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.

MANHATTONIAN PROXIMITY IN A SIMPLE POLYGON

1995 ◽  
Vol 05 (01n02) ◽  
pp. 53-74 ◽  
Author(s):  
ROLF KLEIN ◽  
ANDRZEJ LINGAS
Keyword(s):  
Voronoi Diagram ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Linear Time ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Manhattan Metric ◽  
Optimal Linear ◽  
Planar Polygon

Let P be a simple planar polygon. We present a linear time algorithm for constructing the bounded Voronoi diagram of P in the Manhattan metric, where each point z in P belongs to the region of the closest vertex of P that is visible from z. Among other consequences, the minimum spanning tree of the vertices in the Manhattan metric that is contained in P can be computed within optimal linear time. The same results hold for the L∞-metric.α

DEVELOPING PROXIMITY GRAPHS BY PHYSARUM POLYCEPHALUM: DOES THE PLASMODIUM FOLLOW THE TOUSSAINT HIERARCHY?

2009 ◽  
Vol 19 (01) ◽  
pp. 105-127 ◽  
Author(s):  
ANDREW ADAMATZKY
Keyword(s):  
Foraging Behavior ◽  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Physarum Polycephalum ◽  
Minimum Spanning Tree ◽  
Nearest Neighbour ◽  
Minimal Spanning Tree ◽  
Proximity Graphs ◽  
Relative Neighborhood Graph ◽  
Plasmodium Of Physarum Polycephalum

Plasmodium of Physarum polycephalum spans sources of nutrients and constructs varieties of protoplasmic networks during its foraging behavior. When the plasmodium is placed on a substrate populated with sources of nutrients, it spans the sources with protoplasmic network. The plasmodium optimizes the network to deliver efficiently the nutrients to all parts of its body. How exactly does the protoplasmic network unfold during the plasmodium's foraging behavior? What types of proximity graphs are approximated by the network? Does the plasmodium construct a minimal spanning tree first and then add additional protoplasmic veins to increase reliability and through-capacity of the network? We analyze a possibility that the plasmodium constructs a series of proximity graphs: nearest-neighbour graph (NNG), minimum spanning tree (MST), relative neighborhood graph (RNG), Gabriel graph (GG) and Delaunay triangulation (DT). The graphs can be arranged in the inclusion hierarchy (Toussaint hierarchy): NNG ⊆ MST ⊆ RNG ⊆ GG ⊆ DT . We aim to verify if graphs, where nodes are sources of nutrients and edges are protoplasmic tubes, appear in the development of the plasmodium in the order NNG → MST → RNG → GG → DT , corresponding to inclusion of the proximity graphs.

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

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

scholarly journals FURTHEST SITE ABSTRACT VORONOI DIAGRAMS

2001 ◽  
Vol 11 (06) ◽  
pp. 583-616 ◽  
Author(s):  
KURT MEHLHORN ◽  
STEFAN MEISER ◽  
RONALD RASCH
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Randomized Algorithm ◽  
Linear Size ◽  
Expected Time ◽  
Algorithmic Treatment

Voronoi diagrams were introduced by R. Klein as a unifying approach to Voronoi diagrams. In this paper we study furthest site abstract Voronoi diagrams and give a unified mathematical and algorithmic treatment for them. In particular, we show that furthest site abstract Voronoi diagrams are trees, have linear size, and that, given a set of n sites, the furthest site abstract Voronoi diagram can be computed by a randomized algorithm in expected time O(n log n).

scholarly journals RANDOMIZATION YIELDS SIMPLE O(n log⋆ n) ALGORITHMS FOR DIFFICULT Ω(n) PROBLEMS

1992 ◽  
Vol 02 (01) ◽  
pp. 97-111 ◽  
Author(s):  
OLIVIER DEVILLERS
Keyword(s):  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Simple Polygon ◽  
Complexity Of Algorithms ◽  
Additional Information ◽  
Influence Graph

We use here the results on the influence graph1 to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(n log n) to O(n log ⋆ n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree).

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