Tight Bounds on the Number of Closest Pairs in Vertical Slabs

TL;DR

This paper studies vertical-slab closest-pair queries in constant-dimensional Euclidean spaces. It introduces the function $f_d(n,m)$, the maximum number of distinct closest pairs across all vertical slabs determined by $m+1$ vertical hyperplanes, and provides tight bounds: $f_d(n,m)=\Theta(m^2)$ when $m=O(\sqrt{n})$ and $f_d(n,m)=\Theta(n\log(m/\sqrt{n}))$ when $m=\omega(\sqrt{n})$, with $f_d(n,n)=\Theta(n\log n)$. The authors prove the upper bounds via a Gupta-Sharathkumar lemma extended to constant dimension using a WSPD-based divide-and-conquer, yielding the recurrence $f_d(n,m)\le 2f_d(n/2,m/2)+O(n)$ and its solution. Complementarily, they construct matching lower bounds in $d=2$ across regimes: $\Omega(m^2)$ for $m\le 3\sqrt{n}$ and $\Omega(n\log(m/\sqrt{n}))$ for larger $m$, including $\Omega(n\log n)$ when $m=n$. Building on these results, they present a linear-space data structure that answers vertical-slab closest-pair queries in $O(n^{1/2+\varepsilon})$ time for any $\varepsilon>0$, by combining a shortest-segment reporting structure, axis-parallel range reporting, and a sparsity bound; selecting $m=\sqrt{n}$ achieves the claimed sublinear query time. Overall, the work advances both the combinatorial understanding of vertical-slab closest pairs and practical, space-efficient query data structures in higher dimensions.

Abstract

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $|A(S,m)|$ be the number of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible value of $|A(S,m)|$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $ε>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+ε})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

Figures (2)

• Figure 1: The pairs in $A(S,n)$ with positive slope that cross $\ell$ do not contain a cycle.
• Figure 2: (A), (B), and (C) are the three regions created by a query $\llbracket H_a,H_b \rrbracket$. The rectangle $R_p$ is the range query for the point $p$ with respect to the query $\llbracket H_a,H_b \rrbracket$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2: Callahan and Kosaraju well-separated-2
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 1
• proof