# 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}
}
• Published in
Journal of Computational…
11 June 2015
• Computer Science, Mathematics
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…

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## References

SHOWING 1-10 OF 22 REFERENCES

• K. Bringmann
• Computer Science, Mathematics
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
• 2014
It is shown that the Fréchet distance cannot be computed in strongly subquadratic time, i.e., in time O(n2-&delta;) for any delta > 0.001-approximation, which means that finding faster algorithms is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subaquadratic algorithm can be considered unlikely.
• Computer Science
SIAM J. Comput.
• 2014
This work presents the first subquadratic algorithm for computing the discrete Frechet distance between two sequences of points in the plane, and uses the geometry of the problem in a subtle way to encode legal positions of the frogs as states of a finite automaton.
• Computer Science, Mathematics
Int. 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 $$n^{o(1)}$$).
• Computer Science
ESA
• 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.
• Mathematics, Computer Science
SODA
• 2014
A randomized algorithm to compute the Frechet distance between two polygonal curves in time [EQUATION] on a pointer machine and in time O(n2(log log n)2) on a word RAM is given and there is evidence that the decision problem may not be 3SUM-hard after all.
• Computer Science, Mathematics
Int. J. Comput. Geom. Appl.
• 1995
As a measure for the resemblance of curves in arbitrary dimensions we consider the so-called Frechet-distance, which is compatible with parametrizations of the curves. For polygonal chains P and Q
• Computer Science, Economics
• 1994
A discrete variation of the Fréchet distance that provides good approximations of the continuous measure and can be efficiently computed using a simple algorithm is presented.
• Computer Science
STOC '13
• 2013
This paper presents the first improvement over the diameter approximation algorithm of Aingworth et.
• Computer Science, Mathematics
Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
• 1998
A generalized reduction which is called sub-exponential reduction family (SERF) that preserves sub- Exponential complexity for NP-search problems and shows that Circuit-SAT is SERF-complete for all NP- search problems, and that for any fixed k, k-S AT,k-Colorability, k -Set Cover Independent Set, Clique, Vertex Cover are SERF -complete for the class SNP of search problems expressible by second order existential formulas whose first order
• Computer Science, Mathematics
SODA
• 2015
This paper extends the polynomial method to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.