{"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/an-automatic-speedup-theorem-for-distributed","title":"An Automatic Speedup Theorem for Distributed Problems","arxiv_id":"1902.09958","date":"2019-02-26","proceeding":null,"authors":["Sebastian Brandt"],"abstract":"Recently, Brandt et al. [STOC'16] proved a lower bound for the distributed Lov\\'asz Local Lemma, which has been conjectured to be tight for sufficiently relaxed LLL criteria by Chang and Pettie [FOCS'17]. At the heart of their result lies a speedup technique that, for graphs of girth at least $2t+2$, transforms any $t$-round algorithm for one specific LLL problem into a $(t-1)$-round algorithm for the same problem. We substantially improve on this technique by showing that such a speedup exists for any locally checkable problem $\\Pi$, with the difference that the problem $\\Pi_1$ the inferred $(t-1)$-round algorithm solves is not (necessarily) the same problem as $\\Pi$. Our speedup is automatic in the sense that there is a fixed procedure that transforms a description for $\\Pi$ into a description for $\\Pi_1$ and reversible in the sense that any $(t-1)$-round algorithm for $\\Pi_1$ can be transformed into a $t$-round algorithm for $\\Pi$. In particular, for any locally checkable problem $\\Pi$ with exact deterministic time complexity $T(n, \\Delta) \\leq t$ on graphs with $n$ nodes, maximum node degree $\\Delta$, and girth at least $2t+2$, there is a sequence of problems $\\Pi_1, \\Pi_2, \\dots$ with time complexities $T(n, \\Delta)-1, T(n, \\Delta)-2, \\dots$, that can be inferred from $\\Pi$. As a first application of our generalized speedup, we solve a long-standing open problem of Naor and Stockmeyer [STOC'93]: we show that weak $2$-coloring in odd-degree graphs cannot be solved in $o(\\log^* \\Delta)$ rounds, thereby providing a matching lower bound to their upper bound.","url_abs":"http://arxiv.org/abs/1902.09958v1","url_pdf":"http://arxiv.org/pdf/1902.09958v1.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":"an-automatic-speedup-theorem-for-distributed","repo_url":"https://github.com/olidennis/round-eliminator","is_official":0,"mentioned_in_paper":0,"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}