Approximating The p-Mean Curve of Large Data-Sets

A set of piecewise linear functions, called polylines, $P_1,\ldots,P_L$ each with at most $n$ vertices can be simplified into a polyline $M$ with $k$ vertices, such that the Fr\'echet distances $\epsilon_1,\ldots,\epsilon_L$ to each of these polylines are minimized under the $L_p$ distance. We call $M$ for $L_p$ with $p\geq 1$ a $p$-mean curve ($p$-MC). We discuss $p\geq 1$, for which $L_p$ distance satisfies the triangle inequality and $p$-mean has not been discussed before for most values $p$. Computing the $p$-mean polyline is NP-hard for $L=\Omega(1)$ and some values of $p$, so we discuss approximation algorithms. We give a $O(n^2\log k)$ time exact algorithm for $L=2$ and $p\geq 1$. Also, we reduce the Fr\'echet distance to the discrete Fr\'echet distance which adds a factor $2$ to both $k$ and $\epsilon$. Then we use our exact algorithm to find a $3$-approximation for $L>2$ in $\operatorname{poly}(n,L)$ time. Our method is based on a generalization of the free-space diagram (FSD) for Fr\'echet distance and composable core-sets for approximate summaries.