Balls Hierarchy: A Proximity Query for Metric Spaces and Covering Theorem

In recent years, considerable advances have been made in the field of storing and processing moving data points. Data structures that discretely update moving objects or points are called Kinetic Data Structures. A particular type of data structure that constructs geometric spanner for moving point sets is called Deformable Spanner. This paper explores the underlying philosophy of deformable spanner and describes its rudiments. Deformable Spanners create a multi-resolution hierarchical subgraph that reflects a great deal about the original graph while maintaining all the existing proximity information in the graph. This proximity information plays a crucial role in wide variety of application areas, such as fast similarity queries in metric spaces, clustering in both dynamic and static multidimensional data, and cartographic representation of the complex networks. The motivation behind our work is twofold: first, to provide comprehensive understanding of the deformable spanner with sample illustrations without sacrificing depth of coverage; second, to analyze and improve deformable spanner's covering property for 2D case in Euclidean space.