Fréchet distance with speed limits

In this paper, we introduce a new generalization of the well-known Fréchet distance between two polygonal curves, and provide an efficient algorithm for computing it. The classical Fréchet distance between two polygonal curves corresponds to the maximum distance between two point objects that traverse the curves with arbitrary non-negative speeds. Here, we consider a problem instance in which the speed of traversal along each segment of the curves is restricted to be within a specified range. We provide an efficient algorithm that decides in O(n logn) time whether the Fréchet distance with speed limits between two polygonal curves is at most ε, where n is the number of segments in the curves, and ε > 0 is an input parameter. We then use our solution to this decision problem to find the exact Fréchet distance with speed limits in O(n log n) time.