Kinetic $k$-Semi-Yao Graph and its Applications

TL;DR

This paper introduces the kinetic $k$-Semi-Yao graph ($k$-SYG), a proximity graph that extends the theta graph, and develops efficient kinetic data structures for maintaining $k$-nearest neighbors and reverse $k$-nearest neighbor queries among moving points in high-dimensional space.

Contribution

It presents the first kinetic data structure for the $k$-Semi-Yao graph and applies it to maintain $k$-nearest neighbors and reverse $k$-nearest neighbor queries for moving points in $ eal^d$, improving previous methods.

Findings

First KDS for $k$-SYG in $ eal^d$

Deterministic KDS for all $k$-nearest neighbors

Efficient KDS for reverse $k$-nearest neighbor queries

Abstract

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 $k\geq 1$.