Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails

TL;DR

This paper establishes that computing the Frechet distance between two curves cannot be done in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails, indicating inherent computational difficulty.

Contribution

The paper provides the first conditional lower bounds for the exact and approximate computation of the Frechet distance based on SETH, extending to various variants and input assumptions.

Findings

No strongly subquadratic algorithms for Frechet distance unless SETH fails

Conditional lower bounds for 1.001-approximation of Frechet distance

Results apply to continuous, discrete, and c-packed curves

Abstract

The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Frechet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n^2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Frechet distance cannot be computed in strongly subquadratic time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. Our result holds for both the continuous and the discrete Frechet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves).