Following a curve with the discrete Fréchet distance

Abstract

Finding the similarity between curves is an important problem that comes up in many areas such as 3D modeling, GIS applications, ordering, and reachability. A related problem is to find one of the curves given a measure of similarity and another curve. Given a set of points S, a polygonal curve P, and an ε>0, the discrete set-chain matching problem is to find another polygonal curve Q such that the nodes of Q are points in S and dF(P,Q)≤ε. Here, dF is the discrete Fréchet distance between the two polygonal curves. For the first time we study the set-chain matching problem based on the discrete Fréchet distance rather than the continuous Fréchet distance. We further extend the problem based on unique or non-unique nodes and on limiting the number of points used. We prove that three of the variations of the set-chain matching problem are NP-complete. For the version of the problem that is polynomial, we give an O(|P||S|) time greedy solution.