Computing the exact distribution function of the stochastic longest path length in a dag

Ei Ando, Hirotaka Ono, Kunihiko Sadakane, Masafumi Yamashita

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

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 e > 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 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 σ p ≤ (d/dx) p Fij(x) for any non-negative integer p. It runs in polynomial time in n, and its error is bounded by e, when x, e, σ and k can be regarded as constants.

Original languageEnglish
Title of host publicationTheory and Applications of Models of Computation - 6th Annual Conference, TAMC 2009, Proceedings
Pages98-107
Number of pages10
DOIs
Publication statusPublished - 2009
Event6th Annual Conference on Theory and Applications of Models of Computation, TAMC 2009 - Changsha, China
Duration: May 18 2009May 22 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5532 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other6th Annual Conference on Theory and Applications of Models of Computation, TAMC 2009
Country/TerritoryChina
CityChangsha
Period5/18/095/22/09

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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