{"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/tight-bounds-for-repeated-balls-into-bins","title":"Tight Bounds for Repeated Balls-into-Bins","arxiv_id":"2203.12400","date":"2022-03-23","proceeding":null,"authors":["Dimitrios Los","Thomas Sauerwald"],"abstract":"We study the repeated balls-into-bins process introduced by Becchetti, Clementi, Natale, Pasquale and Posta (2019). This process starts with $m$ balls arbitrarily distributed across $n$ bins. At each round $t=1,2,\\ldots$, one ball is selected from each non-empty bin, and then placed it into a bin chosen independently and uniformly at random. We prove the following results: $\\quad \\bullet$ For any $n \\leq m \\leq \\mathrm{poly}(n)$, we prove a lower bound of $\\Omega(m/n \\cdot \\log n)$ on the maximum load. For the special case $m=n$, this matches the upper bound of $O(\\log n)$, as shown in [BCNPP19]. It also provides a positive answer to the conjecture in [BCNPP19] that for $m=n$ the maximum load is $\\omega(\\log n/ \\log \\log n)$ at least once in a polynomially large time interval. For $m\\in [\\omega(n),n\\log n]$, our new lower bound disproves the conjecture in [BCNPP19] that the maximum load remains $O(\\log n)$. $\\quad \\bullet$ For any $n\\leq m\\leq\\mathrm{poly}(n)$, we prove an upper bound of $O(m/n\\cdot\\log n)$ on the maximum load for all steps of a polynomially large time interval. This matches our lower bound up to multiplicative constants. $\\quad \\bullet$ For any $m\\geq n$, our analysis also implies an $O(m^2/n)$ waiting time to reach a configuration with a $O(m/n\\cdot\\log m)$ maximum load, even for worst-case initial distributions. $\\quad \\bullet$ For any $m \\geq n$, we show that every ball visits every bin in $O(m\\log m)$ rounds. For $m = n$, this improves the previous upper bound of $O(n \\log^2 n)$ in [BCNPP19]. We also prove that the upper bound is tight up to multiplicative constants for any $n \\leq m \\leq \\mathrm{poly}(n)$.","url_abs":"https://arxiv.org/abs/2203.12400v2","url_pdf":"https://arxiv.org/pdf/2203.12400v2.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":"tight-bounds-for-repeated-balls-into-bins","repo_url":"https://github.com/Dim131/RBB","is_official":1,"mentioned_in_paper":0,"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}