# Approximability of the discrete Fréchet distance

@inproceedings{Bringmann2015ApproximabilityOT, title={Approximability of the discrete Fr{\'e}chet distance}, author={Karl Bringmann and Wolfgang Mulzer}, booktitle={Journal of Computational Geometry}, year={2015} }

The Frechet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds. In this paper, we study the approximability of the discrete Frechet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new…

## 70 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.

### 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 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.

### 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.

### 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 with a Retractable Leash

- Computer ScienceESA
- 2013

This work presents a novel approach that avoids the detour through the decision version of the Fréchet distance between polygonal curves and gives the first quadratic time algorithm.

### A Subquadratic $n^\epsilon$-approximation for the Continuous Fr\'echet Distance

- Computer Science, Mathematics
- 2022

An O ( α )-approximate algorithm that runs in O (( n + mn/α ) log 3 n ) time for any α ∈ [1, n ], assuming m ≤ n and constant dimension d .

### 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.

### Tight Bounds for Approximate Near Neighbor Searching for Time Series under the Fréchet Distance

- Computer Science, MathematicsSODA
- 2022

Holds hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which is achieved by proving hardness of the one- sided sparse version for any super-constant value of k.

### An improved approximation algorithm for the discrete Fréchet distance

- Computer ScienceInf. Process. Lett.
- 2018

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