Algorithms in computational geometry and geometric graphs

计算几何和几何图的算法

• 批准号: RGPIN-2020-03959
• 负责人: Lubiw, Anna
• 金额: $ 3.5万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=750215
• 关键词: Algorithms; computational; geometry; geometric; graphs

My research is in design and analysis of algorithms, specifically for problems involving geometry and graphs. Currently, I focus on reconfiguration of geometric structures and graphs: How can one structure be changed to another, either through continuous motion or through discrete changes?  Examples in popular culture include Rubik's cubes and transformers; in mathematics, the topic has a vast and deep history, for example knot theory, and mechanical linkages. Reconfiguration can often be accomplished via discrete steps. The questions I propose answering are ones of: existence (can an initial structure be reconfigured to a target structure); distance (how many steps are needed for reconfiguration); and efficiency (is there an efficient algorithm to test existence or find the distance). These problems can be modelled as connectivity and shortest path problems in an exponentially large "reconfiguration graph'' where a vertex represents a configuration and an edge represents a reconfiguration step.  I propose to study the structure of such reconfiguration graphs, building on previous work with PhD students on reconfiguration of triangulations of a point set in the plane. Triangulations of a point set are heavily used in applications such as meshing, and the basic reconfiguration step, a "flip", is well-studied. In the process of studying flips in the edge-labelled setting, we discovered new topological properties of the reconfiguration complex, an enhancement of the reconfiguration graph. I will extend this to other settings, with the goal of furthering our understanding of the structure of reconfiguration graphs. "Morphing" is one kind of reconfiguration, and I will continue to work on problems of morphing graph drawings. Given two planar drawings of the same graph with points for vertices, and straight line segments for edges, the goal is to move continuously from the first drawing to the second, remaining planar throughout the motion. This problem has many applications in visualization and animation. We have developed a theoretically satisfactory algorithm to find a piece-wise-linear morph but many practical issues such as preventing vertices from coming too close to each other remain open. My work on reconfiguration in a more geometric setting focuses on unfolding polyhedra, a problem with applications in manufacturing 3D shapes out of metal, cardboard or plastic. One famous open question is whether we can cut some edges of any convex polyhedron to give a non-overlapping "net" in the plane. In practical applications we may cut across faces, and nets are known to exist for this relaxation. However, some nets are better than others - I propose to find efficient algorithms to solve associated optimization problems of minimizing the length of the cuts or the size of the minimum disc enclosing the net, both of which are relevant in applications.

我的研究是对几何结构和图形的设计和分析：如何将一种结构更改为另一个结构，要么通过流行文化中的thoss剧院连续运动？具有庞大的历史，并且我提出的问题是：找到这些问题。基本的重新配置步骤是在研究边缘设置中研究翻转的过程，我们发现了重新配置的重新配置的新拓扑特性。是一种在形变图的问题上的一种重新配置，是在整个运动中连续移动到第二个图形，剩余的平面。线性变形，但实际上是彼此关闭的诸如PRES顶点，在更加几何的环境中，我的重新配置的工作集中在polyhedra上，这是制造3D形状的问题我们是否可以切入任何凸的多面体，以在实用应用中提供非重叠的“净”。切口的长度或最小圆盘的大小封闭了网络，两者在应用中都相关。