{"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/computing-the-exact-packedness-of-a-curve","title":"Computing The Packedness of Curves","arxiv_id":"2012.04403","date":"2020-12-08","proceeding":null,"authors":["Sepideh Aghamolaei","Vahideh Keikha","Mohammad Ghodsi","Ali Mohades"],"abstract":"A polygonal curve $P$ with $n$ vertices is $c$-packed, if the sum of the lengths of the parts of the edges of the curve that are inside any disk of radius $r$ is at most $cr$, for any $r>0$. Similarly, the concept of $c$-packedness can be defined for any scaling of a given shape. Assuming $L$ is the diameter of $P$ and $\\delta$ is the minimum distance between points on disjoint edges of $P$, we show the approximation factor of the existing $O(\\frac{\\log (L/\\delta)}{\\epsilon}n^3)$ time algorithm is $1+\\epsilon$-approximation algorithm. The massively parallel versions of these algorithms run in $O(\\log (L/\\delta))$ rounds. We improve the existing $O((\\frac{n}{\\epsilon^3})^{\\frac 4 3}\\polylog \\frac n \\epsilon)$ time $(6+\\epsilon)$-approximation algorithm by providing a $(4+\\epsilon)$-approximation $O(n(\\log^2 n)(\\log^2 \\frac{1}{\\epsilon})+\\frac{n}{\\epsilon})$ time algorithm, and the existing $O(n^2)$ time $2$-approximation algorithm improving the existing $O(n^2\\log n)$ time $2$-approximation algorithm. Our exact $c$-packedness algorithm takes $O(n^5)$ time, which is the first exact algorithm for disks. We show using $\\alpha$-fat shapes instead of disks adds a factor $\\alpha^2$ to the approximation. We also give a data-structure for computing the curve-length inside query disks. It has $O(n^6\\log n)$ construction time, uses $O(n^6)$ space, and has query time $O(\\log n+k)$, where $k$ is the number of intersected segments with the query shape. We also give a massively parallel algorithm for relative $c$-packedness with $O(1)$ rounds.","url_abs":"https://arxiv.org/abs/2012.04403v3","url_pdf":"https://arxiv.org/pdf/2012.04403v3.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":"computing-the-exact-packedness-of-a-curve","repo_url":"https://github.com/saghamolaei/Packedness","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}