changsha, China, May 1822, 2009. Proceedings
 Pages
 pp 98107
 Copyright
 2009
 DOI
 10.1007/9783642020179_13
 Print ISBN
 9783642020162
 Online ISBN
 9783642020179
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 5532
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 Springer Berlin Heidelberg
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 Editors

 Jianer Chen ^{(16)}
 S. Barry Cooper ^{(17)}
 Editor Affiliations

 16. Department of Computer Science and Engineering, Texas A&M University
 17. School of Mathematics, University of Leeds
 Authors

 Ei Ando ^{(18)}
 Hirotaka Ono ^{(18)} ^{(19)}
 Kunihiko Sadakane ^{(18)}
 Masafumi Yamashita ^{(18)} ^{(19)}
 Author Affiliations

 18. Department of Computer Science and Communication Engineering,Graduate School of Information Science and Electrical Engineering, Kyushu University,
 19. Institute of Systems, Information Technologies and Nanotechnologies,
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