Geometric Data Structures Approximations for Network Optimisation Problems MILOŠ ŠEDA, TOMÁŠ BŘEZINA Institute of Automation and Computer Science Brno University of Technology Technická 2, 616 69 Brno CZECH REPUBLIC seda@fme.vutbr.cz, brezina@fme.vutbr.cz Abstract: - A frequent task in transportation, routing, robotics, and communications applications is to find the shortest path between two positions. In robot motion planning, the robot should pass around the obstacles touching none of them, i.e. the goal is to find a collision-free path from a starting to a target position. Research of path planning has yielded many fundamentally different approaches to its solution, mainly based on various decomposition and roadmap methods. In this paper, we show a possible use of geometric data structures in point-to-point motion planning in the Euclidean plane and present an approach using generalised Voronoi diagrams that decreases the probability of collisions with obstacles and generate smooth trajectories. The second application area, investigated here, is focused on problems of finding minimal networks connecting a set of given points in the Euclidean plane and their approximations using the Delaunay triangulation. Key-Words: - motion planning, spanning tree, Steiner tree, Delaunay triangulation, Voronoi diagram 1 Introduction In recent years a number of new data structures and algorithmic techniques have been developed that have improved and simplified many of the previous approaches used in network optimisation [6], robot motion planning [3], [9], [15], etc. Geometric data structures defined in computational geometry have a surprising variety of uses [1], [2], [10]. Computational geometry emerged from the field of algorithm design and analysis in the late 1970s. It has many application domains including computer graphics, geographic information systems (GIS), robotics, and others in which geometric algorithms play a fundamental role. Computational geometry deals with specific geometric data structures, the most important ones being Voronoi diagrams, Delaunay triangulation, visibility graph and convex hull. Before we study examples of their applications, we will introduce them and summarise the basic definitions. 2 Basic Notions A Voronoi diagram of a set of points (called sites) in the Euclidean plane is a collection of regions that divide up the plane. Each region corresponds to one of the sites and all the points in one region are closer to the site representing the region than to any other site. More formally [1], [2], [7], [10]: Definition 1 Let P be a set of n points in the plane. For two distinct sites p i , p j ∈ P, the dominance of p i over p j is defined as the subset of the plane that is at least as close to p i as to p j . Formally, dom(p i , p j )={x∈ℜ 2 | d(x, p i ) ≤ d(x, p j )}, (1) where d denotes the Euclidean distance.  Clearly, dom(p i , p j ) is a closed half-plane bounded by the perpendicular bisector of p i and p j . Definition 2 Voronoi region (or Voronoi polytope, Voronoi cell, Voronoi face, Dirichlet polygon, Thiessen polygon) of a site p i ∈P is a close or open area V(p i ) of points in the plane such that p i ∈ V(p i ) for each p i , and any point x∈ V(p i ) is at least as close to p i as to any other sites in P (i.e. V(p i ) is the area lying in all of the dominances of p i over the remaining sites in P). Formally, { } ) dom( }) { ( : ) , ( ) , ( | ) ( } { 2 ,q p p P q q x d p x d x P V i p P q i i i i I − ∈ = = − ∈ ∀ ≤ ℜ ∈ = (2)  Proceedings of the 11th WSEAS Int. Conf. on MATHEMATICAL METHODS, COMPUTATIONAL TECHNIQUES AND INTELLIGENT SYSTEMS ISSN: 1790-2769 158 ISBN: 978-960-474-094-9