The prefix Fréchet similarity

Abstract

We present the prefix Fréchet similarity as a new measure for similarity of curves which is for instance motivated by evacuation analysis and defined as follows. Given two (polygonal) curves T and T′, we ask for two prefix curves of T and T′ which have a Fréchet distance no larger than a given distance threshold δ≥0 with respect to the L1 metric such that the sum of the lengths of the prefix curves is maximal. As parameterized Fréchet measures as for example the prefix Fréchet similarity are highly unstable regarding the value of the distance threshold δ, we give an algorithm that computes exactly the profile of the prefix Fréchet similarity which means the complete functional relation between δ and the prefix Fréchet similarity of T and T′. This is the first efficient algorithm for computing exactly the whole profile of a parametrized Fréchet distance.While the running time of our algorithm for computing the profile of the prefix Fréchet similarity is O(n3log⁡n), we provide a lower bound of Ω(n2) for the running time of any algorithm computing the profile of the prefix Fréchet similarity, where n denotes the number of segments on T and T′. This implies that our running time is at most a near linear factor away from being optimal.