Optimal Algorithms for Geometric Centers and Depth
📝 Original Paper Info
- Title: Optimal Algorithms for Geometric Centers and Depth- ArXiv ID: 1912.01639
- Date: 2021-12-24
- Authors: Timothy M. Chan and Sariel Har-Peled and Mitchell Jones
📝 Abstract
$\renewcommand{\Re}{\mathbb{R}}$ We develop a general randomized technique for solving "implic it" linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many cases, the structure of the implicitly defined constraints can be exploited in order to obtain efficient linear program solvers. We apply this technique to obtain near-optimal algorithms for a variety of fundamental problems in geometry. For a given point set $P$ of size $n$ in $\Re^d$, we develop algorithms for computing geometric centers of a point set, including the centerpoint and the Tukey median, and several other more involved measures of centrality. For $d=2$, the new algorithms run in $O(n\log n)$ expected time, which is optimal, and for higher constant $d>2$, the expected time bound is within one logarithmic factor of $O(n^{d-1})$, which is also likely near optimal for some of the problems.💡 Summary & Analysis
This paper focuses on developing optimal algorithms for calculating geometric centers in point sets, specifically addressing issues related to centerpoints and Tukey medians. The authors introduce a randomized technique that can efficiently solve "implicit" linear programming problems with constraints defined implicitly by an underlying set of elements. This method is then applied to derive near-optimal algorithms for various geometric centrality measures.The problem at hand involves determining the most accurate representation of the center in a point set, which often requires complex measurements beyond simple averages or medians. The proposed solution leverages randomization to achieve efficient computation times. In two dimensions, the new algorithm runs in $O(n\log n)$ expected time, while for higher dimensions (up to constant $d>2$), it performs within a logarithmic factor of $O(n^{d-1})$. These algorithms significantly improve upon previous methods by reducing computational complexity and providing faster solutions.
The significance of this work lies in its practical applications. Geometric center calculations are crucial in numerous fields such as image processing, data analysis, and machine learning. By providing efficient and optimal algorithms for these tasks, the research contributes to more effective tools and methodologies that can enhance performance and accuracy across various domains.
📄 Full Paper Content (ArXiv Source)
📊 논문 시각자료 (Figures)
