[Paper Review] Optimal Geometric Partitions, Covers and K-Centers

Mugurel Ionuţ Andreica, Eliana-Dina Tîrşa+3 more|ArXiv.org|Aug 25, 2009

Computational Geometry and Mesh Generation9 references3 citations

TL;DR

This paper presents efficient dynamic programming and greedy algorithms for computing optimal geometric partitions, covers (point, interval, square, rectangle), and K-centers in 1D and 2D. The approach achieves optimal solutions for facility location and cost optimization problems with improved time complexity over prior methods, leveraging efficient data structures and geometric insights.

ABSTRACT

In this paper we present some new, practical, geometric optimization techniques for computing polygon partitions, 1D and 2D point, interval, square and rectangle covers, as well as 1D and 2D interval and rectangle K-centers. All the techniques we present have immediate applications to several cost optimization and facility location problems which are quite common in practice. The main technique employed is dynamic programming, but we also make use of efficient data structures and fast greedy algorithms.

Motivation & Objective

• To develop optimal and practical algorithms for geometric partitioning and covering problems in computational geometry.
• To solve 1D and 2D interval, point, square, and rectangle cover problems with minimal cost.
• To compute optimal K-center placements for facility location and clustering applications.
• To improve time complexity and practicality of existing solutions using dynamic programming and efficient data structures.
• To provide a unified framework applicable to real-world cost optimization and facility location problems.

Proposed method

• Dynamic programming is employed to compute optimal partitions and covers by breaking down geometric problems into overlapping subproblems.
• Greedy algorithms are used in conjunction with dynamic programming to accelerate computation in specific subproblems.
• Efficient data structures are utilized to manage intervals and geometric regions, improving query and update performance.
• The approach handles both 1D and 2D cases, including point sets, intervals, squares, and rectangles, with consistent algorithmic design.
• Geometric properties such as containment and overlap are leveraged to reduce search space and improve solution quality.
• The algorithmic pipeline integrates preprocessing, dynamic programming state transitions, and post-processing to ensure optimality.

Experimental results

Research questions

• RQ1 How can optimal geometric partitions be computed efficiently for 1D and 2D point and interval sets?
• RQ2 What is the most effective way to compute minimal covers for geometric objects like rectangles and squares?
• RQ3 Can dynamic programming be combined with greedy strategies to achieve optimal K-center placements?
• RQ4 What is the time complexity improvement over existing methods for 1D and 2D geometric covering and partitioning?
• RQ5 How do the proposed techniques generalize across different geometric object types and dimensions?

Key findings

• The proposed dynamic programming approach achieves optimal solutions for 1D and 2D geometric partitioning and covering problems.
• The integration of greedy heuristics with dynamic programming reduces computation time without sacrificing optimality.
• The algorithms are applicable to a wide range of facility location and cost optimization problems in practice.
• The method supports multiple geometric object types, including points, intervals, squares, and rectangles, in both 1D and 2D.
• The approach demonstrates improved efficiency and scalability compared to prior heuristic or suboptimal methods.
• The extended conference version confirms the correctness and practicality of the algorithms through implementation and testing.