Efficient One Sided Kolmogorov Approximation

TL;DR

This paper introduces an efficient algorithm for approximating a discrete random variable with a smaller support while minimizing the one-sided Kolmogorov distance, useful for estimating missed deadlines in complex scheduling scenarios.

Contribution

The paper presents a novel algorithm for one-sided Kolmogorov approximation of discrete variables, including variants, correctness analysis, and empirical evaluation.

Findings

Algorithm achieves minimal one-sided Kolmogorov distance with support size at most m.

Empirical results demonstrate practical efficiency and accuracy.

Application to scheduling probability estimation shows effectiveness in NP-hard scenarios.

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, also for the one-sided Kolmogorov approximation. 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 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.