{"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/linear-time-runs-over-general-ordered","title":"Linear Time Runs over General Ordered Alphabets","arxiv_id":"2102.08670","date":"2021-02-17","proceeding":null,"authors":["Jonas Ellert","Johannes Fischer"],"abstract":"A run in a string is a maximal periodic substring. For example, the string $\\texttt{bananatree}$ contains the runs $\\texttt{anana} = (\\texttt{an})^{3/2}$ and $\\texttt{ee} = \\texttt{e}^2$. There are less than $n$ runs in any length-$n$ string, and computing all runs for a string over a linearly-sortable alphabet takes $\\mathcal{O}(n)$ time (Bannai et al., SODA 2015). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an $\\mathcal{O}(n\\alpha(n))$ time algorithm (where $\\alpha(n)$ is the extremely slowly growing inverse Ackermann function). We show how to achieve $\\mathcal{O}(n)$ time by exploiting combinatorial properties of the Lyndon array, thus proving Kosolobov's conjecture.","url_abs":"https://arxiv.org/abs/2102.08670v1","url_pdf":"https://arxiv.org/pdf/2102.08670v1.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":"linear-time-runs-over-general-ordered","repo_url":"https://github.com/jonas-ellert/linear-time-runs","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}