Approximability of the discrete Fréchet distance

The Frechet 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 Frechet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound that strongly subquadratic algorithms for the discrete Frechet 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 Frechet 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 2^Theta(n). Moreover, we design an alpha-approximation algorithm that runs in time O(n log n + n^2 / alpha), for any alpha in [1, n]. Hence, an n^epsilon-approximation of the Frechet distance can be computed in strongly subquadratic time, for any epsilon > 0.