Computing the discrete Fréchet distance upper bound of imprecise input is NP-hard

The Fréchet distance is a natural measure of similarity between two curves [4]. The Fréchet distance between two curves is often referred to as the “dog-leash distance”. Alt and Godau [4] presented an algorithm to compute the Fréchet distance between two polygonal curves of n and m vertices in O(nm log(nm)) time. There has been a lot of applications using the Fréchet distance to do pattern/curve matching. A slightly simpler version of the Fréchet distance is the discrete Fréchet distance, where only the vertices of polygonal curves are considered. It takes O(mn) time to compute the discrete Fréchet distance using a standard dynamic programming technique [6]. The discrete Fréchet distance with shortcuts was also studied by Avraham et al. recently [5]. A novel technique, based on distance selection, was designed to compute the discrete Fréchet distance with shortcuts efficiently. The computational geometry with imprecise objects has drawn much interest to researchers since a few years ago. There are two models: one is the continuous model, where a precise point is selected from an erroneous region (say a disk, or rectangle) [7]; the other is the discrete or color-spanning model, where a precise point is selected from several discrete objects with the same color and all colors must be selected [1]. A lot of algorithms have been designed to handle imprecise geometric problems on both models. Ahn et al. studied the problem of computing the discrete Fréchet distance between two imprecise point sequences, and gave an efficient algorithm for computing the lower bound (of the distance) and efficient approximation algorithms for the corresponding upper bound (under a realistic assumption) [2,3]. It is unknown whether computing the discrete Fréchet distance upper bound for imprecise input is solvable in polynomial time or not, so Ahn et al. left that as an open problem [2,3]. In this paper, we proved that the problem is in fact NP-hard. We also consider the same problem under the discrete Fréchet distance with shortcuts and give efficient polynomial-time solutions. Main results: We study the problem of computing the upper bound of the discrete Fréchet distance for imprecise input, and prove that the problem is NP-hard. This solves an open problem posted in 2010 by Ahn et al. If shortcuts are allowed, we show that the upper bound of the discrete Fréchet distance with shortcuts for imprecise input can be computed in polynomial time and we present several efficient algorithms1.