{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8670","name":"Approximate Hamming Distance in a Stream","abstract":"We consider the problem of computing a (1+epsilon)-approximation of the Hamming distance between a pattern of length n and successive substrings of a stream. We first look at the one-way randomised communication complexity of this problem. We show the following:\r\n\r\n- If Alice and Bob both share the pattern and Alice has the first half of the stream and Bob the second half, then there is an O(epsilon^{-4}*log^2(n)) bit randomised one-way communication protocol.\r\n\r\n- If Alice has the pattern, Bob the first half of the stream and Charlie the second half, then there is an O(epsilon^{-2}*sqrt(n)*log(n)) bit randomised one-way communication protocol. We then go on to develop small space streaming algorithms for (1 + epsilon)-approximate Hamming distance which give worst case running time guarantees per arriving symbol.\r\n\r\n- For binary input alphabets there is an O(epsilon^{-3}*sqrt(n)*log^2(n)) space and O(epsilon^{-2}*log(n)) time streaming\r\n(1 + epsilon)-approximate Hamming distance algorithm.\r\n\r\n- For general input alphabets there is an O(epsilon^{-5}*sqrt(n)*log^4(n)) space and O(epsilon^{-4}*log^3(n)) time streaming\r\n(1 + epsilon)-approximate Hamming distance algorithm.","keywords":["Hamming distance","communication complexity","data stream model"],"author":[{"@type":"Person","name":"Clifford, Rapha\u00ebl","givenName":"Rapha\u00ebl","familyName":"Clifford"},{"@type":"Person","name":"Starikovskaya, Tatiana","givenName":"Tatiana","familyName":"Starikovskaya"}],"position":20,"pageStart":"20:1","pageEnd":"20:14","dateCreated":"2016-08-23","datePublished":"2016-08-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Clifford, Rapha\u00ebl","givenName":"Rapha\u00ebl","familyName":"Clifford"},{"@type":"Person","name":"Starikovskaya, Tatiana","givenName":"Tatiana","familyName":"Starikovskaya"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.20","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/arxiv.org\/abs\/1109.5269","isPartOf":{"@type":"PublicationVolume","@id":"#volume6258","volumeNumber":55,"name":"43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)","dateCreated":"2016-08-23","datePublished":"2016-08-23","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Mitzenmacher, Michael","givenName":"Michael","familyName":"Mitzenmacher"},{"@type":"Person","name":"Rabani, Yuval","givenName":"Yuval","familyName":"Rabani"},{"@type":"Person","name":"Sangiorgi, Davide","givenName":"Davide","familyName":"Sangiorgi"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8670","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":"#volume6258"}}}