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

@article{Buchin2017FourSW, title={Four Soviets Walk the Dog: Improved Bounds for Computing the Fr{\'e}chet Distance}, author={Kevin Buchin and Maike Buchin and Wouter Meulemans and Wolfgang Mulzer}, journal={Discrete \& Computational Geometry}, year={2017}, volume={58}, pages={180-216} }

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 $$O(n^2 \log n)$$O(n2logn) 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…

## 36 Citations

### 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.

### 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.

### The k-Fréchet Distance: How to Walk Your Dog While Teleporting

- Computer Science, MathematicsISAAC
- 2019

A new distance measure for comparing polygonal chains: the k-Frechet distance, closely related to the well-studied Frechet distance but detects similarities between curves that resemble each other only piecewise.

### 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.

### Fast fréchet distance between curves with long edges

- Computer Science, MathematicsIWISC
- 2018

It is shown that for curves with long edges the Fréchet distance computations become easier, and three main results are proved for the case when all edges of both curves are long compared to the FrÉchet distance.

### Approximating the (continuous) Fréchet distance

- Computer Science, MathematicsSoCG
- 2021

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

### Computing the Fréchet Distance Between Uncertain Curves in One Dimension

- Mathematics, Computer ScienceWADS
- 2021

While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance – and indeed it can easily prove that the problem is NP-hard in 2D – the optimal placed vertices in 1D can be computed in polynomial time.

### 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.

### On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance

- Computer ScienceSoCG
- 2018

It is shown that computing an optimal simplification using the undirected Hausdorff distance is NP-hard, and how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible is analyzed.

### On the Discrete Fréchet Distance in a Graph

- Computer Science, MathematicsSoCG
- 2022

A conditional lower bound is provided showing that the Fréchet distance, or even its 1.01-approximation, between arbitrary paths in a weighted planar graph cannot be computed in O((|P | · |Q|)1−δ) time for any δ > 0 unless the Orthogonal Vector Hypothesis fails.

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