Approximating the integral Fréchet distance

We present a pseudo-polynomial time (1+ε)-approximation algorithm for computing the integral and average Fréchet distance between two given polygonal curves T1 and T2. The running time is in O(ζ4n4/ε2) where n is the complexity of T1 and T2 and ζ is the maximal ratio of the lengths of any pair of segments from T1 and T2. 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. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity, I.3.5 Computational Geometry and Object Modeling