Approximate Matching of Curves to Point Sets

Let P be a polygonal curve in R of length n, and S be a point set of size k. The Curve/Point Set Matching problem consists of finding a polygonal curve Q on S such that the Fréchet distance from P is less than a given ε. We consider eight variations of the problem based on the distance metric used and the omittability and repeatability of the points. We provide closure to a recent series of complexity results for the case where S consists of precise points. We also formulate a more realistic version of the problem that takes into account measurement errors and attempts to match a given curve to a set of imprecise points. We show that all three variations of the problem that are in P when S consists of precise points become NP-complete when S consists of imprecise points. Finally, we present a 3-factor approximation algorithm for a version of the problem.