Kinetic Data Structures for the Semi-Yao Graph and All Nearest Neighbors in R^d

TL;DR

This paper introduces a simple, deterministic kinetic data structure for efficiently maintaining all nearest neighbors of moving points in high-dimensional space, improving previous randomized methods in complexity and simplicity.

Contribution

It presents the first kinetic data structure for the Semi-Yao graph in any dimension, simplifying and enhancing prior results for nearest neighbor maintenance.

Findings

Processes $O(n^2eta_{2s+2}^2(n)\log n)$ events

Achieves $O(n^2\beta_{2s+2}(n)\log^{d+1} n)$ total cost

Improves previous results by a factor of $\log^d n$ in event count

Abstract

This paper presents a simple kinetic data structure for maintaining all the nearest neighbors of a set of $n$ moving points in $\mathbb{R}^d$, where the trajectory of each point is an algebraic function of at most constant degree $s$. The approach is based on maintaining the edges of the Semi-Yao graph, a sparse graph whose edge set includes the pairs of nearest neighbors as a subset. Our kinetic data structure (KDS) for maintaining all the nearest neighbors is deterministic. It processes $O(n^2\beta_{2s+2}^2(n)\log n)$ events with a total cost of $O(n^2\beta_{2s+2}(n)\log^{d+1} n)$. Here, $\beta_s(n)$ is an extremely slow-growing function. The best previous KDS for all the nearest neighbors in $ \mathbb{R}^d$ is by Agarwal, Kaplan, and Sharir (TALG 2008). It is a randomized result. Our structure and analysis are simpler than theirs. Also, we improve their result by a factor of $\log^d n$ in the number of events and by a $\log n$ factor in the total cost. This paper generalizes and improves the 2013 work of Rahmati, King and Whitesides (SoCG 2013) on maintaining the Semi-Yao graph in $\mathbb{R}^2$; its new technique provides the first KDS for the Semi-Yao graph in $\mathbb{R}^d$. Our KDS is local in the worst case, meaning that only a constant number of events is associated with any one point at any time. For maintaining all the nearest neighbors, neither our KDS nor the KDS by Agarwal~\etal~is local, and furthermore, each event in our KDS and in their KDS is handled in polylogarithmic time in an amortized sense. Finally, in this paper, we also give a KDS for maintenance of all the $(1+\epsilon)$-nearest neighbors which is local and each event can be handled in a polylogarithmic worst-case time.