Collaborative Research: AF: Medium: Algorithms for Geometric Graphs

这个项目研究的是几何图形。这些是实现组合图的关系的几何结构，也就是一组被称为“节点”或“顶点”的元素以及它们之间的一组成对关系，例如将由社会网络或道路网络确定。几何图的应用非常广泛，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，这样图形的节点就是空间中的点，节点之间的关系用连接节点对的线段或曲线来表示。这些组合图的几何实现可以根据它们达到各种参数的程度来衡量，例如面积、边长、角度间隔等。事实上，图形绘制的研究领域专门专注于生成良好的(忠实的和有代表性的)图的几何实现的算法。改进的几何图形处理方法可以使任何生成或使用几何图形的应用程序受益，例如数据可视化或汽车导航。本项目的目标大致围绕以下两个主题组织：(1)生成图形几何实现的算法。本主题针对产生图的几何实现的算法和复杂性界限，包括对诸如面积、边长度、边弯曲、边交叉等复杂性度量的考虑。(2)几何图上的算法。这一主题是针对以几何图形作为输入的算法，例如道路网络，目标是实现比一般图形可能改善的复杂性界限。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.