scholarly journals Dynamic Algorithms for Visibility Polygons in Simple Polygons

2020 ◽  
Vol 30 (01) ◽  
pp. 51-78
Author(s):  
R. Inkulu ◽  
K. Sowmya ◽  
Nitish P. Thakur
Keyword(s):  
Line Segment ◽  
Time Complexity ◽  
Simple Polygon ◽  
Incremental Algorithm ◽  
Dynamic Algorithms ◽  
Insertions And Deletions ◽  
Shortest Path Trees ◽  
Query Algorithm ◽  
Fixed Line ◽  
Visibility Polygons

We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and deletions to the simple polygon. A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point [Formula: see text] due to the insertion (resp. deletion) of vertex [Formula: see text] to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial changes needed to the visibility polygon of [Formula: see text] due to the insertion (resp. deletion) of [Formula: see text]. An output-sensitive query algorithm to answer the visibility polygon query corresponding to any point [Formula: see text] in [Formula: see text] amid vertex insertions and deletions to the simple polygon. If [Formula: see text] is not exterior to the current simple polygon, then the visibility polygon of [Formula: see text] is computed. Otherwise, our algorithm outputs the visibility polygon corresponding to the exterior visibility of [Formula: see text]. An incremental algorithm to maintain the weak visibility polygon of a fixed-line segment located interior to the simple polygon amid vertex insertions to the simple polygon. The time complexity to update the weak visibility polygon of a line segment [Formula: see text] due to the insertion of vertex [Formula: see text] to the current simple polygon is expressed in terms of the sum of the number of combinatorial updates needed to the geodesic shortest path trees rooted at [Formula: see text] and [Formula: see text] due to the insertion of [Formula: see text]. An output-sensitive algorithm to compute the weak visibility polygon corresponding to any query line segment located interior to the simple polygon amid both the vertex insertions and deletions to the simple polygon. Each of these algorithms requires preprocessing the initial simple polygon. And, the algorithms that maintain the visibility polygon (resp. weak visibility polygon) compute the visibility polygon (resp. weak visibility polygon) with respect to the initial simple polygon during the preprocessing phase.

FINDING AN OPTIMAL BRIDGE BETWEEN TWO POLYGONS

2002 ◽  
Vol 12 (03) ◽  
pp. 249-261 ◽  
Author(s):  
XUEHOU TAN
Keyword(s):  
Hierarchical Structure ◽  
Shortest Path ◽  
Line Segment ◽  
Euclidean Distance ◽  
Geodesic Distance ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Search Trees ◽  
Bridge Problem ◽  
Path Queries

Let π(a,b) denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a,b), denoted by gd(a,b), in contrast with the Euclidean distance between a and b in the plane, denoted by d(a,b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge) that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′,p), d(p,q) and gd(q,q′), with any p′ ∈ P and any q′ ∈ Q, is minimized. We present an O(n log 3 n) time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n2) time bound. Our result is obtained by making substantial use of a hierarchical structure that consists of segment trees, range trees and persistent search trees, and a structure that supports dynamic ray shooting and shortest path queries as well.

A New Algorithm for Euclidean Shortest Path of Visiting Segments in a Polygon

2014 ◽  
Vol 644-650 ◽  
pp. 1891-1894
Author(s):  
Li Juan Wang ◽  
An Sheng Deng ◽  
Bo Jiang ◽  
Qi Wei
Keyword(s):  
Test Data ◽  
Shortest Path ◽  
Time Complexity ◽  
Simple Polygon ◽  
Rubber Band ◽  
Crossing Over ◽  
Euclidean Shortest Path ◽  
Adjacent Segments ◽  
Improved Algorithm

Let s and t be two points on the boundary of a simple polygon, how to compute the Euclidean shortest path between s and t which visits a sequence of segments given in the simple polygon is the problem to be discussed, especially, the situation of the adjacent segments intersect is the focus of our study. In this paper, we first analyze the degeneration applying rubber-band algorithm to solve the problem. Then based on rubber-band algorithm, we present an improved algorithm which can solve the degeneration by the method of crossing over two segments to deal with intersection and in our algorithm the adjacent segments order can be changed when they intersect. Particularly, we have implemented the algorithm and have applied a large of test data to test it. The experiments demonstrate that our algorithm is correct and efficient, and it has the same time complexity as the rubber-band algorithm.

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

1995 ◽  
Vol 05 (03) ◽  
pp. 243-256 ◽  
Author(s):  
DAVID RAPPAPORT
Keyword(s):  
Convex Hull ◽  
General Problem ◽  
Line Segment ◽  
Simple Polygon ◽  
Fixed Number ◽  
Line Segments ◽  
Geometric Objects ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

THE ANCHORED VORONOI DIAGRAM: STATIC, DYNAMIC VERSIONS AND APPLICATIONS

2006 ◽  
Vol 16 (04) ◽  
pp. 315-332
Author(s):  
J. M. DÍAZ-BÁÑEZ ◽  
F. GÓMEZ ◽  
I. VENTURA
Keyword(s):  
Fixed Point ◽  
Voronoi Diagram ◽  
Line Segment ◽  
Time Complexity ◽  
Optimization Problems ◽  
Dynamic Version

For a given set S of n points in the plane and a fixed point o, we introduce the Voronoi diagram of S anchored at o. It will be defined as an abstract Voronoi diagram that uses as bisectors the following curves. For each pair of points p, q in S, the bisecting curve between p and q is the locus of points x in the plane such that the line segment [Formula: see text] is equidistant to both p and q. We show that those bisectors have nice properties and, therefore, this new structure can be computed in O(n log n) time and O(n) space both for nearest-site and furthest-site versions. Also, we prove that the dynamic version of this diagram can be built in O(n2λ6s+2(n) log n) time complexity, where s is a constant depending on the function that describes the motion of the points. Finally, we show how to use these structures for solving several locational optimization problems.

scholarly journals TRIANGULATING A REGION BETWEEN ARBITRARY POLYGONS

2017 ◽  
pp. 160-165
Author(s):  
Vasyl Tereshchenko ◽  
Yaroslav Tereshchenko
Keyword(s):  
Data Structure ◽  
Convex Hull ◽  
Efficient Algorithm ◽  
Time Complexity ◽  
Optimal Algorithm ◽  
Simple Polygon ◽  
Overall Design ◽  
Simple Implementation ◽  
Monotone Polygons ◽  
Two Stages

The paper presents an optimal algorithm for triangulating a region between arbitrary polygons on the plane with time complexity O(N log⁡N ). An efficient algorithm is received by reducing the problem to the triangulation of simple polygons with holes. A simple polygon with holes is triangulated using the method of monotone chains and keeping overall design of the algorithm simple. The problem is solved in two stages. In the first stage a convex hull for m polygons is constructed by Graham’s method. As a result, a simple polygon with holes is received. Thus, the problem of triangulating a region between arbitrary polygons is reduced to the triangulation of a simple polygon with holes. In the next stage the simple polygon with holes is triangulated using an approach based on procedure of splitting polygon onto monotone polygons using the method of chains [15]. An efficient triangulating algorithm is received. The proposed algorithm is characterized by a very simple implementation, and the elements (triangles) of the resulting triangulation can be presented in the form of simple and fast data structure: a tree of triangles [17].

SCALABLE ALGORITHMS FOR BICHROMATIC LINE SEGMENT INTERSECTION PROBLEMS ON COARSE GRAINED MULTICOMPUTERS

1996 ◽  
Vol 06 (04) ◽  
pp. 487-506 ◽  
Author(s):  
ANDREAS FABRI ◽  
OLIVIER DEVILLERS
Keyword(s):  
Line Segment ◽  
Time Complexity ◽  
Coarse Grained ◽  
Log P ◽  
Scalable Algorithms ◽  
Interprocessor Communication ◽  
Additional Time ◽  
Sequential Computation ◽  
Communication Time ◽  
Line Segment Intersection

We present output-sensitive scalable parallel algorithms for bichromatic line segment intersection problems for the coarse grained multicomputer model. Under the assumption that n≥p2, where n is the number of line segments and p the number of processors, we obtain an intersection counting algorithm with a time complexity of [Formula: see text], where Ts(m, p) is the time used to sort m items on a p processor machine. The first term captures the time spent in sequential computation performed locally by each processor. The second term captures the interprocessor communication time. An additional [Formula: see text] time in sequential computation is spent on the reporting of the k intersections. As the sequential time complexity is O(n log n) for counting and an additional time O(k) for reporting, we obtain a speedup of [Formula: see text] in the sequential part of the algorithm. The speedup in the communication part obviously depends on the underlying architecture. For example for a hypercube it ranges between [Formula: see text] and [Formula: see text] depending on the ratio of n and p. As the reporting does not involve more interprocessor communication than the counting, the algorithm achieves a full speedup of p for k≥ O( max (n log n log p, n log 3 p)) even on a hypercube.

AN INCREMENTAL ALGORITHM FOR CONSTRUCTING SHORTEST WATCHMAN ROUTES

1993 ◽  
Vol 03 (04) ◽  
pp. 351-365 ◽  
Author(s):  
XUEHOU TAN ◽  
TOMIO HIRATA ◽  
YASUYOSHI INAGAKI
Keyword(s):  
Simple Polygon ◽  
Incremental Algorithm

The problem of finding the shortest watchman route in a simple polygon P through a point s on its boundary is considered. A route is a watchman route if every point inside P can be seen from at least one point along the route. We present an incremental algorithm that constructs the shortest watchman route in O(n3) time for a simple polygon with n edges. This improves the previous O(n4) bound.

scholarly journals An Incremental Algorithm for Concept Lattice Based on SSIM

2021 ◽  
Author(s):  
Yu Hu ◽  
Yan Zhu Hu ◽  
Zhong Su ◽  
Xiao Li Li ◽  
Zhen Meng ◽  
...  
Keyword(s):  
Machine Learning ◽  
Data Analysis ◽  
Formal Concept Analysis ◽  
Time Complexity ◽  
Complexity Analysis ◽  
Concept Lattice ◽  
Incremental Algorithm ◽  
Formal Concept ◽  
Image Structure ◽  
Structure Similarity

Abstract As an effective tool for data analysis, Formal Concept Analysis (FCA) is widely used in software engineering and machine learning. The construction of concept lattice is a key step of the FCA. How to effectively update the concept lattice is still an open, interesting and important issue. The main aim of this paper is to provide a solution to this problem. So, we propose an incremental algorithm for concept lattice based on image structure similarity (SsimAddExtent). In addition, we perform time complexity analysis and experiments to show effectiveness of algorithm.

Perceptual Salience of Borders between Coloured Regions

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 251-251
Author(s):  
L T Maloney ◽  
J Yang
Keyword(s):  
Line Segment ◽  
Measurement Model ◽  
Power Functions ◽  
Colour Space ◽  
Perceptual Salience ◽  
Difference Measurement ◽  
Valuable Method ◽  
Fixed Line

Difference measurement is a little-used but potentially valuable method for studying suprathreshold vision. We used it to investigate the perceptual salience of borders between adjacent uniformly coloured regions. In these experiments, the observer viewed pairs of colours C1, C2 presented simultaneously in two rectangles, sharing a common border, centred on a neutral background. On each temporal 2AFC trial, the observer saw two colour pairs ( C1, C2) and ( C3, C4), and was asked to judge which of the two colour borders was more salient (stronger). In a single condition of the experiment, the colours C1, C2, C3, C4 were drawn from a set of ten colours constrained to lie along a fixed line segment in LMS space extending from the neutral background point (NBP) in a particular direction in LMS colour space. Let D i denote the distance of C i from the NBP in LMS space. A difference measurement model accounts for such data if and only if there is an increasing ‘scale’ f( D) such that ( C1, C2) is judged more salient than ( C3, C4) precisely when f( D2) - f( D1) > f( D4) - f( D3). For each of three observers and eight directions, we found that estimated functions f( D) were concave, resembling power functions with exponents between 0.64 and 1.

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