Algorithmic Studies in Applied Geometry

计算几何的方法将应用于设计、分析、实现和测试算法，以解决几个应用领域中出现的问题，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、制图和图形学。主要的项目目标是发展几何算法的基本进展。此外，该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，以便准确地制定他们的算法需求，并提供可用的算法工具、理论结果的见解和实验研究的软件。这四个问题领域是：(a)。几何优化和网络——几何环境下的最优路径和网络设计，包括TSP变体、车辆路径、约束生成树、最小权重细分、各种约束下的最优路径规划和可生存网络设计；(b)。传感器网络和蜂群机器人——传感器定位，传感器覆盖和部署，数据现场监控，以及固定或移动（机器人）传感器的自组织网络；(c)。空中交通管理——面对天气和交通拥堵引起的动态和不确定约束，空域的优化利用，分区（负载均衡），以及国家空域系统网络结构的优化；(d)。形状逼近、虚拟模型和制造——形状逼近、碰撞检测、虚拟原型和制造工艺规划。这些问题在两个方面受到攻击：(1)应用形式化算法分析，试图证明最坏情况或平均情况下的时间/空间的最紧可能边界（上和下），或问题的近似比率；(2)开发简单、快速、实用的溶液技术，并进行了实验比较。

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.