The Fréchet Distance Problem in Weighted Regions

where α and β range over all continuous non-decreasing functions with α(0) = β(0) = 0 and α(1) = β(1) = 1, and d′ is a distance metric between points. The functions α and β are also refereed to as reparametrizations. We call (α, β) a matching between P and Q. The Fréchet distance is described intuitively by a man walking a dog on a leash. The man follows a curve (path), and the dog follows another path. Both can control their speed but backtracking is not allowed. The Fréchet distance between the curves is the length of the shortest leash that is sufficient for the man and the dog to walk their paths from start to end. In this paper, we study the Fréchet distance problem in weighted regions in R2. Given a weighted subdivision R = {R1, R2, . . . , Rm} of the plane with a total of n edges and two parameterized polygonal chains P and Q, approximate the Fréchet distance between P and Q, where the (weighted) distance between two points P (s) and Q(t) on P and Q, respectively, is defined as S(P (s)Q(t)) = ∑m i=1 wi ∗Ri(P (s)Q(t)), with Ri(P (s)Q(t)) the length of link P (s)Q(t) within region Ri, and wi a positive weight associated with Ri. For simplicity, we assume R is triangulated and P,Q lie on the boundaries of the weighted regions. Fig. 1 depicts an example of this problem.