Reverse spline interpolation

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Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the mathematical method called to do this minimizing reverse spline interpolation?

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I think you are looking for the Ramer–Douglas–Peucker algorithm. It takes an arbitrary polygon curve and simplifies it according to some given precision.