Computing the Discrete Fréchet Distance with Imprecise Input

Abstract

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time $2^{O(d^2)} m^2n^2\log^2(mn)$ the Fréchet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O( mn log²(mn) + (m ² + n ²)log(mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L _∞ distance, we give an O(dmn log(dmn))-time algorithm.

We also give efficient O(dmn)-time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in “realistic” settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).