TY - GEN

T1 - Computing the exact distribution function of the stochastic longest path length in a dag

AU - Ando, Ei

AU - Ono, Hirotaka

AU - Sadakane, Kunihiko

AU - Yamashita, Masafumi

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-642-02017-9_13

DO - 10.1007/978-3-642-02017-9_13

M3 - Conference contribution

AN - SCOPUS:67650227268

SN - 9783642020162

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 98

EP - 107

BT - Theory and Applications of Models of Computation - 6th Annual Conference, TAMC 2009, Proceedings

T2 - 6th Annual Conference on Theory and Applications of Models of Computation, TAMC 2009

Y2 - 18 May 2009 through 22 May 2009

ER -