Approximating the Integral Fréchet Distance

Abstract

We present a pseudo-polynomial time (1 + epsilon)-approximation algorithm for computing the integral and average Frechet distance between two given polygonal curves T_1 and T_2. The running time is in O(zeta^{4}n^4/epsilon^2) where n is the complexity of T_1 and T_2 and zeta is the maximal ratio of the lengths of any pair of segments from 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 Frechet similarity of segments for all leash lengths. These two aspects are related to each other and are of independent interest.