Related papers: Geodesic Fr\'echet Distance Inside a Simple Polygo…
The Fr\'echet distance is a popular similarity measure that is well-understood for polygonal curves in $\mathbb{R}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved…
Let $P$ be a polygon with $k$ vertices. Let $R$ and $B$ be two simple, interior disjoint curves on the boundary of $P$, with $n$ and $m$ vertices. We show how to compute the Fr\'echet distance between $R$ and $B$ using the geodesic…
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…
In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the…
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…
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…
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fr\'echet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely:…
Let $m$ and $n$ be the numbers of vertices of two polygonal curves in $\mathbb{R}^d$ for any fixed $d$ such that $m \leq n$. Since it was known in 1995 how to compute the Fr\'{e}chet distance of these two curves in $O(mn\log (mn))$ time, it…
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…
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.…
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…
Since its introduction to computational geometry by Alt and Godau in 1992, the Fr\'echet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal…
Computing the Fr\'{e}chet distance for surfaces is a surprisingly hard problem and the only known algorithm is limited to computing it between flat surfaces. We adapt this algorithm to create one for computing the Fr\'{e}chet distance for a…
The discrete Fr\'echet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fr\'echet distance under translation, which enables detection of similar movement patterns in different spatial…
We study several polygonal curve problems under the Fr\'{e}chet distance via algebraic geometric methods. Let $\mathbb{X}_m^d$ and $\mathbb{X}_k^d$ be the spaces of all polygonal curves of $m$ and $k$ vertices in $\mathbb{R}^d$,…
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'…
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…
We study the problem of sub-trajectory nearest-neighbor queries on polygonal curves under the continuous Fr\'echet distance. Given an $n$ vertex trajectory $P$ and an $m$ vertex query trajectory $Q$, we seek to report a vertex-aligned…
The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric…
We study the shortcut Fr\'{e}chet distance, a natural variant of the Fr\'{e}chet distance, that allows us to take shortcuts from and to any point along one of the curves. The classic Fr\'echet distance is a bottle-neck distance measure and…