Best piecewise constant approximation of a function of single variable (Q1111948)

Let F(I) be a class of piecewise constant functions on an interval I. The following two problems are studied: (1) Given \(f\in F(I)\) and a positive constant \(\epsilon\), find \(g\in F(I)\) such that \(\| g-f\| \leq \epsilon\) and g has a minimal number of constant pieces. (2) Given \(f\in F(I)\) and a positive integer k, find \(g\in F(I)\) which minimizes \(\| g-f\|\) among all functions with the number of constant pieces less then k. These problems are solved by using shortest path and dynamic programming techniques, and polynomial time algorithms are proposed. Some applications are briefly described.