Theory and Applications of Models of Computation
Volume 5532 of the series Lecture Notes in Computer Science pp 98107
Computing the Exact Distribution Function of the Stochastic Longest Path Length in a DAG
 Ei AndoAffiliated withDepartment of Computer Science and Communication Engineering,Graduate School of Information Science and Electrical Engineering, Kyushu University
 , Hirotaka OnoAffiliated withDepartment of Computer Science and Communication Engineering,Graduate School of Information Science and Electrical Engineering, Kyushu UniversityInstitute of Systems, Information Technologies and Nanotechnologies
 , Kunihiko SadakaneAffiliated withDepartment of Computer Science and Communication Engineering,Graduate School of Information Science and Electrical Engineering, Kyushu University
 , Masafumi YamashitaAffiliated withDepartment of Computer Science and Communication Engineering,Graduate School of Information Science and Electrical Engineering, Kyushu UniversityInstitute of Systems, Information Technologies and Nanotechnologies
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.
 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
 SpringerVerlag 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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