scholarly journals A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time

2001 ◽  
Vol 26 (2) ◽  
pp. 245-265 ◽  
Author(s):  
N. M. Amato ◽  
M. T. Goodrich ◽  
E. A. Ramos
Keyword(s):  
Linear Time ◽  
Randomized Algorithm ◽  
Simple Polygon

A LINEAR-TIME RANDOMIZED ALGORITHM FOR THE BOUNDED VORONOI DIAGRAM OF A SIMPLE POLYGON

1996 ◽  
Vol 06 (03) ◽  
pp. 263-278 ◽  
Author(s):  
ROLF KLEIN ◽  
ANDRZEJ LINGAS
Keyword(s):  
Voronoi Diagram ◽  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Linear Time ◽  
Randomized Algorithm ◽  
Simple Polygon ◽  
Minimal Spanning Tree ◽  
Expected Time

For a polygon P, the bounded Voronoi diagram of P is a partition of P into regions assigned to the vertices of P. A point p inside P belongs to the region of a vertex v if and only if v is the closest vertex of P visible from p. We present a randomized algorithm that builds the bounded Voronoi diagram of a simple polygon in linear expected time. Among other applications, we can construct within the same time bound the generalized Delaunay triangulation of P and the minimal spanning tree on P’s vertices that is contained in P.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

scholarly journals Finding the Medial Axis of a Simple Polygon in Linear Time

1999 ◽  
Vol 21 (3) ◽  
pp. 405-420 ◽  
Author(s):  
F. Chin ◽  
J. Snoeyink ◽  
C. A. Wang
Keyword(s):  
Linear Time ◽  
Medial Axis ◽  
Simple Polygon

scholarly journals Finding the convex hull of a simple polygon in linear time

1986 ◽  
Vol 19 (6) ◽  
pp. 453-458 ◽  
Author(s):  
S.Y. Shin ◽  
T.C. Woo
Keyword(s):  
Convex Hull ◽  
Linear Time ◽  
Simple Polygon

On the Euclidean Minimum Spanning Tree Problem

2005 ◽  
Vol 1 (1) ◽  
pp. 11-14 ◽  
Author(s):  
Sanguthevar Rajasekaran
Keyword(s):  
Spanning Tree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Linear Time ◽  
Randomized Algorithm ◽  
Weighted Graph ◽  
Deterministic Algorithm ◽  
Probabilistic Algorithms ◽  
Tree Algorithms ◽  
Minimum Spanning Tree Problem

Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In this note we point out that it is possible to devise simple algorithms for EMSTP in k- dimensions (for any constant k) whose expected run time is O(n), under the assumption that the points are uniformly distributed in the space of interest.CR Categories: F2.2 Nonnumerical Algorithms and Problems; G.3 Probabilistic Algorithms

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