Discretely Following a Curve

Finding the similarity between paths is an important problem that comes up in many areas such as 3D modeling, GIS applications, ordering, and reachability. Given a set of points S, a polygonal curve P, and an e > 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 d f (P,Q) ≤ e. Here, d F is the discrete Frechet distance between the two polygonal curves. For the first time we study the set-chain matching problem based on the discrete Frechet distance rather than the continuous Frechet 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 we prove is polynomial, we give the optimal substructure and the recurrence for a dynamic programming solution.

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