Approximating the Integral Fréchet Distance

Abstract

A pseudo-polynomial time $(1+\epsilon )$-approximation algorithm is presented for computing the integral and average Fréchet distance between two given polygonal curves $T_1$ and $T_2$. In particular, the running time is upper-bounded by $\mathcal{O}(\hat{I}{\P }^4{n}^4/{\epsilon }^2)$ where $n$ is the complexity of $T_1$ and $T_2$ and $\hat{I}\P$ is the maximal ratio of the lengths of any pair of segments from $T_1$ and $T_2$. The Fréchet distance captures the minimal cost of a continuous deformation of $T_1$ into $T_2$ and vice versa and defines the cost of a deformation as the maximal distance between two points that are related. The integral Fréchet distance defines the cost of a deformation as the integral of the distances between points that are related. The average Fréchet distance is defined as the integral Fréchet distance divided by the lengths of $T_1$ and $T_2$. Furthermore, we give relations between weighted shortest paths inside a single parameter cell $C$ and the monotone free space axis of $C$. As a result we present a simple construction of weighted shortest paths inside a parameter cell. Additionally, such a shortest path provides an optimal solution for the partial Fréchet similarity of segments for all leash lengths. These two aspects are related to each other and are of independent interest.