scholarly journals Fast Fréchet Distance Between Curves with Long Edges

2019 ◽  
Vol 29 (02) ◽  
pp. 161-187
Author(s):  
Joachim Gudmundsson ◽  
Majid Mirzanezhad ◽  
Ali Mohades ◽  
Carola Wenk
Keyword(s):  
Data Structure ◽  
Approximation Algorithm ◽  
Special Class ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Fréchet Distance ◽  
Frechet Distance ◽  
Polygonal Curves

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

2012 ◽  
Vol 22 (01) ◽  
pp. 27-44 ◽  
Author(s):  
HEE-KAP AHN ◽  
CHRISTIAN KNAUER ◽  
MARC SCHERFENBERG ◽  
LENA SCHLIPF ◽  
ANTOINE VIGNERON
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Fréchet Distance ◽  
Planar Case ◽  
Running Time ◽  
Frechet Distance ◽  
Axis Parallel ◽  
Polygonal Curves ◽  
Orthogonal Case ◽  
Improved Algorithm

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

AN OPTIMAL DATA STRUCTURE FOR SHORTEST RECTILINEAR PATH QUERIES IN A SIMPLE RECTILINEAR POLYGON

1996 ◽  
Vol 06 (02) ◽  
pp. 205-225 ◽  
Author(s):  
SVEN SCHUIERER
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
Linear Time ◽  
Time Algorithm ◽  
Query Time ◽  
Linear Time Algorithm ◽  
Link Length ◽  
N Storage ◽  
Path Queries ◽  
Rectilinear Polygon

We present a data structure that allows to preprocess a rectilinear polygon with n vertices such that, for any two query points, the shortest path in the rectilinear link or L1-metric can be reported in time O( log n+k) where k is the link length of the shortest path. If only the distance is of interest, the query time reduces to O( log n). Furthermore, if the query points are two vertices, the distance can be reported in time O(1) and a shortest path can be constructed in time O(1+k). The data structure can be computed in time O(n) and needs O(n) storage. As an application we present a linear time algorithm to compute the diameter of a simple rectilinear polygon w.r.t. the L1-metric.

scholarly journals FRÉCHET DISTANCE PROBLEMS IN WEIGHTED REGIONS

2010 ◽  
Vol 02 (02) ◽  
pp. 161-179 ◽  
Author(s):  
YAM KI CHEUNG ◽  
OVIDIU DAESCU
Keyword(s):  
Approximation Algorithm ◽  
Shortest Path ◽  
Line Segment ◽  
Fréchet Distance ◽  
Weighted Regions ◽  
Frechet Distance ◽  
Polygonal Curves ◽  
Polyhedral Obstacles ◽  
Distance Problem ◽  
Planar Subdivisions

We discuss two versions of the Fréchet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases, we give algorithms for finding a (1 + ∊)-factor approximation of the Fréchet distance between two polygonal curves. We also consider the Fréchet distance between two polygonal curves among polyhedral obstacles in [Formula: see text] (1/∞ weighted region problem) and present a (1 + ∊)-factor approximation algorithm.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

A Linear time algorithm for deciding security

1976 ◽  
Author(s):  
A. K. Jones ◽  
R. J. Lipton ◽  
L. Snyder
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm

AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

2000 ◽  
Vol 11 (03) ◽  
pp. 365-371 ◽  
Author(s):  
LJUBOMIR PERKOVIĆ ◽  
BRUCE REED
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Tree Decomposition ◽  
Linear Time Algorithm ◽  
Input Graph ◽  
Primary Motivation ◽  
Small Width ◽  
Tree Decompositions ◽  
Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

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