7 research outputs found
Fast Frechet Distance Between Curves With Long Edges
Computing the Fr\'echet distance between two polygonal curves takes roughly
quadratic time. In this paper, we show that for a special class of curves the
Fr\'echet distance computations become easier. Let and be two polygonal
curves in with and vertices, respectively. We prove four
results for the case when all edges of both curves are long compared to the
Fr\'echet distance between them: (1) a linear-time algorithm for deciding the
Fr\'echet distance between two curves, (2) an algorithm that computes the
Fr\'echet distance in time, (3) a linear-time
-approximation algorithm, and (4) a data structure that supports
-time decision queries, where is the number of vertices of
the query curve and the number of vertices of the preprocessed curve
Realizability of Free Spaces of Curves
The free space diagram is a popular tool to compute the well-known Fr\'echet
distance. As the Fr\'echet distance is used in many different fields, many
variants have been established to cover the specific needs of these
applications. Often, the question arises whether a certain pattern in the free
space diagram is "realizable", i.e., whether there exists a pair of polygonal
chains whose free space diagram corresponds to it. The answer to this question
may help in deciding the computational complexity of these distance measures,
as well as allowing to design more efficient algorithms for restricted input
classes that avoid certain free space patterns. Therefore, we study the inverse
problem: Given a potential free space diagram, do there exist curves that
generate this diagram?
Our problem of interest is closely tied to the classic Distance Geometry
problem. We settle the complexity of Distance Geometry in ,
showing -hardness. We use this to show that for curves in
, the realizability problem is
-complete, both for continuous and for discrete Fr\'echet
distance. We prove that the continuous case in is only weakly
NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is
fixed-parameter tractable. Interestingly, for the discrete case in
, we show that the problem becomes solvable in polynomial time.Comment: 26 pages, 12 figures, 1 table, International Symposium on Algorithms
And Computations (ISAAC 2023
Global Curve Simplification
Due to its many applications, curve simplification is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve P with n vertices, the goal is to find another polygonal curve P' with a smaller number of vertices such that P' is sufficiently similar to P. Quality guarantees of a simplification are usually given in a local sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work we aim to provide a systematic overview of curve simplification problems under global distance measures that bound the distance between P and P'. We consider six different curve distance measures: three variants of the Hausdorff distance and three variants of the Fréchet distance. And we study different restrictions on the choice of vertices for P'. We provide polynomial-time algorithms for some variants of the global curve simplification problem, and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area
On optimal min-# curve simplification problem
In this paper we consider the classical min--\# curve simplification problem in three different variants. Let , be a polygonal curve with vertices in , and be a distance measure. We aim to simplify by another polygonal curve with minimum number of vertices satisfying . We obtain three main results for this problem: (1) An -time algorithm when is the Fr\'echet distance and vertices in are selected from a subsequence of vertices in . (2) An NP-hardness result for the case that is the directed Hausdorff distance from to and the vertices of can lie anywhere on while respecting the order of edges along . (3) For any , an -time algorithm that computes whose vertices can lie anywhere in the space and whose Fr\'echet distance to is at most with at most links, where is the number of links in the optimal simplified curve and hides polynomial factors of
Global curve simplification
Due to its many applications, curve simplification is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve P with n vertices, the goal is to find another polygonal curve P' with a smaller number of vertices such that P' is sufficiently similar to P. Quality guarantees of a simplification are usually given in a local sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work we aim to provide a systematic overview of curve simplification problems under global distance measures that bound the distance between P and P'. We consider six different curve distance measures: three variants of the Hausdorff distance and three variants of the Fréchet distance. And we study different restrictions on the choice of vertices for P'. We provide polynomial-time algorithms for some variants of the global curve simplification problem, and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area