{"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-height-optimal-tangles-faster","title":"Computing Height-Optimal Tangles Faster","arxiv_id":"1901.06548","date":"2019-01-19","proceeding":null,"authors":["Oksana Firman","Philipp Kindermann","Alexander Ravsky","Alexander Wolff","Johannes Zink"],"abstract":"We study the following combinatorial problem. Given a set of $n$ y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of numbers between 1 and $n$) and an initial order of the wires, a tangle realizes $L$ if each pair of wires changes its order exactly as many times as specified by $L$. The aim is to find a tangle that realizes $L$ using the smallest number of layers. We show that this problem is NP-hard, and we give an algorithm that computes an optimal tangle for $n$ wires and a given list $L$ of swaps in $O((2|L|/n^2+1)^{n^2/2} \\cdot \\varphi^n \\cdot n)$ time, where $\\varphi \\approx 1.618$ is the golden ratio. We can treat lists where every swap occurs at most once in $O(n!\\varphi^n)$ time. We implemented the algorithm for the general case and compared it to an existing algorithm. Finally, we discuss feasibility for lists with a simple structure.","url_abs":"https://arxiv.org/abs/1901.06548v6","url_pdf":"https://arxiv.org/pdf/1901.06548v6.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-height-optimal-tangles-faster","repo_url":"https://github.com/PhKindermann/chaotic-attractors","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}