# Walking the Dog Fast in Practice: Algorithm Engineering of the Fr\'echet Distance

## TL;DR

This paper introduces a highly efficient, certifying implementation for the Fréchet distance decision procedure, significantly improving practical performance and addressing the gap between theoretical hardness and real-world applications.

## Contribution

It presents a fast, certifying algorithm for the Fréchet distance decision problem, enhancing practical efficiency and empirical analysis on realistic data.

## Key findings

- Up to 100x faster decision procedure compared to previous methods.
- Up to 30x faster queries in near-neighbor data structures.
- Empirical validation on diverse datasets including handwritten characters and GPS trajectories.

## Abstract

The Fr\'echet distance provides a natural and intuitive measure for the popular task of computing the similarity of two (polygonal) curves. While a simple algorithm computes it in near-quadratic time, a strongly subquadratic algorithm cannot exist unless the Strong Exponential Time Hypothesis fails. Still, fast practical implementations of the Fr\'echet distance, in particular for realistic input curves, are highly desirable. This has even lead to a designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge was to implement a near-neighbor data structure under the Fr\'echet distance. The bottleneck of the top three implementations turned out to be precisely the decision procedure for the Fr\'echet distance. In this work, we present a fast, certifying implementation for deciding the Fr\'echet distance, in order to (1) complement its pessimistic worst-case hardness by an empirical analysis on realistic input data and to (2) improve the state of the art for the GIS Cup challenge. We experimentally evaluate our implementation on a large benchmark consisting of several data sets (including handwritten characters and GPS trajectories). Compared to the winning implementation of the GIS Cup, we obtain running time improvements of up to more than two orders of magnitude for the decision procedure and of up to a factor of 30 for queries to the near-neighbor data structure.