Efficient Optimal Approximation of Discrete Random Variables for Estimation of Probabilities of Missing Deadlines

Abstract

We present an efficient algorithm that, given a discrete random variable X and a number m, computes a random variable whose support is of size at most m and whose Kolmogorov distance from X is minimal. We present some variants of the algorithm, analyse their correctness and computational complexity, and present a detailed empirical evaluation that shows how they performs in practice. The main application that we examine, which is our motivation for this work, is estimation of the probability of missing deadlines in series-parallel schedules. Since exact computation of these probabilities is NP-hard, we propose to use the algorithms described in this paper to obtain an approximation.

Estimating the probability of meeting a deadline in schedules and plans

“…It is interesting to look for optimal approximations. In [25], we showed that an optimal approximation of a single random variable can be obtained in polynomial time. Specifically, we showed that given a random variable X and a target support size m, we can find the minimal ε * and a variable X such that X has support of size m and X ≺ ε * X .…”

An optimal approximation of discrete random variables with respect to the Kolmogorov distance

“…al. [4,5]. These papers study approximations of random variables in the context of estimating deadlines.…”

“…al. in [4,5] handle this reduction using weaker or sub-optimal notion of approximation than ours, as discussed in Section 2.…”

“…We estimated the probability for meeting deadlines in plans, as described in [4,5], and experimented with four different methods of approximation. The first two, OptTrim [4] and the Trim [5], are taken from the repository provided by the authors and are designed for achieving only a one-sided Kolmogorov approximation -a weaker notion of approximation then the Kolmogorov approximation analyzed in this work. The third method is a simple sampling scheme also described in [5] and the fourth is our Kolmogorov approximation obtained by the proposed KolmogorovApprox algorithm.…”

“…Many different approaches to approximation of probability distributions are studied in the literature [4,5,[11][12][13]16]. These approaches vary in the types random variables considered, how they are represented, and in the criteria used for evaluation of the quality of the approximations.…”