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The Fr\'echet distance is a similarity measure between two curves $A$ and $B$: Informally, it is the minimum length of a leash required to connect a dog, constrained to be on $A$, and its owner, constrained to be on $B$, as they walk…

Computational Geometry · Computer Science 2012-04-25 Pankaj K. Agarwal , Rinat Ben Avraham , Haim Kaplan , Micha Sharir

The similarity of two polygonal curves can be measured using the Fr\'echet distance. We introduce the notion of a more robust Fr\'echet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural…

Computational Geometry · Computer Science 2013-06-19 Anne Driemel , Sariel Har-Peled

The Fr\'{e}chet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network),…

Computational Geometry · Computer Science 2024-11-20 Anne Driemel , Ivor van der Hoog , Eva Rotenberg

The Frechet 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…

Computational Geometry · Computer Science 2014-08-01 Karl Bringmann

We show that a variant of the continuous Frechet distance between polygonal curves can be computed using essentially the same algorithm used to solve the discrete version. The new variant is not necessarily monotone, but this shortcoming…

Computational Geometry · Computer Science 2026-01-01 Sariel Har-Peled , Benjamin Raichel , Eliot W. Robson

We introduce the discrete Fr\'echet gap and its variants as an alternative measure of similarity between polygonal curves. We believe that for some applications the new measure (and its variants) may better reflect our intuitive notion of…

Computational Geometry · Computer Science 2015-06-17 Omrit Filtser , Matthew J. Katz

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…

Computational Geometry · Computer Science 2019-01-08 Karl Bringmann , Marvin Künnemann , André Nusser

The discrete Fr\'echet distance is a useful similarity measure for comparing two sequences of points $P=(p_1,\ldots, p_m)$ and $Q=(q_1,\ldots,q_n)$. In many applications, the quality of the matching can be improved if we let $Q$ undergo…

Computational Geometry · Computer Science 2015-01-16 Rinat Ben Avraham , Haim Kaplan , Micha Sharir

The classical measure of similarity between two polygonal chains in Euclidean space is the Fr\'echet distance, which corresponds to the coordinated motion of two mobile agents along the chains while minimizing their maximum distance. As…

Computational Geometry · Computer Science 2019-06-20 Hugo A. Akitaya , Leonie Ryvkin , Csaba D. Tóth

The Fr\'echet distance is a computational mainstay for comparing polygonal curves. The Fr\'echet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in…

Computational Geometry · Computer Science 2025-01-23 Lotte Blank , Jacobus Conradi , Anne Driemel , Benedikt Kolbe , André Nusser , Marena Richter

All known algorithms for the Fr\'echet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values.…

Computational Geometry · Computer Science 2016-08-11 Kevin Buchin , Maike Buchin , Rolf van Leusden , Wouter Meulemans , Wolfgang Mulzer

Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves'…

Computational Geometry · Computer Science 2020-08-18 Karl Bringmann , Marvin Künnemann , André Nusser

The Fr\'echet distance is a commonly used similarity measure between curves. It is known how to compute the continuous Fr\'echet distance between two polylines with $m$ and $n$ vertices in $\mathbb{R}^d$ in $O(mn (\log \log n)^2)$ time;…

Computational Geometry · Computer Science 2022-08-29 Thijs van der Horst , Marc van Kreveld , Tim Ophelders , Bettina Speckmann

The \emph{Fr\'echet distance} is a well studied similarity measures between curves. The \emph{discrete Fr\'echet distance} is an analogous similarity measure, defined for a sequence $A$ of $m$ points and a sequence $B$ of $n$ points, where…

Computational Geometry · Computer Science 2016-09-09 Rinat Ben Avraham , Omrit Filtser , Haim Kaplan , Matthew J. Katz , Micha Sharir

The Fr\'echet distance is a distance measure between trajectories in $\Bbb{R}^d$ or walks in a graph $G$. Given constant-time shortest path queries, the Discrete Fr\'echet distance $D_G(P, Q)$ between two walks $P$ and $Q$ can be computed…

Computational Geometry · Computer Science 2025-07-08 Ivor van der Hoog , Thijs van der Horst , Eva Rotenberg , Lasse Wulf

The Fr\'echet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min-max formulation that considers all direction-preserving continuous bijections of the two curves. Because of its susceptibility to…

Computational Geometry · Computer Science 2022-02-24 Jacobus Conradi , Anne Driemel

We describe the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fr\'echet distance between two polygonal chains. Specifically, let $P$ and $Q$ be two polygonal chains with…

Computational Geometry · Computer Science 2021-03-30 Connor Colombe , Kyle Fox

The Fr\'{e}chet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently…

Computational Geometry · Computer Science 2013-04-24 Jean-Lou De Carufel , Amin Gheibi , Anil Maheshwari , Jörg-Rüdiger Sack , Christian Scheffer

The Fr\'echet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Fr\'echet distance between them. This variant is called the Translation…

Computational Geometry · Computer Science 2021-08-13 Joachim Gudmundsson , André van Renssen , Zeinab Saeidi , Sampson Wong

We study the problem of constructing a data structure that can store a two-dimensional polygonal curve $P$, such that for any query segment $\overline{ab}$ one can efficiently compute the Fr\'{e}chet distance between $P$ and…

Computational Geometry · Computer Science 2022-03-04 Maike Buchin , Ivor van der Hoog , Tim Ophelders , Lena Schlipf , Rodrigo I. Silveira , Frank Staals
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