In these theses are pepresented the Voronoi diagram and Network Voronoi diagram. The shortest path Dijkstra’s algorithm was modified in this way that calculates shortest paths from several Voronoi generators at the same time. The first result - partition of the nodes of the network. The seond result - arcs of the network are attributed to the generators, considering especially their direction and asymmetric costs. Applications allow compare Network Voronoi diagrams to Voronoi diagrams. For this puspose we modified Fortune algorithm. We made particular product for Taxi depot. The user can make his own implementation.

In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.

In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.

In this work Voronoi diagrams which are drawn by the cellular automaton are discussed. The square and hexagon cellular automata were created and used for drawing Voronoi diagrams. As a result of using the created programs Voronoi diagrams, which are obtained in case of two and more dots, are observed. The following results of the research were achieved: § Voronoi diagram can be obtained by the cellular automaton. § Voronoi diagrams, which were obtained, are not precise due to different speed of movement in different directions. § In square - cell case the obtained diagrams depend on the chosen situation of the neighbors. § In hexagon - cell case the obtained Voronoi diagrams are more but not completely precise. The mathematic calculations are not being made while creating Voronoi diagrams by the cellular automaton.. The diagrams are obtained in short period of time. It is possible to watch the process of the diagram creation. A lot of computer's operation time is lost not during the calculation but for re-drawing the obtained image. This is the reason why it is necessary to optimize the image creating algorithm.

In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.

In this work, we focus on two kinds of problems involving the proximity of geometric objects. The first part revolves around intersection detection problems. In this setting, we are given two (or more) geometric objects and we are allowed to preprocess them. Then, the objects are translated and rotated within a geometric space, and we need to efficiently test if they intersect in these new positions. We develop representations of convex polytopes in any (constant) dimension that allow us to perform this intersection test in logarithmic time.In the second part of this work, we turn our attention to facility location problems. In this setting, we are given a set of sites in a geometric space and we want to place a facility at a specific place in such a way that the distance between the facility and its farthest site is minimized. We study first the constrained version of the problem, in which the facility can only be place within a given geometric domain. We then study the facility location problem under the geodesic metric. In this setting, we consider a different way to measure distances: Given a simple polygon, we say that the distance between two points is the length of the shortest path that connects them while staying within the given polygon. In both cases, we present algorithms to find the optimal location of the facility.In the process of solving facility location problems, we rely heavily on geometric structures called Voronoi diagrams. These structures summarize the proximity information of a set of ``simple'' geometric objects in the plane and encode it as a decomposition of the plane into interior disjoint regions whose boundaries define a plane graph. We study the problem of constructing Voronoi diagrams incrementally by analyzing the number of edge insertions and deletions needed to maintain its combinatorial structure as new sites are added. / Option Informatique du Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.

This research is motivated by an issue frequently encountered in water quality monitoring and ecological assessment. One concern for researchers and watershed resource managers is how the biological community in a watershed is affected by human activities. The conventional single model approach based on regression and logistic regression usually fails to adequately model the relationship between biological responses and environmental stressors since the study samples are collected over a large spatial region and the response-stressor relationships are usually weak in this situation. In this dissertation, we propose two alternative modeling approaches to partition the whole region of study into disjoint subregions and model the response-stressor relationships within subregions simultaneously. In our examples, these modeling approaches found stronger relationships within subregions and should help the resource managers improve impairment assessment and decision making. The first approach is an adjusted Bayesian classification and regression tree (ABCART). It is based on the Bayesian classification and regression tree approach (BCART) and is modified to accommodate spatial partitions in ecological studies. The second approach is a Voronoi diagram based partition approach. This approach uses the Voronoi diagram technique to randomly partition the whole region into subregions with predetermined minimum sample size. The optimal partition/cluster is selected by Monte Carlo simulation. We propose several model selection criteria for optimal partitioning and modeling according to the nature of the study and extend it to multivariate analysis to find the underlying structure of response-stressor relationships. We also propose a multivariate hotspot detection approach (MHDM) to find the region where the response-stressor relationship is the strongest according to an R-square-like criterion. Several sets of ecological data are studied in this dissertation to illustrate the implementation of the above partition modeling approaches. The findings from these studies are consistent with other studies. / Ph. D.

Mobile robots are increasingly being used to perform tasks in unknown environments. The potential of robots to undertake such tasks lies in their ability to intelligently and efficiently locate and interact with objects in their environment. My research focuses on developing algorithms to plan paths for mobile robots in a partially known environment observed by an overhead camera. The environment consists of dynamic obstacles and targets. A new methodology, Extrapolated Artificial Potential Field, is proposed for real time robot path planning. An algorithm for probabilistic collision detection and avoidance is used to enhance the planner. The aim of the robot is to select avoidance maneuvers to avoid the dynamic obstacles. The navigation of a mobile robot in a real-world dynamic environment is a complex and daunting task. Consider the case of a mobile robot working in an office environment. It has to avoid the static obstacles such as desks, chairs and cupboards and it also has to consider dynamic obstacles such as humans. In the presence of dynamic obstacles, the robot has to predict the motion of the obstacles. Humans inherently have an intuitive motion prediction scheme when planning a path in a crowded environment. A technique has been developed which predicts the possible future positions of obstacles. This technique coupled with the generalized Voronoi diagram enables the robot to safely navigate in a given environment.

This thesis provides strategies for multiple vehicles to explore unknown environments in a cooperative and systematic manner. These strategies are called Simultaneous Cooperative Exploration and Networking (SCENT) strategies. As the basis for development of SCENT strategies, we first tackle the motion control and planning for one vehicle with range sensors. In particular, we develop the curve-tracking controllers for autonomous vehicles with rigidly mounted range sensors, and a provably complete exploration strategy is proposed so that one vehicle with range sensors builds a topological map of an environment. The SCENT algorithms introduced in this thesis extend the exploration strategy for one vehicle to multiple vehicles. The enabling idea of the SCENT algorithms is to construct a topological map of the environment, which is considered completely explored if the map corresponds to a complete Voronoi diagram of the environment. To achieve this, each vehicle explores its local area by incrementally expanding the already visited areas of the environment. At the same time, every vehicle deploys communication devices at selected locations and, as a result, a communication network is created concurrently with a topological map. This additional network allows the vehicles to share information in a distributed manner resulting in an efficient exploration of the workspace. The efficiency of the proposed SCENT algorithms is verified through theoretical investigations as well as experiments using mobile robots. Moreover, the resulting networks and the topological maps are used to solve coordinated multi-robot tasks, such as capturing intruders.