Following a curve with the discrete Fréchet distance

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>0ε>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)≤εdF(P,Q)≤ε. Here, dFdF 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|)O(|P||S|) time greedy solution.