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。

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.