Optimal Covering and Hitting of Line Segments by Two Axis-Parallel Squares

2017 ◽  
pp. 457-468 ◽  
Author(s):  
Sanjib Sadhu ◽  
Sasanka Roy ◽  
Subhas C. Nandy ◽  
Suchismita Roy
Keyword(s):  
Line Segments ◽  
Axis Parallel

scholarly journals FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS

2011 ◽  
Vol 21 (02) ◽  
pp. 189-213 ◽  
Author(s):  
VLADIMIR ESTIVILL-CASTRO ◽  
APICHAT HEEDNACRAM ◽  
FRANCIS SURAWEERA
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Fixed Parameter Tractable ◽  
Line Segments ◽  
Fpt Algorithms ◽  
Fixed Parameter ◽  
Axis Parallel ◽  
Np Complete ◽  
Number Of Bends ◽  
Straight Lines

This paper discusses the κ-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer κ, and we are asked whether we can travel in straight lines through these n points with at most κ bends. There are a number of applications where minimizing the number of bends in the tour is desirable because bends are considered very costly. We prove that this problem is fixed-parameter tractable (FPT). The proof is based on the kernelization approach. We also consider the RECTILINEAR κ-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel. 1 Note that a rectilinear tour with κ bends is a cover with κ-line segments, and therefore a cover by lines. We introduce two types of constraints derived from the distinction between line-segments and lines. We derive FPT-algorithms with different techniques and improved time complexity for these cases.

THE MINIMUM GUARDING TREE PROBLEM

2014 ◽  
Vol 06 (01) ◽  
pp. 1450011 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
JOSEPH S. B. MITCHELL ◽  
PAWEL ŻYLIŃSKI
Keyword(s):  
Approximation Algorithms ◽  
Nonempty Subset ◽  
Parallel Line ◽  
Minimum Length ◽  
Np Hard ◽  
Orthogonal Axis ◽  
Line Segments ◽  
Parallel Lines ◽  
Simple Alternative ◽  
Axis Parallel

Given a set ℒ of non-parallel lines in the plane and a nonempty subset ℒ′ ⊆ ℒ, a guarding tree for ℒ′ is a tree contained in the union of the lines in ℒ such that if a mobile guard (agent) runs on the edges of the tree, all lines in ℒ′ are visited by the guard. Similarly, given a connected arrangement 𝒮 of line segments in the plane and a nonempty subset 𝒮′ ⊆ 𝒮, we define a guarding tree for 𝒮′. The minimum guarding tree problem for a given set of lines or line segments is to find a minimum-length guarding tree for the input set. We provide a simple alternative (to [N. Xu, Complexity of minimum corridor guarding problems, Inf. Process. Lett.112(17–18) (2012) 691–696.]) proof of the problem of finding a guarding tree of minimum length for a set of orthogonal (axis-parallel) line segments in the plane. Then, we present two approximation algorithms with factors 2 and 3.98, respectively, for computing a minimum guarding tree for a subset of a set of n arbitrary non-parallel lines in the plane; their running times are O(n8) and O(n6 log n), respectively. Finally, we show that this problem is NP-hard for lines in 3-space.

scholarly journals CONSTRUCTING THE CITY VORONOI DIAGRAM FASTER

2008 ◽  
Vol 18 (04) ◽  
pp. 275-294 ◽  
Author(s):  
ROBERT GÖRKE ◽  
CHAN-SU SHIN ◽  
ALEXANDER WOLFF
Keyword(s):  
Voronoi Diagram ◽  
Transportation Network ◽  
Parallel Line ◽  
Line Segments ◽  
Quickest Path ◽  
Manhattan Metric ◽  
Voronoi Regions ◽  
Axis Parallel ◽  
The City ◽  
And Storage

Given a set P of n point sites in the plane, the city Voronoi diagram subdivides the plane into the Voronoi regions of the sites, with respect to the city metric. This metric is induced by quickest paths according to the Manhattan metric and an accelerating transportation network that consists of c non-intersecting axis-parallel line segments. We describe an algorithm that constructs the city Voronoi diagram (including quickest path information) using O((c+n) polylog (c+n)) time and storage by means of a wavefront expansion. For [Formula: see text] our algorithm is faster than an algorithm by Aichholzer et al., which takes O(n log n + c2 log c) time.

scholarly journals Linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares

2019 ◽  
Vol 769 ◽  
pp. 63-74 ◽  
Author(s):  
Sanjib Sadhu ◽  
Sasanka Roy ◽  
Subhas C. Nandy ◽  
Suchismita Roy
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Line Segments ◽  
Axis Parallel

On Covering Points with Minimum Turns

2015 ◽  
Vol 25 (01) ◽  
pp. 1-9 ◽  
Author(s):  
Minghui Jiang
Keyword(s):  
Np Hard ◽  
Line Segments ◽  
Polygonal Chain ◽  
Axis Parallel ◽  
A Chain ◽  
Set Of Points ◽  
The Given

We study the problem of finding a polygonal chain of line segments to cover a set of points in ℝd, d≥2, with the goal of minimizing the number of links or turns in the chain. A chain of line segments that covers all points in the given set is called a covering tour if the chain is closed, and is called a covering path if the chain is open. A covering tour or a covering path is rectilinear if all segments in the chain are axis-parallel. We prove that the two problems Minimum-Link Rectilinear Covering Tour and Minimum-Link Rectilinear Covering Path are both NP-hard in ℝ10.

The limits of strain and lattice parameter measurements by CBED

1989 ◽  
Vol 47 ◽  
pp. 518-519
Author(s):  
Hamish L. Fraser
Keyword(s):  
Electron Beam ◽  
Mirror Symmetry ◽  
Local Symmetry ◽  
Lattice Parameter ◽  
Growth Direction ◽  
Surface Relaxation ◽  
Strained Layer ◽  
Axis Parallel ◽  
Local Lattice ◽  
Strained Layer Superlattice

The topic of strain and lattice parameter measurements using CBED is discussed by reference to several examples. In this paper, only one of these examples is referenced because of the limitation of length. In this technique, scattering in the higher order Laue zones is used to determine local lattice parameters. Work (e.g. 1) has concentrated on a model strained-layer superlattice, namely Si/Gex-Si1-x. In bulk samples, the strain is expected to be tetragonal in nature with the unique axis parallel to [100], the growth direction. When CBED patterns are recorded from the alloy epi-layers, the symmetries exhibited by the patterns are not tetragonal, but are in fact distorted from this to lower symmetries. The spatial variation of the distortion close to a strained-layer interface has been assessed. This is most readily noted by consideration of Fig. 1(a-c), which show enlargements of CBED patterns for various locations and compositions of Ge. Thus, Fig. 1(a) was obtained with the electron beam positioned in the center of a 5Ge epilayer and the distortion is consistent with an orthorhombic distortion. When the beam is situated at about 150 nm from the interface, the same part of the CBED pattern is shown in Fig. 1(b); clearly, the symmetry exhibited by the mirror planes in Fig. 1 is broken. Finally, when the electron beam is positioned in the center of a 10Ge epilayer, the CBED pattern yields the result shown in Fig. 1(c). In this case, the break in the mirror symmetry is independent of distance form the heterointerface, as might be expected from the increase in the mismatch between 5 and 10%Ge, i.e. 0.2 to 0.4%, respectively. From computer simulation, Fig.2, the apparent monocline distortion corresponding to the 5Ge epilayer is quantified as a100 = 0.5443 nm, a010 = 0.5429 nm and a001 = 0.5440 nm (all ± 0.0001 nm), and α = β = 90°, γ = 89.96 ± 0.02°. These local symmetry changes are most likely due to surface relaxation phenomena.

Discrimination of Occluded Line Segments by Pigeons

2009 ◽  
Author(s):  
Robert G. Cook ◽  
Carl Erick Hagmann
Keyword(s):  
Line Segments
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