Determining Weak Visibility of a Polygon from an Edge in Parallel

1998 ◽  
Vol 08 (03) ◽  
pp. 277-304
Author(s):  
Danny Z. Chen
Keyword(s):  
Computational Model ◽  
Parallel Algorithm ◽  
Simple Polygon ◽  
Sequential Algorithms ◽  
Simple Polygons ◽  
Crew Pram

The problem of determining the weak visibility of an n-vertex simple polygon P from an edge e of P is that of deciding whether every point in P is weakly visible from e. In this paper we present an optimal parallel algorithm for solving this problem. Our algorithm runs in O( log n) time using O(n/ log n) processors in the CREW PRAM computational model, and is very different from the sequential algorithms for this problem. We also show how to solve optimally, in parallel, several other problems that are related to the weak visibility of simple polygons.

scholarly journals AN OPTIMAL PARALLEL ALGORITHM FOR DETECTING WEAK VISIBILITY OF A SIMPLE POLYGON

1995 ◽  
Vol 05 (01n02) ◽  
pp. 93-124 ◽  
Author(s):  
DANNY Z. CHEN
Keyword(s):  
Computational Model ◽  
Parallel Algorithm ◽  
Simple Polygon ◽  
Sequential Algorithms ◽  
Crew Pram

The problem of detecting the weak visibility of an n-vertex simple polygon P is that of finding whether P is weakly visible from one of its edges and (if it is) identifying every edge from which P is weakly visible. In this paper, we present an optimal parallel algorithm for solving this problem. Our algorithm runs in O(log n) time using O(n/log n) processors in the CREW PRAM computational model, and is very different from the sequential algorithms for this problem. Based on this algorithm, several other problems related to weak visibility can be optimally solved in parallel.

PARALLEL COMPUTATION OF INTERNAL AND EXTERNAL FARTHEST NEIGHBORS IN SIMPLE POLYGONS

1992 ◽  
Vol 02 (02) ◽  
pp. 175-190 ◽  
Author(s):  
SUMANTA GUHA
Keyword(s):  
Parallel Algorithms ◽  
Parallel Computation ◽  
Shortest Path ◽  
Simple Polygon ◽  
Divide And Conquer ◽  
Simple Polygons ◽  
Crew Pram ◽  
Serial Algorithm

We present efficient parallel algorithms for two problems in simple polygons: the all-farthest neighbors problem and the external all-farthest neighbors problem. The all-farthest neighbors problem is that of computing, for each vertex p of a simple polygon P, a point ψ(p) in P farthest from p when the distance between p and ψ(p) is measured by the shortest path between them constrained to lie inside P. The external all-farthest neighbors problem is that of computing, for each vertex p of a simple polygon P, a point ϕ(p) on (the boundary of) P farthest from p when the distance between p and ϕ(p) is measured by the shortest path between them constrained to lie outside (the interior of) P. Both our algorithms run in O( log 2 n) time on a CREW PRAM with O(n) processors. Our divide-and-conquer method for the external all-farthest neighbors problem, in fact, leads to a new O(n log n) time serial algorithm that matches the currently best serial algorithm for this problem, but is simpler.

OPTIMAL PARALLEL ALGORITHMS FOR TRIANGULATED SIMPLE POLYGONS

1995 ◽  
Vol 05 (01n02) ◽  
pp. 145-170 ◽  
Author(s):  
JOHN HERSHBERGER
Keyword(s):  
Convex Hull ◽  
Parallel Algorithms ◽  
Shortest Path ◽  
Simple Polygon ◽  
Implicit Representation ◽  
Shortest Path Tree ◽  
Simple Polygons ◽  
Pram Model ◽  
Crew Pram ◽  
Path Queries

We provide optimal parallel solutions to several shortest path and visibility problems set in triangulated simple polygons. Let P be a triangulated simple polygon with n vertices, preprocessed to support shortest path queries. We can find the shortest path tree from any point inside P in O(log n) time using O(n/log n) processors. In the game bounds, we can preprocess P for shooting queries (a query can be answered in O(log n) time by a uniprocessor). Given a set S of m points inside P, we can find an implicit representation of the relative convex hull of S in O(log(nm)) time with O(m) processors. If the relative convex hull has k edges, we can explicitly produce these edges in O(log(nm)) time with O(k/log(nm)) processors. All of these algorithms are deterministic and use the CREW PRAM model.

PARALLEL ALGORITHMS FOR LONGEST INCREASING CHAINS IN THE PLANE AND RELATED PROBLEMS

1999 ◽  
Vol 09 (04) ◽  
pp. 511-520 ◽  
Author(s):  
MIKHAIL J. ATALLAH ◽  
DANNY Z. CHEN ◽  
KEVIN S. KLENK
Keyword(s):  
Parallel Algorithms ◽  
Computational Model ◽  
Parallel Algorithm ◽  
Related Problem ◽  
Single Source ◽  
Crew Pram ◽  
Longest Paths ◽  
Nonnegative Weight

Given a set [Formula: see text] of n points in the plane such that each point in [Formula: see text] is asscociated with a nonnegative weight, we consider the problem of computing the single-source longest increasing chains among the points in [Formula: see text] This problem is a generalization of the plannar maximal layers problem. In this paper, we present a parallel algorithm that computes the single-source longest incresing chains in the plane in [Formula: see text] time using [Formula: see text] processors in the CREW PRAM computational model. We also solve a related problem of computing the all-pairs longest paths in an n-node weighted planar st-graph, in [Formula: see text] time using [Formula: see text] CREW PRAM processors. Both of our parallel algorithms are improvement over the previously best known results.

DETERMINING THE SEPARATION OF SIMPLE POLYGONS

1994 ◽  
Vol 04 (04) ◽  
pp. 457-474 ◽  
Author(s):  
NANCY M. AMATO
Keyword(s):  
Parallel Algorithm ◽  
Time Complexity ◽  
Minimum Distance ◽  
Sequential Algorithm ◽  
Parallel Methods ◽  
Simple Polygons ◽  
Closest Pair ◽  
Time Optimal ◽  
Crew Pram

Given simple polygons P and Q, their separation, denoted by σ(P, Q), is defined to be the minimum distance between their boundaries. We present a parallel algorithm for finding a closest pair among all pairs (p, q), p ∈ P and q ∈ Q. The algorithm runs in O ( log n) time using O(n) processors on a CREW PRAM, where n = |P| + |Q|. This algorithm is time-optimal and improves by a factor of O ( log n) on the time complexity of previous parallel methods. The algorithm can be implemented serially in Θ (n) time, which gives the first optimal sequential algorithm for determining the separation of simple polygons. Our results are obtained by providing a unified treatment of the separation and the closest visible vertex problems for simple polygons.

A parallel algorithm for solving the n-queens problem based on inspired computational model

Biosystems ◽  
2015 ◽  
Vol 131 ◽  
pp. 22-29 ◽  
Author(s):  
Zhaocai Wang ◽  
Dongmei Huang ◽  
Jian Tan ◽  
Taigang Liu ◽  
Kai Zhao ◽  
...  
Keyword(s):  
Computational Model ◽  
Parallel Algorithm

Two-Guard Walkability of Simple Polygons

1998 ◽  
Vol 08 (01) ◽  
pp. 85-116 ◽  
Author(s):  
L. H. Tseng ◽  
P. Heffernan ◽  
D. T. Lee
Keyword(s):  
Linear Space ◽  
Simple Polygon ◽  
The Other ◽  
Simple Polygons

A pair of points s and g on the boundary of a simple polygon P admits a walk if two guards can simultaneously walk along the two boundary chains of P from s to g such that they are always visible to each other. The walk is a counter-walk if one guard moves from s to g while the other moves from g to s in the same direction along the boundary and they are always visible to each other. The (counter-)walk is straight if no backtracking is necessary during the (counter-)walk. In this paper, we show that, given a polygon with n vertices, to test if there exists (s,g) that admits a (straight) (counter-)walk can be solved in time O(n log n) and in linear space. Also we compute all (s,g)'s that admit a (straight) walk in O(n log n) time and all vertex pairs that admit a (straight) counter-walk in O(n log n + m), where m is O(n2).

scholarly journals NLP Formulation for Polygon Optimization Problems

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 24 ◽  
Author(s):  
Saeed Asaeedi ◽  
Farzad Didehvar ◽  
Ali Mohades
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problems ◽  
Theoretical Perspective ◽  
Simple Polygon ◽  
Programming Models ◽  
Minimum Area ◽  
Simple Polygons ◽  
Maximum Angle ◽  
Optimal Approach ◽  
Set Of Points

In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by α + π where α ( 0 ≤ α ≤ π ) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate these three generalized problems as nonlinear programming models, and then present a genetic algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.

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