Incremental Planar Nearest Neighbor Queries with Optimal Query Time

TL;DR

The paper delivers the first data-structure family for dynamic 2D nearest neighbor queries with optimal $O(\log n)$ query time and polylogarithmic update time in several regimes. It introduces a novel, geometry-aware variant of fractional cascading built on planar separators, Voronoi diagrams, and a Jump primitive that navigates a multi-level partition of the input while exploiting convex-hull interactions. The main contributions include an insertion-only structure with $O(\log n)$ queries and $O(\log^3 n)$ amortized insertions, plus semi-online, offline fully dynamic, and offline partially persistent extensions that preserve $O(\log n)$ query time under polylogarithmic updates. The techniques hinge on carefully designed sampling and hull-based regions (Hull$_i(j,p_i)$) to avoid paying full $O(\log n)$ searches across all levels, enabling efficient dynamic planar nearest neighbor search with broad applicability to related dynamic geometric problems.

Abstract

In this paper we show that two-dimensional nearest neighbor queries can be answered in optimal $O(\log n)$ time while supporting insertions in $O(\log^{1+\varepsilon}n)$ time. No previous data structure was known that supports $O(\log n)$-time queries and polylog-time insertions. In order to achieve logarithmic queries our data structure uses a new technique related to fractional cascading that leverages the inherent geometry of this problem. Our method can be also used in other semi-dynamic scenarios.

Figures (4)

• Figure 1: Part of a Voronoi diagram for a point set $T_j$. Two elements of $\mathit{Pieces}_j(k)$ have been highlighted, one in striped blue, call it $\mathit{Piece}_j^1(k)$, and one in striped green, call it $\mathit{Piece}_j^2(k)$. For each piece, the cells of fringe vertices are shaded red. Thus, the set $\mathit{Sample}_j(k)$ are the red verticies, and the region $\mathit{Cells}(T_j,\mathit{Sample}_j(k))$ is shaded red. The green-and-red shaded region is $\mathit{Cells}(T_j,\mathit{Sep}_j^2(k))$ and the green-but-not-red shaded region is $\mathit{Cells}(T_j,\overline{\mathit{Sep}}_j^2(k))$
• Figure 2: High complexity cells can occur in Voronoi diagrams. Such cells must be included in fringe verticies $\mathit{Sample}_j(k)$, illustrated in red for some point set $T_j$. This results in the complexity of the boundary of the interior sets $\mathit{Cells}(T_j,\overline{\mathit{Sep}}\_j^\ell(k))$, the connected components of white Voronoi cells, are of complexity $O(d^{4(j-i)})$ by Lemma \ref{['l:inner']}.
• Figure 3: Two iterations of the Jump procedure. The query point $q$ is shown in red. Points $p_i=NN(T_i,q)$, $p_{i+3}=NN(T_{i+3},q)$, and edges $e_i$ and $e_{i+3}$ are shown in blue.
• Figure 4: Illustration of the computation of $\mathit{Hull}_i(j,p_i)$ and $\overline{\mathit{Hull}}_i(j,p_i)$. Observe that $\mathit{Hull}_i(j,p_i)$ is the convex hull of those parts of $\mathit{Cells}(T_j,\mathit{Sample}_j(j-i))$ (shaded pink) that are inside $\mathit{Cell}(T_i,p_i)$ (shaded tan). $\overline{\mathit{Hull}}_i(j,p_i)$ is simply the remainder of $\mathit{Cell}(T_i,p_i)$, and has two connected components, including a small one at the bottom. By Lemma \ref{['lemma:hull']}, the closest point in $T_i \cup T_j$ to all points in $\mathit{Hull}_i(j,p_i)$ is $p_i$.

Theorems & Definitions (31)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6