In this paper, a new algorithm for solving MEB problem is proposed based on new understandings on the geometry property of minimal enclosing ball problem. A substitution of Ritter's algorithm is proposed to get approximate results with higher precision, and a 1+ϵ approximation algorithm is presented to get the approximation with specified precision within much less time comparing with present algorithms.
Like Ritter's algorithm, this algorithm iterates over all points and increase the radius gradually. However, the algorithm does not try to cover all seen points in each step, instead, it will create a new ball (or circle in 2D case) to just touch the new point and cover half of the existing ball. This approach makes sure that the new ball is always increasing in its size and still be smaller than the optimal ball. And finally, a Ritter's algorithm is applied to ensure every point is covered.
The result is an approximate solution to the MEB problem. The radius is usually just slightly bigger than the optimal solution (around 1%) instead (5~20% with Ritter's algorithm).
This paper also explained how to compute 1+ϵ approximation solution, where ϵ is specified to a given precision.
Table of Contents
1. Introduction
2. Related Works
Ritter's bounding sphere
Bădoiu 's Core set based 1+ϵ approximation
Kumar's Core set based 1+ϵ approximation
3. Preliminary
4. Basic Bouncing Bubble algorithm
5. Pruning Bouncing Bubble algorithm
6. Experimental Results
7. Conclusion
Research Objectives and Key Topics
The primary goal of this paper is to introduce more efficient approximation algorithms for the Minimal Enclosing Ball (MEB) problem. The author proposes a "Basic Bouncing Bubble" algorithm as a high-precision, low-latency alternative to Ritter's approach, as well as a "Pruning Bouncing Bubble" algorithm designed to achieve arbitrary precision in significantly less time than existing state-of-the-art solvers.
- Minimal Enclosing Ball (MEB) problem geometry
- Development of the Bouncing Bubble algorithm
- 1+ϵ approximation techniques
- Pruning strategies for optimized computation
- Performance benchmarking against exact and approximate solvers
Excerpt from the Book
4. Basic Bouncing Bubble algorithm
Algorithm 1. Basic Bouncing Bubble
Procedure BouncingBubbleBasic(S)
begin
B1≔ any point p∈S;
for i :=1 to 2
for each p∈S
if ‖pC‖ > r
construct new Bubble Bi+1 according to Lemma 5
Call Ritter's enclosing approach;
end;
This algorithm is easy to be implemented as there is no LP, nor SCOP. It uses only simple iterations to compute a approximation. Thus the computational complexity is minimized. In fact, it walks through the data set only 3 times (same as Ritter's algorithm does).
As simple as it is, in all cases we tested it produces better results than Ritter's approach. In most cases, the error is less than 2%, where Ritter's approach usually result in around 15% error. Thus we propose this algorithm as a substitution of Ritter's algorithm.
Summary of Chapters
1. Introduction: Defines the Minimal Enclosing Ball problem and outlines existing challenges in high-dimensional data mining and computational geometry.
2. Related Works: Provides an overview of established approaches like Ritter’s bounding sphere, Bădoiu’s core-set method, and Kumar’s incremental precision approach.
3. Preliminary: Establishes the mathematical definitions and geometric properties of a "Bubble" used as a basis for the new algorithms.
4. Basic Bouncing Bubble algorithm: Presents a simple, iterative algorithm that acts as an efficient alternative to Ritter's method for better accuracy.
5. Pruning Bouncing Bubble algorithm: Introduces a sophisticated version of the algorithm that prunes points to significantly improve performance for high-precision requirements.
6. Experimental Results: Details the empirical evaluation of the proposed algorithms against different data distributions, dimensions, and size constraints.
7. Conclusion: Summarizes the effectiveness of the proposed approximation algorithms in reducing computation time while maintaining precision.
Keywords
Minimal Enclosing Ball, MEB, Bouncing Bubble algorithm, Bounding sphere, Core set, 1+epsilon approximation, Data mining, Computational geometry, Ritter's algorithm, Pruning, High-dimensional data, Computational complexity, Optimization, Approximation algorithm, Algorithm performance.
Frequently Asked Questions
What is the core focus of this research paper?
The paper focuses on developing faster and more precise approximation algorithms for the Minimal Enclosing Ball (MEB) problem, which is essential for tasks like clustering and nearest neighbor searching.
What are the primary themes discussed?
The research explores geometric properties of MEB, the construction of "bubbles" for covering data points, and the optimization of computational time through pruning and iterative refinement.
What is the main objective or research question?
The objective is to replace existing, less precise or computationally expensive solvers with a "Bouncing Bubble" approach that achieves high precision within limited time constraints.
Which scientific methodology is utilized?
The author uses a geometric approach, deriving lemmas to construct expanding balls ("bubbles") and implementing a pruning technique to eliminate redundant data points during the solving process.
What is covered in the main body of the work?
The main body covers mathematical preliminaries, the basic Bouncing Bubble algorithm, the advanced Pruning Bouncing Bubble algorithm, and extensive experimental validation using various data distributions.
Which keywords best characterize this work?
Key terms include Minimal Enclosing Ball (MEB), 1+ϵ approximation, computational complexity, Bouncing Bubble, and pruning.
Why is the "Bouncing Bubble" name used?
The algorithm is called "Bouncing Bubble" because, during the inflation process, the center of the ball shifts dynamically, appearing to bounce among the data points until it converges.
How does the Pruning Bouncing Bubble algorithm differ from the Basic version?
While the basic version is a simple, iterative substitution for Ritter's algorithm, the pruning version adds a mechanism to remove points that are already inside the calculated boundary, which significantly boosts efficiency for large datasets.
What do the experimental results indicate about execution time?
Results show that execution time scales linearly with the dimension of the space and is remarkably insensitive to the size of the dataset when high precision is required, thanks to the pruning strategy.
- Citation du texte
- Bo Tian (Auteur), 2012, Bouncing Bubble: A fast algorithm for Minimal Enclosing Ball problem, Munich, GRIN Verlag, https://www.grin.com/document/204869
