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ACM"],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>\n            Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of\n            <jats:italic>n<\/jats:italic>\n            items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in\n            <jats:italic>n<\/jats:italic>\n            . Given the sketch and a query item\n            <jats:italic>y<\/jats:italic>\n            , one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to\n            <jats:italic>y<\/jats:italic>\n            .\n          <\/jats:p>\n          <jats:p>\n            Most works to date focused on additive \u03b5\n            <jats:italic>n<\/jats:italic>\n            error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This article investigates\n            <jats:italic>multiplicative<\/jats:italic>\n            (1\u00b1 \u03b5)-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values.\n          <\/jats:p>\n          <jats:p>\n            The most space-efficient algorithms due to prior work store either O(log (\u03b5\n            <jats:sup>2<\/jats:sup>\n            <jats:italic>n<\/jats:italic>\n            )\/\u03b5\n            <jats:sup>2<\/jats:sup>\n            ) or\n            <jats:italic>O<\/jats:italic>\n            (log\n            <jats:sup>3<\/jats:sup>\n            (\u03b5\n            <jats:italic>n<\/jats:italic>\n            )\/\u03b5) universe items. We present a randomized sketch storing\n            <jats:italic>O<\/jats:italic>\n            (log\n            <jats:sup>1.5<\/jats:sup>\n            (\u03b5\n            <jats:italic>n<\/jats:italic>\n            )\/\u03b5) items that can (1\u00b1 \u03b5)-approximate the rank of each universe item with high constant probability; this space bound is within an\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(O(\\sqrt {\\log (\\varepsilon n)})\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments.\n          <\/jats:p>","DOI":"10.1145\/3617891","type":"journal-article","created":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T11:13:46Z","timestamp":1693480426000},"page":"1-48","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Relative Error Streaming Quantiles"],"prefix":"10.1145","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0698-0922","authenticated-orcid":false,"given":"Graham","family":"Cormode","sequence":"first","affiliation":[{"name":"University of Warwick, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-0108-289X","authenticated-orcid":false,"given":"Zohar","family":"Karnin","sequence":"additional","affiliation":[{"name":"Amazon, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3132-2785","authenticated-orcid":false,"given":"Edo","family":"Liberty","sequence":"additional","affiliation":[{"name":"Pinecone, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2234-6225","authenticated-orcid":false,"given":"Justin","family":"Thaler","sequence":"additional","affiliation":[{"name":"Georgetown University, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1169-7934","authenticated-orcid":false,"given":"Pavel","family":"Vesel\u00fd","sequence":"additional","affiliation":[{"name":"Charles University, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2023,10,16]]},"reference":[{"key":"e_1_3_4_2_2","first-page":"26","article-title":"Mergeable summaries","volume":"38","author":"Agarwal Pankaj K.","year":"2013","unstructured":"Pankaj K. 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IEEE, 51\u201351."}],"container-title":["Journal of the ACM"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3617891","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3617891","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T16:37:57Z","timestamp":1750178277000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3617891"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,16]]},"references-count":28,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10,31]]}},"alternative-id":["10.1145\/3617891"],"URL":"https:\/\/doi.org\/10.1145\/3617891","relation":{},"ISSN":["0004-5411","1557-735X"],"issn-type":[{"value":"0004-5411","type":"print"},{"value":"1557-735X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,16]]},"assertion":[{"value":"2021-12-07","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-08-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-10-16","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}