{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8596","name":"Efficient Summing over Sliding Windows","abstract":"This paper considers the problem of maintaining statistic aggregates over the last W elements of a data stream. First, the problem of counting the number of 1's in the last W bits of a binary stream is considered. A lower bound of Omega(1\/epsilon + log(W)) memory bits for Wepsilon-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1\/epsilon + log(W)) bits, indicating that the algorithm is optimal and that the bound is tight. Next, the more general problem of maintaining a sum of the last W integers, each in the range of {0, 1, ..., R}, is addressed. The paper shows that approximating the sum within an additive error of RW epsilon can also be done using Theta(1\/epsilon + log(W)) bits for epsilon = Omega(1\/W). For epsilon = o(1\/W), we present a succinct algorithm which uses B(1 + o(1)) bits, where B = Theta(W*log(1\/(W*epsilon))) is the derived lower bound. We show that all lower bounds generalize to randomized algorithms as well. All algorithms process new elements and answer queries in O(1) worst-case time.","keywords":["Streaming","Statistics","Lower Bounds"],"author":[{"@type":"Person","name":"Ben Basat, Ran","givenName":"Ran","familyName":"Ben Basat"},{"@type":"Person","name":"Einziger, Gil","givenName":"Gil","familyName":"Einziger"},{"@type":"Person","name":"Friedman, Roy","givenName":"Roy","familyName":"Friedman"},{"@type":"Person","name":"Kassner, Yaron","givenName":"Yaron","familyName":"Kassner"}],"position":11,"pageStart":"11:1","pageEnd":"11:14","dateCreated":"2016-06-22","datePublished":"2016-06-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ben Basat, Ran","givenName":"Ran","familyName":"Ben Basat"},{"@type":"Person","name":"Einziger, Gil","givenName":"Gil","familyName":"Einziger"},{"@type":"Person","name":"Friedman, Roy","givenName":"Roy","familyName":"Friedman"},{"@type":"Person","name":"Kassner, Yaron","givenName":"Yaron","familyName":"Kassner"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SWAT.2016.11","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1604.02450","http:\/\/dx.doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2014.96","http:\/\/dx.doi.org\/10.1007\/978-3-540-78773-0_60","http:\/\/dx.doi.org\/10.1007\/978-3-642-45030-3_48"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6256","volumeNumber":53,"name":"15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)","dateCreated":"2016-06-22","datePublished":"2016-06-22","editor":{"@type":"Person","name":"Pagh, Rasmus","givenName":"Rasmus","familyName":"Pagh"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8596","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6256"}}}