Weighted Minimum Backward Frechet Distance

The minimum backward Fréchet distance (MBFD) problem is a natural optimization problem for the weak Fréchet distance, a variant of the well-known Fréchet distance. In this problem, a threshold ε and two polygonal curves, T1 and T2, are given. The objective is to find a pair of walks on T1 and T2, which minimizes the union of the portions of backward movements while the distance between the moving entities, at any time, is at most ε. In this paper, we generalize this model to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous. More specifically, each edge of T1 and T2 has an associated nonnegative weight. The cost of backtracking on an edge is the Euclidean length of backward movement on that edge multiplied by the corresponding weight. The objective is to find a pair of walks that minimizes the sum of the costs on the edges of the curves, while guaranteeing that the curves remain at weak Fréchet distance ε. We propose an exact algorithm whose run time and space complexity is O(n3), where n is the maximum number of the edges of T1 and T2.