Dynamic Nearest-Neighbor Searching Under General Metrics in ${\mathbb R}^3$ and Its Applications

2026-03-27

AI Summary

This work addresses the lack of efficient data structures for dynamic nearest neighbor and intersection queries under general metrics induced by homothetic scaled convex bodies in three dimensions. The paper presents the first sublinear-time dynamic structure supporting insertions, deletions, and graph traversal operations. By leveraging hierarchical geometric partitioning, a shrink-and-bifurcate technique, and refined handling of semi-algebraic sets, the structure achieves expected query time $O^*(n/s^{1/3})$ and update time $O^*(s/n)$. This result establishes the first near-optimal complexity for dynamic proximity and intersection queries under general metrics in $\mathbb{R}^3$. Furthermore, it accelerates fundamental graph algorithms—such as intersection graph traversal, Dijkstra’s, and Prim’s—to expected time $O^*(n^{3/2})$, and reverse shortest paths to $O^*(n^{62/39})$.

Abstract

Let $K$ be a compact, centrally-symmetric, strictly-convex region in ${\mathbb R}^3$, which is a semi-algebraic set of constant complexity, i.e. the unit ball of a corresponding metric, denoted as $\|\cdot\|_K$. Let ${\mathcal{K}}$ be a set of $n$ homothetic copies of $K$. This paper contains two main sets of results: (i) For a storage parameter $s\in[n,n^3]$, ${\mathcal{K}}$ can be preprocessed in $O^*(s)$ expected time into a data structure of size $O^*(s)$, so that for a query homothet $K_0$ of $K$, an intersection-detection query (determine whether $K_0$ intersects any member of ${\mathcal{K}}$, and if so, report such a member) or a nearest-neighbor query (return the member of ${\mathcal{K}}$ whose $\|\cdot\|_K$-distance from $K_0$ is smallest) can be answered in $O^*(n/s^{1/3})$ time; all $k$ homothets of ${\mathcal{K}}$ intersecting $K_0$ can be reported in additional $O(k)$ time. In addition, the data structure supports insertions/deletions in $O^*(s/n)$ amortized expected time per operation. Here the $O^*(\cdot)$ notation hides factors of the form $n^\varepsilon$, where $\varepsilon>0$ is an arbitrarily small constant, and the constant of proportionality depends on $\varepsilon$. (ii) Let $\mathcal{G}(\mathcal{K})$ denote the intersection graph of ${\mathcal{K}}$. Using the above data structure, breadth-first or depth-first search on $\mathcal{G}(\mathcal{K})$ can be performed in $O^*(n^{3/2})$ expected time. Combining this result with the so-called shrink-and-bifurcate technique, the reverse-shortest-path problem in a suitably defined proximity graph of ${\mathcal{K}}$ can be solved in $O^*(n^{62/39})$ expected time. Dijkstra's shortest-path algorithm, as well as Prim's MST algorithm, on a $\|\cdot\|_K$-proximity graph on $n$ points in ${\mathbb R}^3$, with edges weighted by $\|\cdot\|_K$, can also be performed in $O^*(n^{3/2})$ time.

Problem

• dynamic nearest-neighbor searching
• general metrics
• semi-algebraic sets
• intersection graph
• proximity graph

Innovation

• dynamic nearest neighbor
• general metric
• semi-algebraic convex body
• intersection graph
• proximity graph