706 research outputs found
Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions
Let be a set of points and a convex -gon in .
We analyze in detail the topological (or discrete) changes in the structure of
the Voronoi diagram and the Delaunay triangulation of , under the convex
distance function defined by , as the points of move along prespecified
continuous trajectories. Assuming that each point of moves along an
algebraic trajectory of bounded degree, we establish an upper bound of
on the number of topological changes experienced by the
diagrams throughout the motion; here is the maximum length of an
-Davenport-Schinzel sequence, and is a constant depending on the
algebraic degree of the motion of the points. Finally, we describe an algorithm
for efficiently maintaining the above structures, using the kinetic data
structure (KDS) framework
Farthest-Polygon Voronoi Diagrams
Given a family of k disjoint connected polygonal sites in general position
and of total complexity n, we consider the farthest-site Voronoi diagram of
these sites, where the distance to a site is the distance to a closest point on
it. We show that the complexity of this diagram is O(n), and give an O(n log^3
n) time algorithm to compute it. We also prove a number of structural
properties of this diagram. In particular, a Voronoi region may consist of k-1
connected components, but if one component is bounded, then it is equal to the
entire region
Kinetic Geodesic Voronoi Diagrams in a Simple Polygon
We study the geodesic Voronoi diagram of a set S of n linearly moving sites inside a static simple polygon P with m vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such events, and then develop a kinetic data structure (KDS) that maintains the geodesic Voronoi diagram as the sites move. To this end, we first analyze how often a single bisector, defined by two sites, or a single Voronoi center, defined by three sites, can change. For both these structures we prove that the number of such changes is at most O(m³), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient kinetic data structures for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector KDS handles each event in O(log m) time, and our Voronoi center handles each event in O(log² m) time. Both structures can be extended to efficiently support updating the movement of the sites as well. Using these data structures as building blocks we obtain a compact KDS for maintaining the full geodesic Voronoi diagram
Voronoi-based space partitioning for coordinated multi-robot exploration
Recent multi-robot exploration algorithms usually rely on occupancy grids as their core world representation. However, those grids are not appropriate for environments that are very large or whose boundaries are not well delimited from the beginning of the exploration. In contrast, polygonal representations do not have such limitations. Previously, the authors have proposed a new exploration algorithm based on partitioning unknown space into as many regions as available robots by applying K-Means clustering to an occupancy grid representation, and have shown that this approach leads to higher robot dispersion than other approaches, which is potentially beneficial for quick coverage of wide areas. In this paper, the original K-Means clustering applied over grid cells, which is the most expensive stage of the aforementioned exploration algorithm, is substituted for a Voronoi-based partitioning algorithm applied to polygons. The computational cost of the exploration algorithm is thus significantly reduced for large maps. An empirical evaluation and comparison of both partitioning approaches is presented.This work is partially supported by the Government of Spain under MCYT DPI2004-07993-C03-03. Ling Wu is supported by a FPI scholarship from the Spanish Ministry of Education and Science
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