Approximability of the Discrete Fréchet Distance

The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds. In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399. This raises the question of how well we can approximate the Fréchet distance (of two given d-dimensional point sequences of length n) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be 2Θ(n). Moreover, we design an α-approximation algorithm that runs in time O(n logn + n2/α), for any α ∈ [1, n]. Hence, an n-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any ε > 0. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems – Geometrical problems and computations