Curve Simplification under the L2-norm

With the increased production and availability of large data sets comes the opportunity to glean valuable information about behaviors that would previously have been difficult to study formally. One such data set describes the trajectory of various objects, such as that generated by a GPS. Naturally, we often wish to compare these trajectories. To simplify this process, we seek an efficient yet accurate summary of a trajectory. In this instance, we consider the problem of finding such a summary, a k-piece, continuous, piecewise linear function which minimizes distance from the underlying trajectory in the L2-norm. We present an exact algorithm which is exponential in k, as well as a (1 + ) approximation algorithm which runs in polynomial time.