{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:08:58Z","timestamp":1750219738111,"version":"3.41.0"},"reference-count":63,"publisher":"Association for Computing Machinery (ACM)","issue":"6","license":[{"start":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T00:00:00Z","timestamp":1701302400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2023,12,31]]},"abstract":"<jats:p>\n            We consider the allocation of\n            <jats:italic>m<\/jats:italic>\n            balls (jobs) into\n            <jats:italic>n<\/jats:italic>\n            bins (servers). In the standard\n            <jats:sc>Two-Choice<\/jats:sc>\n            process, at each step\n            <jats:italic>t<\/jats:italic>\n            =1,2,... ,m we first sample\n            <jats:italic>two<\/jats:italic>\n            randomly chosen bins, compare their two loads and then place a ball in the least loaded bin. It is well-known that for any\n            <jats:italic>m<\/jats:italic>\n            \u2a7e\n            <jats:italic>n<\/jats:italic>\n            , this results in a gap (difference between the maximum and average load) of log\n            <jats:sub>2<\/jats:sub>\n            log\n            <jats:italic>n<\/jats:italic>\n            + \u0398 (1) (with high probability).\n          <\/jats:p>\n          <jats:p>\n            In this work, we consider\n            <jats:sc>Two-Choice<\/jats:sc>\n            in different settings with noisy load comparisons. One key setting involves an adaptive adversary whose power is limited by some threshold\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(g \\in \\mathbb {N}\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            . In each step, such adversary can determine the result of any load comparison between two bins whose loads differ by at most\n            <jats:italic>g<\/jats:italic>\n            , while if the load difference is greater than\n            <jats:italic>g<\/jats:italic>\n            , the comparison is correct.\n          <\/jats:p>\n          <jats:p>\n            For this adversarial setting, we first prove that for any\n            <jats:italic>m<\/jats:italic>\n            \u2a7e\n            <jats:italic>n<\/jats:italic>\n            the gap is\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(\\mathcal {O}(g+\\log n)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            with high probability. Then through a refined analysis we prove that if\n            <jats:italic>g<\/jats:italic>\n            \u2a7d log\n            <jats:italic>n<\/jats:italic>\n            , then for any\n            <jats:italic>m<\/jats:italic>\n            \u2a7e\n            <jats:italic>n<\/jats:italic>\n            the gap is\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(\\mathcal {O}(\\frac{g}{\\log g} \\cdot \\log \\log n)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            . For constant values of\n            <jats:italic>g<\/jats:italic>\n            , this generalizes the heavily loaded analysis of [\n            <jats:xref ref-type=\"bibr\">19<\/jats:xref>\n            ,\n            <jats:xref ref-type=\"bibr\">61<\/jats:xref>\n            ] for the\n            <jats:sc>Two-Choice<\/jats:sc>\n            process, and establishes that asymptotically the same gap bound holds even if load comparisons among \u201csimilarly loaded\u201d bins are wrong. Finally, we complement these upper bounds with tight lower bounds, which establish an interesting phase transition on how the parameter\n            <jats:italic>g<\/jats:italic>\n            impacts the gap.\n          <\/jats:p>\n          <jats:p>\n            The analysis also applies to settings with outdated and delayed information. For example, for the setting of [\n            <jats:xref ref-type=\"bibr\">18<\/jats:xref>\n            ]\u00a0where balls are allocated in consecutive batches of size\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(b = n\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            , we present an improved and tight gap bound of\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(\\Theta (\\frac{\\log n}{\\log \\log n})\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            . This bound also extends for a range of values of\n            <jats:italic>b<\/jats:italic>\n            and applies to a relaxed setting where the reported load of a bin can be any load value from the last\n            <jats:italic>b<\/jats:italic>\n            steps.\n          <\/jats:p>","DOI":"10.1145\/3625386","type":"journal-article","created":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T08:35:29Z","timestamp":1695803729000},"page":"1-84","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Balanced Allocations with the Choice of Noise"],"prefix":"10.1145","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1574-5166","authenticated-orcid":false,"given":"Dimitrios","family":"Los","sequence":"first","affiliation":[{"name":"Department of Computer Science and Technology, University of Cambridge, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0882-283X","authenticated-orcid":false,"given":"Thomas","family":"Sauerwald","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Technology, University of Cambridge, UK"}]}],"member":"320","published-online":{"date-parts":[[2023,11,30]]},"reference":[{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199809)13:2<159::AID-RSA3>3.0.CO;2-Q"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1145\/1060590.1060692"},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975031.144"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/3210377.3210411"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.4230\/LIPIcs.OPODIS.2021.14"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.4230\/LIPIcs.ICALP.2020.7"},{"key":"e_1_3_3_8_2","doi-asserted-by":"publisher","DOI":"10.1214\/09-AAP656"},{"key":"e_1_3_3_9_2","unstructured":"Yossi Azar Andrei Z. 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