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)

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 FMAX(x) be the distribution function of the longest path length. We first represent FMAX(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(2n) 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 (vi,vj) ∈ E is non-negative, (2) the Taylor series of its distribution function Fij(x) converges to Fij(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.

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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 

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