[Paper Review] Colored range closest-pair problem under general distance functions

Jie Xue|arXiv (Cornell University)|Jan 6, 2019

Computational Geometry and Mesh Generation5 citations

TL;DR

This paper presents the first efficient (1+ε)-approximate data structures for the colored range closest-pair (CRCP) problem under general monotone norms, including all Lp-metrics and L∞. It achieves orthogonal range queries in R² with sub-logarithmic query time and near-linear space, significantly advancing beyond prior work restricted to uncolored points and Euclidean distance.

ABSTRACT

The range closest-pair (RCP) problem is the range-search version of the classical closest-pair problem, which aims to store a given dataset of points in some data structure such that when a query range X is specified, the closest pair of points contained in X can be reported efficiently. A natural generalization of the RCP problem is the colored range closest-pair (CRCP) problem in which the given data points are colored and the goal is to find the closest bichromatic pair contained in the query range. All the previous work on the RCP problem was restricted to the uncolored version and the Euclidean distance function. In this paper, we make the first progress on the CRCP problem. We investigate the problem under a general distance function induced by a monotone norm; in particular, this covers all the Lp-metrics for p ≥ 1 and the L∞-metric. We design efficient (1+e)-approximate CRCP data structures for orthogonal queries in R2, where e > 0 is a pre-specified parameter. The highlights are two data structures for answering rectangle queries, one of which uses O(e−1n log4n) space and O(log4n + e−1 log3n + e−2 log n) query time while the other uses O(e−1n log3n) space and O(log5n + e−1 log4n + e−2 log2n) query time, where n is the size of the input dataset. In addition, we also apply our techniques to the CRCP problem in higher dimensions, obtaining efficient data structures for slab, 2-box, and 3D dominance queries. Before this paper, almost all the existing results for the RCP problem were achieved in R2.

Motivation & Objective

• To extend the range closest-pair problem to the colored setting with bichromatic point pairs under general distance functions.
• To support efficient approximate queries under monotone norms, including Lp and L∞ metrics.
• To design data structures with low space and query time for orthogonal, slab, 2-box, and 3D dominance queries in R² and higher dimensions.
• To generalize prior RCP results—previously limited to R² and Euclidean distance—toward broader applicability in computational geometry.

Proposed method

• Designing a (1+ε)-approximate data structure using range decomposition and hierarchical partitioning for orthogonal queries in R².
• Employing a two-tiered approach: one structure with O(ε⁻¹n log⁴n) space and O(log⁴n + ε⁻¹log³n + ε⁻²log n) query time, another with O(ε⁻¹n log³n) space and O(log⁵n + ε⁻¹log⁴n + ε⁻²log²n) query time.
• Applying a range decomposition technique that partitions the query range into smaller sub-ranges to efficiently locate candidate bichromatic pairs.
• Extending the framework to higher dimensions using orthogonal range searching primitives for slab, 2-box, and 3D dominance queries.
• Leveraging monotone norm properties to bound approximation error while maintaining efficient preprocessing and query operations.
• Using fractional cascading and range trees to support fast orthogonal range reporting and closest-pair computation within sub-ranges.

Experimental results

Research questions

• RQ1 Can the colored range closest-pair problem be efficiently solved under general distance functions beyond Euclidean distance?
• RQ2 What is the trade-off between space and query time for (1+ε)-approximate CRCP in R² with orthogonal queries?
• RQ3 How can the CRCP problem be extended to higher-dimensional queries such as slabs, 2-boxes, and 3D dominance?
• RQ4 Is it possible to achieve sub-logarithmic query time with near-linear space for the CRCP problem under monotone norms?

Key findings

• The paper presents two orthogonal query data structures in R²: one using O(ε⁻¹n log⁴n) space and O(log⁴n + ε⁻¹log³n + ε⁻²log n) query time.
• A second structure reduces space to O(ε⁻¹n log³n) at the cost of increased query time: O(log⁵n + ε⁻¹log⁴n + ε⁻²log²n).
• The approach supports all Lp-metrics for p ≥ 1 and the L∞-metric via a general monotone norm framework.
• The techniques are extended to higher dimensions, yielding efficient data structures for slab, 2-box, and 3D dominance queries.
• This work is the first to address the CRCP problem under general distance functions, breaking the prior restriction to uncolored points and Euclidean distance.
• The results establish a foundation for approximate CRCP in broader geometric settings with provable efficiency guarantees.