Design and Analysis of Algorithms for Problems in Computational Geometry

我研究几何问题的框架内的设计和算法分析。我的长期研究目标是为需要空间信息的问题设计、分析和实现算法，并且(a)是基本的，(b)具有实际意义，(c)在理论上具有挑战性并有可能提出新问题，(d)有助于理解方法或概念，以及(e)导致可应用于其他问题的通用技术和工具的发展。这需要开发有效的算法解决方案，建立组合和几何性质，设计适当的数据结构，提供正确性证明和复杂性界限，以及可能的实现。在短期内，我的研究重点是三个相互关联的领域：几何图形，几何位置分析和几何路径问题。我在几何图形方面的工作集中在建立图论和潜在空间配置的几何属性，这些属性可以导致有效的算法解决方案。在几何位置分析中，我们设计算法来确定设施的最佳位置。我计划通过探索在二维和三维环境中的变体和应用来继续我对加权最短路径的研究工作。***考虑单位磁盘图（UDG）来说明我的研究方向。UDG是一类著名的几何图，通常用于对自组织无线通信网络的拓扑结构进行建模。每个节点，在网络中建模一个全向天线，被视为平面上的一个点。如果对应的点在单位距离内，则两个节点之间有一条边（超出此范围的信号强度被认为是微弱的）。如果一个节点之间存在中间节点，那么信号就可以从一个节点传播到另一个节点，这样信号就可以从一个节点跳到另一个节点，每个节点都在其前一个节点的单位距离内，直到到达目的地节点。为了检查任意一对节点之间的连通性，我们可以使用图连通性算法，所需的时间与图的大小成正比。根据输入节点的空间配置，UDG可以非常稀疏，也可以非常密集。因此，密集图对应的配置运行时间明显高于稀疏图对应的配置运行时间。通过利用几何特性，我们可以设计一种算法来测试UDG的连通性，并确保其运行时间与边的数量无关。如何有效地找到一对查询节点之间的最短跳路径，如何有效地确定直径，以及如何有效地确定与udg和相关无线网络中的覆盖、连通性和干扰问题相关的各种图参数等基本问题尚未解决。* * * * *

I study geometric problems within the framework of the design and analysis of algorithms. My long-term research objectives are to design, analyze and implement algorithms for problems that require spatial information and (a) are fundamental, (b) have practical significance, (c) are theoretically challenging and have the potential to raise new questions, (d) contribute to an understanding of a methodology or concept, and (e) lead to the development of generic techniques and tools that can be applied to other problems. This requires developing efficient algorithmic solutions, establishing combinatorial and geometric properties, designing appropriate data structures, providing correctness proofs and complexity bounds, and possibly implementations. In the short term, my research focus is in three interrelated areas: geometric graphs, geometric location analysis and geometric path problems. My work in geometric graphs is centred around establishing graph theoretic and geometric properties of the underlying spatial configuration that can lead to efficient algorithmic solutions. In geometric location analysis, we design algorithms to determine an optimal location for facilities. I plan to continue my research work on weighted shortest paths by exploring variants and applications in two and three-dimensional settings.***As an illustration of my research directions consider the unit-disk graphs (UDG). UDG is a well-known class of geometric graphs that is often used to model the topology of ad hoc wireless communication networks. Each node, modelling an omni-directional antennae in the network, is viewed as a point in a plane. There is an edge between two nodes if the corresponding points are within a unit distance (signal strength outside this range is considered feeble). A signal can travel from one node to another if there are intermediate nodes between them so that a signal can hop from one node to other, each within a unit distance of its predecessor until it reaches the destination node. To check the connectivity between any pair of nodes, we can use a graph connectivity algorithm, and the time required is proportional to the size of the graph. Depending on the spatial configuration of the input nodes, a UDG can be anywhere from being very sparse to very dense. Therefore, the runtime of the configurations corresponding to the dense graphs is significantly higher than those corresponding to sparse graphs. By exploiting the geometric properties, we can design an algorithm to test the connectivity of UDG and ensure that its run-time is independent of the number of edges. Fundamental questions such as how to efficiently find a shortest hop path between a pair of query nodes, how to efficiently determine the diameter, and how to efficiently determine various graph parameters related to coverage, connectivity and interference questions in UDGs and the associated wireless networks remain unresolved. *** **