Four Soviets Walk the DogImproved Bounds for Computing the Fréchet Distance
Abstract
Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original $O(n^2 \log n)$ algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here, $n$ denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUMhard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ on a pointer machine and in time $O(n^2(\log\log n)^2)$ on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth $O(n^{2\varepsilon})$, for some $\varepsilon > 0$. We believe that this reveals an intriguing new aspect of this wellstudied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fréchet distance on a word RAM.
 Publication:

arXiv eprints
 Pub Date:
 September 2012
 arXiv:
 arXiv:1209.4403
 Bibcode:
 2012arXiv1209.4403B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms
 EPrint:
 34 pages, 15 figures. A preliminary version appeared in SODA 2014