{"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/balanced-allocation-in-batches-the-tower-of","title":"Balanced Allocations in Batches: The Tower of Two Choices","arxiv_id":"2302.04399","date":"2023-02-09","proceeding":null,"authors":["Dimitrios Los","Thomas Sauerwald"],"abstract":"In balanced allocations, the goal is to place $m$ balls into $n$ bins, so as to minimize the gap (difference of max to average load). The One-Choice process places each ball to a bin sampled independently and uniformly at random. The Two-Choice process places balls in the least loaded of two sampled bins. Finally, the $(1+\\beta)$-process mixes these processes, meaning each ball is allocated using Two-Choice with probability $\\beta\\in(0,1)$, and using One-Choice otherwise. Despite Two-Choice being optimal in the sequential setting, it has been observed in practice that it does not perform well in a parallel environment, where load information may be outdated. Following [BCEFN12], we study such a parallel setting where balls are allocated in batches of size $b$, and balls within the same batch are allocated with the same strategy and based on the same load information. For small batch sizes $b\\in[n,n\\log n]$, it was shown in [LS22a] that Two-Choice achieves an asymptotically optimal gap among all processes with a constant number of samples. In this work, we focus on larger batch sizes $b\\in[n\\log n,n^3]$. It was proved in [LS22c] that Two-Choice leads to a gap of $\\Theta(b/n)$. As our main result, we prove that the gap reduces to $O(\\sqrt{(b/n)\\cdot\\log n})$, if one runs the $(1+\\beta)$-process with an appropriately chosen $\\beta$ (in fact this result holds for a larger class of processes). This not only proves the phenomenon that Two-Choice is not the best (leading to the formation of \"towers\" over previously light bins), but also that mixing two processes (One-Choice and Two-Choice) leads to a process which achieves a gap that is asymptotically smaller than both. We also derive a matching lower bound of $\\Omega(\\sqrt{(b/n)\\cdot\\log n})$ for any allocation process, which demonstrates that the above $(1+\\beta)$-process is asymptotically optimal.","url_abs":"https://arxiv.org/abs/2302.04399v2","url_pdf":"https://arxiv.org/pdf/2302.04399v2.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":"balanced-allocation-in-batches-the-tower-of","repo_url":"https://github.com/Dim131/Batched-23","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}