Kinetic Reverse kk-Nearest Neighbor Problem

This paper provides the first solution to the kinetic reverse $k$-nearest neighbor (\rknn) problem in $\mathbb{R}^d$, which is defined as follows: Given a set $P$ of $n$ moving points in arbitrary but fixed dimension $d$, an integer $k$, and a query point $q\notin P$ at any time $t$, report all the points $p\in P$ for which $q$ is one of the $k$-nearest neighbors of $p$

Kinetic kk-Semi-Yao Graph and its Applications

This paper introduces a new proximity graph, called the $k$-Semi-Yao graph ($k$-SYG), on a set $P$ of points in $\mathbb{R}^d$, which is a supergraph of the $k$-nearest neighbor graph ($k$-NNG) of $P$. 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 (\ie, $1$-SYG) in $\mathbb{R}^d$. It generalizes and improves on previous work on maintaining the theta graph in $\mathbb{R}^2$. As an application, we use the kinetic $k$-SYG to provide the first KDS for maintenance of all the $k$-nearest neighbors in $\mathbb{R}^d$, for any $k\geq 1$. 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 $\mathbb{R}^d$ 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 $(1+\epsilon)$-nearest neighbors, which in fact gives better performance than previous KDS's for maintenance of all the exact $1$-nearest neighbors. As another application, we present the first KDS for answering reverse $k$-nearest neighbor queries on moving points in $\mathbb{R}^d$, for any $k\geq 1$.Comment: arXiv admin note: text overlap with arXiv:1307.2700, arXiv:1406.555