Design and Analysis of Algorithms for Problems in Computational Geometry

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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. *** **