[Paper Review] Bottleneck Partial-Matching Voronoi Diagrams

Matthias Henze, Rafel Jaume|arXiv (Cornell University)|May 5, 2014

Computational Geometry and Mesh Generation10 references3 citations

TL;DR

This paper introduces bottleneck partial-matching Voronoi diagrams to minimize the lexicographic bottleneck distance between a smaller point set and a subset of equal size from a larger set under translation. By relating the problem to two novel Voronoi-type diagrams, the authors establish optimal polynomial combinatorial complexity bounds and present efficient construction algorithms that solve the minimization problem and enable solutions to related geometric matching problems.

ABSTRACT

Abstract. Given two point sets in the plane, we study the minimiza-tion of the (lexicographic) bottleneck distance between the smaller set and an equally sized subset of the larger set under translations. We re-late this problem to two Voronoi-type diagrams and derive polynomial bounds for their combinatorial complexity that are optimal in the size of the larger set. We devise efficient construction algorithms for these dia-grams which are used to solve the minimization problem and, moreover, can be used to solve other related problems. 1.

Motivation & Objective

• To minimize the lexicographic bottleneck distance between a smaller point set and an equally sized subset of a larger point set under translation.
• To relate this geometric matching problem to two new types of Voronoi diagrams with provable combinatorial complexity.
• To derive tight polynomial bounds on the complexity of these diagrams that are optimal in the size of the larger point set.
• To design efficient algorithms for constructing the diagrams, enabling solutions to the bottleneck matching problem and related geometric optimization tasks.

Proposed method

• Model the bottleneck matching problem as a geometric optimization under translation, focusing on minimizing the largest distance in a matching.
• Introduce two Voronoi-type diagrams that encode the bottleneck distance behavior across translations.
• Prove that the combinatorial complexity of these diagrams is polynomial and optimal in the size of the larger point set.
• Develop efficient divide-and-conquer and plane-sweep based algorithms for constructing the diagrams in near-optimal time.
• Use the diagrams to compute the translation that minimizes the bottleneck distance between the smaller set and a subset of the larger set.
• Leverage the structural properties of the diagrams to solve related geometric matching and optimization problems.

Experimental results

Research questions

• RQ1 What is the combinatorial complexity of the Voronoi-type diagrams that capture bottleneck partial-matching behavior under translation?
• RQ2 Can the bottleneck distance between a smaller point set and a subset of a larger set be minimized efficiently using geometric diagram structures?
• RQ3 How can the structure of these diagrams be exploited to solve the bottleneck matching problem under translation?
• RQ4 Are the proposed diagrams optimal in terms of combinatorial complexity with respect to the size of the larger point set?
• RQ5 What algorithmic techniques enable efficient construction of these diagrams for practical use in geometric optimization?

Key findings

• The proposed bottleneck partial-matching Voronoi diagrams achieve optimal polynomial combinatorial complexity in the size of the larger point set.
• The diagrams can be constructed efficiently, enabling the solution of the bottleneck matching problem under translation in near-optimal time.
• The lexicographic bottleneck distance minimization problem is solved via the structural properties of the diagrams.
• The framework supports solving other related geometric matching problems beyond the primary optimization task.
• The theoretical bounds on complexity are tight and match the worst-case size of the larger input set.