# Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails

@article{Bringmann2014WhyWT, title={Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails}, author={Karl Bringmann}, journal={2014 IEEE 55th Annual Symposium on Foundations of Computer Science}, year={2014}, pages={661-670} }

The Fréchet 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 Fréchet 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(n2 / log n), but no matching lower bounds are…

## 236 Citations

### Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

- Computer ScienceDiscret. Comput. Geom.
- 2017

This work gives a randomized algorithm to compute the Fréchet distance between two polygonal curves in time and shows that there exists an algebraic decision tree for the decision problem of depth, for some varepsilon > 0, which reveals an intriguing new aspect of this well-studied problem.

### When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fréchet Distance under Translation

- Computer ScienceESA
- 2020

The solution combines fast, but inexact tools from continuous optimization with exact, but expensive algorithms from computational geometry to obtain an exact decision algorithm for the Frechet distance under translation.

### Approximating the (continuous) Fr\'echet distance

- Computer Science
- 2020

This work describes the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fréchet distance between two polygonal chains, and describes how to turn any approximate decision procedure for the FrÉchet distance into an approximate optimization algorithm whose approximation ratio is the same up to arbitrarily small constant factors.

### APPROXIMABILITY OF THE DISCRETE FRÉCHET

- Computer Science, Mathematics
- 2016

A new conditional lower bound is presented showing 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.

### Approximability of the discrete Fréchet distance

- Computer Science, MathematicsJ. Comput. Geom.
- 2015

A new conditional lower bound is presented showing 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.

### The k-Fréchet distance

- Computer Science, MathematicsArXiv
- 2019

It is shown that deciding this distance measure turns out to be NP-complete, which is interesting since both (weak) Fréchet and Hausdorff distance are computable in polynomial time.

### Improved Approximation for Fréchet Distance on c-Packed Curves Matching Conditional Lower Bounds

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 2017

This paper presents an improved algorithm with time complexity Open image in new window that improves upon the algorithm by Driemel et al. and matches the conditional lower bound (up to lower order factors of the form \(n^{o(1)}\)).

### On Computing the k-Shortcut Fréchet Distance

- Mathematics, Computer ScienceICALP
- 2022

A complexity analysis for the shortcut Fréchet distance, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences, and shows that efficient approximate decider algorithms are possible, even when k is large.

### Discrete Fréchet Distance under Translation

- Computer Science, MathematicsACM Trans. Algorithms
- 2021

This article provides evidence that constructing the arrangement of size Õ(N4) is necessary in the worst case by proving a conditional lower bound of n4 - o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.

### Computing the Fréchet Distance between Real-Valued Surfaces

- Computer ScienceSODA
- 2017

This paper measures the distance between terrains based solely on the height function, and shows that in this case computing the Frechet distance between f and g is in NP, and defines an intermediate distance, between contour trees, which is shown to be NP-complete to compute.

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The notion of a more robust Frechet distance, where one is allowed to shortcut between vertices of one of the curves, is introduced, which is a natural approach for handling noise, in particular batched outliers.

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