Covering compact metric spaces greedily

TL;DR

This paper introduces a greedy method for covering compact metric spaces with metric balls, providing a unified analysis and simplified proofs for bounds on coverings of spheres and Euclidean spaces.

Contribution

It presents a general greedy algorithm for metric space coverings and offers a continuous analysis analogous to set cover, with applications to sphere and Euclidean space coverings.

Findings

Efficient coverings of n-dimensional spheres demonstrated.

Simplified proofs for known covering bounds.

Unified greedy approach for metric space coverings.

Abstract

A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvatal's analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give short and transparent proofs of several best known bounds obtained from deterministic constructions in the literature on sphere coverings.