Collaborative Research: AF: Medium: Algorithms for Geometric Graphs

这个项目研究几何图形。这些是实现组合图的关系的几何结构，即，称为“节点”或“顶点”的一组元素以及它们之间的一组成对关系，诸如将由社交网络或道路网络确定的。几何图出现在广泛的应用中，包括物理学、数据可视化、计算生物学和数据取证。任何这样的图都可以在几何空间中实现，使得图的节点是空间中的点，并且节点之间的关系由连接节点对的线段或曲线表示。这些组合图的几何实现，然后可以衡量他们如何实现各种参数，如面积，边长，角度分离等，事实上，图形绘制的研究领域是专门集中在算法产生良好的（忠实的和代表性的）几何实现的图形。改进的几何图处理方法可以使任何生成或使用几何图的应用程序（如数据可视化或汽车导航）受益。本项目的目标大致围绕以下两个主题：（1）生成图的几何实现的算法。本文主要研究图的几何实现的算法和复杂性界，包括面积、边长度、边弯曲、边交叉等复杂性测度。（2）几何图的算法。该奖项旨在研究以道路网络等几何图形为输入的算法，其目标是实现比一般图形更高的复杂度界限。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project studies geometric graphs. These are geometric structures that realize the relationships of a combinatorial graph, that is, a set of elements called “nodes” or “vertices” and a set of pairwise relationships between them, such as would be determined by a social network or road network. Geometric graphs arise in a wide range of applications, including physics, data visualization, computational biology, and data forensics. Any such graph can be realized in a geometric space, so that the nodes of the graph are points in the space and relationships between nodes are represented by line segments or curves connecting pairs of nodes. These geometric realizations of combinatorial graphs can then be measured in terms of how well they achieve various parameters, such as area, edge length, angle separation, etc. Indeed, the research area of graph drawing is exclusively focused on algorithms for producing good (faithful and representative) geometric realizations of graphs. Improved methods for dealing with geometric graphs can benefit any application, such as data visualization or automobile navigation, that generates or uses geometric graphs.The goals of this project are broadly organized around the following two themes: (1) Algorithms for producing geometric realizations of graphs. This theme is directed at algorithms and complexity bounds for producing geometric realizations of graphs, including considerations of complexity measures such as area, edge length, edge bends, edge crossings, etc. (2) Algorithms on geometric graphs. This theme is directed at algorithms that take as input geometric graphs, such as road networks, with the goal of achieving complexity bounds that are improved over those possible for general graphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.