Journey to the Center of the Point Set (1712.02949v2)

Abstract: $\renewcommand{\Re}{\mathbb{R}} \newcommand{\eps}{\varepsilon} \newcommand{\Net}{S} \newcommand{\tldO}{{\widetilde{O}}} \newcommand{\body}{C} $ We revisit an algorithm of Clarkson etal [CEMST96], that computes (roughly) a $1/(4d^2)$-centerpoint in $\tldO(d^9)$ time, for a point set in $\Re^d$, where $\tldO$ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a $1/d^2$-centerpoint with running time $\tldO(d^7)$. While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.

• The paper introduces an enhanced centerpoint algorithm that achieves tighter runtime bounds and improved approximation using a novel random walk approach.
• It employs random sampling and iterative refinement to simplify analysis and outperform previous methods in high-dimensional settings.
• The techniques extend to weak ε-net applications and optimization tasks, offering practical solutions for computational geometry challenges.

Computational Aspects of Centerpoints and Weak $-\epsilon$-nets

This paper revisits and extends upon the algorithm by Clarkson et al. (CEM 96), which involves computing approximate centerpoints of point sets in $\mathbb{R}^d$. Specifically, the authors propose an improved algorithm with a tighter bound on its runtime and a marginally better approximation of the centerpoint. They achieve this improvement using a random walk argument, an approach not traditionally employed in computational geometry. Their contributions also extend to novel applications of these theoretical advancements, highlighting practical implications in algorithmic design.

Centerpoint Computation

A noteworthy contribution is the algorithm that computes a $(1/d)$-centerpoint in $O(d^2)$ time. This is a significant step forward in efficiently approximating the traditional centerpoint—a concept rooted in Helly’s theorem—which indicates that every set of $n$ points in $\mathbb{R}^d$ possesses a $(1/(d+1))$-centerpoint. Notably, while previous methods offer a $(1/(4d))$-centerpoint in similar time complexity, the authors’ variant improves upon this both in terms of centerpoint quality and the simplicity of its analysis. Their approach leverages random sampling and a novel iterative process, described in detail through a random walk analogy, to refine the algorithm's computational efficiency.

Weak $-\epsilon$-net Applications

Weak $-\epsilon$-nets can be computationally prohibitive due to their reliance on exponential or larger size dependencies on $d$. This work proposes alternatives, addressing the curse of dimensionality by introducing "functional nets" and "center nets." Functional nets leverage oracle access to a convex body to determine its weight relative to the point set. The complexity of oracle queries is mitigated using centerpoint approximations, employing a technique that adaptively queries based on precomputed samples, therefore minimizing unnecessary overhead.

On a related note, the "center nets" concept refines weak $-\epsilon$-nets by providing a sample of points containing centerpoints for subsets interacting with convex bodies. Such an approach supports enhanced computational geometry solutions where subset intersection needs rapid approximation.

Algorithm Efficiency and Speculative Extensions

An insightful area of application outlined is the lower bounding of convex functions facilitated by gradient-oriented oracle access. This application draws parallels to classical optimization techniques like the ellipsoid method but with enhanced runtime efficiency due to centerpoint-like selection strategies.

In broader implications, the theoretical refinements and applications presented serve as pivotal steps in geometric optimization, approximation algorithms, and computational geometry. The techniques might inspire further developments in high-dimensional data analysis and optimization procedures, encouraging the combination of randomized strategies with geometric properties.

Conclusion

Ultimately, this work walks the line between theoretical enhancement and practical applicability, offering precise yet implementable improvements to a core computational geometry problem. As a result, the proposed methods hold potential for expanding into multifaceted applications of geometric and combinatorial optimization. Its significance is underscored not only by its theoretical contributions but also by heuristic approaches that can be adapted for complex, real-world data scenarios. Future directions may explore extending these methodologies to more general problems in computational geometry or distributed computing frameworks.