Before revision:  For local directed Hausdorff distance from the input to the simplified curve  , we present an approximation algorithm that computes a curve whose number of vertices is at most twice the number of the vertices of the curve-restricted simplification with the minimum number of vertices .
After revision:  For local directed Hausdorff distance from the input to the simplified curve in \mathbb{R , we present an approximation algorithm that computes a curve whose number of vertices is at most twice the number of the vertices of the curve-restricted simplification with the minimum number of vertices .