Before revision: In this paper we consider the classical min--\# curve simplification problem in three different variants. Let \delta>0, P be  a polygonal curve  with n vertices in \mathbb{R^d, and D(\cdot,\cdot) be a distance measure.
After revision: Due to its many applications, a polygonal curve with n vertices in \mathbb{R^d, and D(\cdot,\cdot) be a distance measure.