Tight Bounds on the Number of Closest Pairs in Vertical Slabs

2025-02-24

AI Summary

This paper investigates the maximum possible number |A(S,m)| of closest-pair distances among n points in ℝᵈ when partitioned by m+1 mutually orthogonal hyperplanes into O(m²) vertical slabs. We establish the first tight asymptotic bound Θ(nm) for this quantity. Building on geometric analysis, divide-and-conquer, slab decomposition, hyperplane arrangements, and extremal combinatorial constructions, we further design the first linear-space O(n)-size data structure supporting closest-pair queries within arbitrary vertical query slabs in O(n^{1/2+ε}) time—breaking the prior barrier of no sublinear-time solutions. Our contributions thus yield both a theoretical tight bound and a practically efficient query mechanism, advancing the state of the art in geometric range searching and proximity problems.

Technology Category

• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Bridging structured and unstructured data

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 $epsilon>0$, 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 $O(n^{1/2+epsilon})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

Problem

• Tight bounds on closest pairs
• Vertical slabs in R^d
• Linear space data structure

Innovation

• Vertical hyperplanes partition optimization
• Linear space data structure
• Sublinear query time algorithm