Kinetic k-Semi-Yao graph and its applications

Abstract

This paper introduces a new supergraph, called the k-Semi-Yao graph (k-SYG), of the k-nearest neighbor graph (k-NNG) of a set P of points in Rd. We provide a kinetic data structure (KDS) to maintain the k-SYG on moving points, where the trajectory of each point is a polynomial function whose degree is bounded by some constant. Our technique gives the first KDS for the theta graph (i.e., 1-SYG) in Rd. It generalizes and improves on previous work on maintaining the theta graph in R2.As an application, we use the kinetic k-SYG to provide the first KDS for maintenance of all the k-nearest neighbors in Rd, for any k≥1. Previous works considered the k=1 case only.Our KDS for all the 1-nearest neighbors is deterministic. The best previous KDS for all the 1-nearest neighbors in Rd is randomized. Our structure and analysis are simpler and improve on this work for the k=1 case. We also provide a KDS for all the (1+ϵ)-nearest neighbors, which in fact gives better performance than previous KDSs for maintenance of all the exact 1-nearest neighbors.As another application, we present the first KDS for answering reversek-nearest neighbor queries on moving points in Rd, for any k≥1.