{"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/improved-lower-bounds-on-the-expected-length","title":"Improved Lower Bounds on the Expected Length of Longest Common Subsequences","arxiv_id":"2407.10925","date":"2024-07-15","proceeding":null,"authors":["George T. Heineman","Chase Miller","Daniel Reichman","Andrew Salls","Gábor Sárközy","Duncan Soiffer"],"abstract":"It has been proven that, when normalized by $n$, the expected length of a longest common subsequence of $d$ random strings of length $n$ over an alphabet of size $\\sigma$ converges to some constant that depends only on $d$ and $\\sigma$. These values are known as the Chv\\'{a}tal-Sankoff constants, and determining their exact values is a well-known open problem. Upper and lower bounds are known for some combinations of $\\sigma$ and $d$, with the best lower and upper bounds for the most studied case, $\\sigma=2, d=2$, at $0.788071$ and $0.826280$, respectively. Building off previous algorithms for lower-bounding the constants, we implement runtime optimizations, parallelization, and an efficient memory reading and writing scheme to obtain an improved lower bound of $0.792665992$ for $\\sigma=2, d=2$. We additionally improve upon almost all previously reported lower bounds for the Chv\\'{a}tal-Sankoff constants when either the size of alphabet, the number of strings, or both are larger than 2.","url_abs":"https://arxiv.org/abs/2407.10925v1","url_pdf":"https://arxiv.org/pdf/2407.10925v1.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":"improved-lower-bounds-on-the-expected-length","repo_url":"https://github.com/statistics-of-subsequences/papermaterials","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}