Algorithmic Studies in Applied Geometry

计算几何的方法将被应用于设计，分析，实施和测试算法的问题出现在几个应用领域，包括几何网络优化，传感器网络，机器人，空中交通管理，几何建模，制造，计算机辅助设计，制图和图形。主要项目目标是几何算法的基本进展的发展。 此外，本项目将努力促进和深化与应用领域和行业研究人员的合作，以精确地制定他们的算法需求，并提供算法工具，理论结果的见解和实验研究的软件。几何优化和网络--几何环境中的最优路由和网络设计，包括TSP变体、车辆路由、约束生成树、最小权重细分、具有各种约束的最优路由规划和可生存网络设计;（B）。传感器网络和群机器人-传感器定位、传感器覆盖和部署、数据场监测以及固定或移动的（机器人）传感器的特设网络;空中交通管理-在面临天气和交通拥挤、分区（负荷平衡）和国家空域系统网络结构优化所引起的动态和不确定制约因素时，优化使用空域;（d）.形状近似、虚拟模型和制造--形状近似、碰撞检测、虚拟原型和制造工艺规划。这些问题从两个方面着手：（1）应用形式化算法分析，试图证明最严格的可能边界（上和下）最坏情况或平均情况的时间/空间，或问题的近似比;（2）开发简单、快速、实用的溶液技术，并进行实验比较。

The methodologies of computational geometry will be applied to design, analyze, implement, and test algorithms for problems that arise in several application areas, including geometric network optimization, sensor networks, robotics, air traffic management, geometric modeling, manufacturing, computer-aided design, cartography, and graphics. The main project goal is the development of fundamental advances in geometric algorithms. Additionally, the project will strive to foster and deepen collaborations with researchers in application domains and industry, in order to formulate their algorithmic needs precisely and to make available algorithmic tools, insights from theoretical results, and software from experimental investigations.The four problem areas are: (a). Geometric Optimization and Networks -- optimal routing and network design in geometric contexts, including TSP variants, vehicle routing, constrained spanning trees, minimum-weight subdivisions, optimal route planning with various constraints, and survivable network design; (b). Sensor Networks and Swarm Robotics -- sensor localization, sensor coverage and deployment, data field monitoring, and ad hoc networking for stationary or mobile (robotic) sensors; (c). Air Traffic Management -- optimal use of airspace in the face of dynamic and uncertain constraints induced by weather and traffic congestion, sectorization (load balancing), and optimization of the network structure of the National Airspace System; (d). Shape Approximation, Virtual Models, and Manufacturing -- shape approximation, collision detection, virtual prototyping, and manufacturing process planning.The problems are attacked on two fronts: (1) Application of formal algorithmic analysis, attempting to prove the tightest possible bounds (upper and lower) on the worst-case or average-case time/space, or approximation ratio for the problem; and (2) Development of solution techniques designed to be simple, fast, and practical, which are compared experimentally.