TY - JOUR

T1 - Approximating the longest path length of a stochastic DAG by a normal distribution in linear time

AU - Ando, Ei

AU - Nakata, Toshio

AU - Yamashita, Masafumi

N1 - Funding Information:
The authors would like to thank the anonymous reviewers for their helpful suggestions. The authors are grateful to Yusuke Matsunaga for introducing us the ITC'99 benchmark set. This work was supported in part by a Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture of Japan. This work was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

PY - 2009/12

Y1 - 2009/12

N2 - This paper presents a linear time algorithm for approximating, in the sense below, the longest path length of a given directed acyclic graph (DAG), where each edge length is given as a normally distributed random variable. Let F (x) be the distribution function of the longest path length of the DAG. Our algorithm computes the mean and the variance of a normal distribution whose distribution function over(F, ̃) (x) satisfies over(F, ̃) (x) ≤ F (x) as long as F (x) ≥ a, given a constant a (1 / 2 ≤ a < 1). In other words, it computes an upper bound 1 - over(F, ̃) (x) on the tail probability 1 - F (x), provided x ≥ F- 1 (a). To evaluate the accuracy of the approximation of F (x) by over(F, ̃) (x), we first conduct two experiments using a standard benchmark set ITC'99 of logical circuits, since a typical application of the algorithm is the delay analysis of logical circuits. We also perform a worst case analysis to derive an upper bound on the difference over(F, ̃)- 1 (a) - F- 1 (a).

AB - This paper presents a linear time algorithm for approximating, in the sense below, the longest path length of a given directed acyclic graph (DAG), where each edge length is given as a normally distributed random variable. Let F (x) be the distribution function of the longest path length of the DAG. Our algorithm computes the mean and the variance of a normal distribution whose distribution function over(F, ̃) (x) satisfies over(F, ̃) (x) ≤ F (x) as long as F (x) ≥ a, given a constant a (1 / 2 ≤ a < 1). In other words, it computes an upper bound 1 - over(F, ̃) (x) on the tail probability 1 - F (x), provided x ≥ F- 1 (a). To evaluate the accuracy of the approximation of F (x) by over(F, ̃) (x), we first conduct two experiments using a standard benchmark set ITC'99 of logical circuits, since a typical application of the algorithm is the delay analysis of logical circuits. We also perform a worst case analysis to derive an upper bound on the difference over(F, ̃)- 1 (a) - F- 1 (a).

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U2 - 10.1016/j.jda.2009.01.001

DO - 10.1016/j.jda.2009.01.001

M3 - Article

AN - SCOPUS:67650278719

SN - 1570-8667

VL - 7

SP - 420

EP - 438

JO - Journal of Discrete Algorithms

JF - Journal of Discrete Algorithms

IS - 4

ER -