scholarly journals Computing the maximum clique in the visibility graph of a simple polygon

2007 ◽  
Vol 5 (3) ◽  
pp. 524-532 ◽  
Author(s):  
Subir Kumar Ghosh ◽  
Thomas Caton Shermer ◽  
Binay Kumar Bhattacharya ◽  
Partha Pratim Goswami
Keyword(s):  
Maximum Clique ◽  
Simple Polygon ◽  
Visibility Graph

scholarly journals A randomized algorithm for finding a maximum clique in the visibility graph of a simple polygon

2015 ◽  
Vol Vol. 17 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Sergio Cabello ◽  
Maria Saumell
Keyword(s):  
Hamiltonian Cycle ◽  
Randomized Algorithm ◽  
Maximum Clique ◽  
Simple Polygon ◽  
Maximum Size ◽  
Visibility Graph ◽  
Probability Of Error ◽  
Discrete Algorithms ◽  
International Audience ◽  
Previous Algorithm

Discrete Algorithms International audience We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.

Efficiently Constructing the Visibility Graph of a Simple Polygon with Obstacles

2000 ◽  
Vol 30 (3) ◽  
pp. 847-871 ◽  
Author(s):  
Sanjiv Kapoor ◽  
S. N. Maheshwari
Keyword(s):  
Simple Polygon ◽  
Visibility Graph

DISTANCE VISIBILITY GRAPHS

1992 ◽  
Vol 02 (04) ◽  
pp. 349-362 ◽  
Author(s):  
COLLETTE COULLARD ◽  
ANNA LUBIW
Keyword(s):  
Simple Polygon ◽  
Weighted Graph ◽  
Visibility Graph ◽  
Necessary Condition ◽  
Connected Component ◽  
Graph Problem ◽  
Visibility Graphs ◽  
Vertex Ordering ◽  
Euclidean Distances ◽  
The Given

A new necessary condition for a graph G to be the visibility graph of a simple polygon is given: each 3-connected component of G must have a vertex ordering in which every vertex is adjacent to a previous 3-clique. This property is used to give an algorithm for the distance visibility graph problem: given an edge-weighted graph G, is it the visibility graph of a simple polygon with the given weights as Euclidean distances?

On Minimum Link Monotone Path Problems

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Xiangzhi Wei ◽  
Ajay Joneja
Keyword(s):  
Path Planning ◽  
Dual Graph ◽  
Simple Polygon ◽  
Visibility Graph ◽  
Polygonal Domain ◽  
Large Set ◽  
The Novel ◽  
Polygonal Domains ◽  
Monotone Path ◽  
Trapezoidal Map

The problem of finding monotone paths between two given points has useful applications in path planning, and in particular, it is useful to look for minimum link paths. We are given a simple polygon P or a polygonal domain D with n vertices and a triplet of input parameters: (s, t, d), where s and t are two points in the plane and d is any direction. We show how to answer a query for the existence of a d-monotone path between s and t inside P (or D) in logarithmic time after preprocessing P in O(En) time, or D in O(En + ERlogR) time, where E is the size of the visibility graph of P (or D), and R is the number of reflex vertices in D. Our approach is based on the novel idea utilizing the dual graph of the trapezoidal map of P (or D). For polygonal domains, our approach uses a trapezoidal map associated with each visibility edge of D, and we show how to compute this large set of trapezoidal maps efficiently. Furthermore, we show how to output a minimum linkd-monotone path between points s and t, for an arbitrary input triplet (s, t, d).

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