{"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/on-two-lz78-style-grammars-compression-bounds","title":"On Two LZ78-style Grammars: Compression Bounds and Compressed-Space Computation","arxiv_id":"1705.09538","date":"2017-05-26","proceeding":null,"authors":["Golnaz Badkobeh","Travis Gagie","Shunsuke Inenaga","Tomasz Kociumaka","Dmitry Kosolobov","Simon J. Puglisi"],"abstract":"We investigate two closely related LZ78-based compression schemes: LZMW (an old scheme by Miller and Wegman) and LZD (a recent variant by Goto et al.). Both LZD and LZMW naturally produce a grammar for a string of length $n$; we show that the size of this grammar can be larger than the size of the smallest grammar by a factor $\\Omega(n^{\\frac{1}3})$ but is always within a factor $O((\\frac{n}{\\log n})^{\\frac{2}{3}})$. In addition, we show that the standard algorithms using $\\Theta(z)$ working space to construct the LZD and LZMW parsings, where $z$ is the size of the parsing, work in $\\Omega(n^{\\frac{5}4})$ time in the worst case. We then describe a new Las Vegas LZD/LZMW parsing algorithm that uses $O (z \\log n)$ space and $O(n + z \\log^2 n)$ time w.h.p..","url_abs":"http://arxiv.org/abs/1705.09538v2","url_pdf":"http://arxiv.org/pdf/1705.09538v2.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":"on-two-lz78-style-grammars-compression-bounds","repo_url":"https://bitbucket.org/dkosolobov/lzd-lzmw","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}