{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-maximum-distance-problem-and-minimal","title":"The Maximum Distance Problem and Minimal Spanning Trees","arxiv_id":"2004.07323","date":"2020-04-15","proceeding":null,"authors":["Enrique G. Alvarado","Bala Krishnamoorthy","Kevin R. Vixie"],"abstract":"Given a compact $E \\subset \\mathbb{R}^n$ and $s > 0$, the maximum distance problem seeks a compact and connected subset of $\\mathbb{R}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $E\\subset \\mathbb{R}^2$, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius $s$, which cover $E$, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $\\Gamma$ in $\\mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $\\Gamma_\\ast$, where $\\mathcal{H}^1(\\Gamma_\\ast)$ is arbitrarily close to $\\mathcal{H}^1(\\Gamma)$. We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at https://github.com/mtdaydream/MDP_MST.","url_abs":"https://arxiv.org/abs/2004.07323v3","url_pdf":"https://arxiv.org/pdf/2004.07323v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-maximum-distance-problem-and-minimal","repo_url":"https://github.com/mtdaydream/MDP_MST","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}