{"id":162033,"date":"2011-01-01T00:00:00","date_gmt":"2011-01-01T00:00:00","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/msr-research-item\/streaming-algorithms-via-precision-sampling\/"},"modified":"2018-10-16T20:07:17","modified_gmt":"2018-10-17T03:07:17","slug":"streaming-algorithms-via-precision-sampling","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/streaming-algorithms-via-precision-sampling\/","title":{"rendered":"Streaming Algorithms via Precision Sampling"},"content":{"rendered":"<div class=\"asset-content\">\n<p>A technique introduced by Indyk and Woodruff (STOC 2005) has inspired several recent advances in data-stream algorithms. We show that a number of these results follow easily from the application of a single probabilistic method called Precision Sampling. Using this method, we obtain simple data-stream algorithms that maintain a randomized sketch of an input vector <i>x=(x<sub>1<\/sub>,x<sub>2<\/sub>,\u2026,x<sub>n<\/sub>)<\/i>, which is useful for the following applications:<\/p>\n<ul>\n<li>Estimating the <i>F<sub>k<\/sub><\/i>-moment of <i>x<\/i>, for <i>k>2<\/i>.<\/li>\n<li>Estimating the <i>\u2113<sub>p<\/sub><\/i>-norm of <i>x<\/i>, for <i>p\u2208[1,2]<\/i>, with small update time.<\/li>\n<li>Estimating cascaded norms <i>\u2113<sub>p<\/sub>(\u2113<sub>q<\/sub>)<\/i> for all <i>p,q>0<\/i>.<\/li>\n<li><i>\u2113<sub>1<\/sub><\/i> sampling, where the goal is to produce an element <i>i<\/i> with probability (approximately) <i>|x<sub>i<\/sub>|\/\\|x\\|<sub>1<\/sub><\/i>. It extends to similarly defined <i>\u2113<sub>p<\/sub><\/i>-sampling, for <i>p\u2208 [1,2]<\/i>.<\/li>\n<\/ul>\n<p>For all these applications the algorithm is essentially the same: scale the vector <i>x<\/i> entry-wise by a well-chosen random vector, and run a heavy-hitter estimation algorithm on the resulting vector. Our sketch is a linear function of <i>x<\/i>, thereby allowing general updates to the vector <i>x<\/i>.<\/p>\n<p>Precision Sampling itself addresses the problem of estimating a sum <i>sum<sub>i=1<\/sub><sup>n<\/sup> a<sub>i<\/sub><\/i> from weak estimates of each real <i>a<sub>i<\/sub>\u2208[0,1]<\/i>. More precisely, the estimator first chooses a desired precision <i>u<sub>i<\/sub>\u2208(0,1]<\/i> for each <i>i\u2208[n]<\/i>, and then it receives an estimate of every <i>a<sub>i<\/sub><\/i> within additive <i>u<sub>i<\/sub><\/i>. Its goal is to provide a good approximation to <i>sum a<sub>i<\/sub><\/i> while keeping a tab on the \u201capproximation cost\u201d <i>sum<sub>i<\/sub> (1\/u<sub>i<\/sub>)<\/i>. Here we refine previous work (Andoni, Krauthgamer, and Onak, FOCS 2010) which shows that as long as <i>sum a<sub>i<\/sub>=\u03a9(1)<\/i>, a good multiplicative approximation can be achieved using total precision of only <i>O(nlog n)<\/i>.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A technique introduced by Indyk and Woodruff (STOC 2005) has inspired several recent advances in data-stream algorithms. We show that a number of these results follow easily from the application of a single probabilistic method called Precision Sampling. Using this method, we obtain simple data-stream algorithms that maintain a randomized sketch of an input vector [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"IEEE","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Symposium on Foundations of Computer Science (FOCS)","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"Symposium on Foundations of Computer Science (FOCS)","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2011-01-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/arxiv.org\/abs\/1011.1263","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":2011,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13561,13546],"msr-publication-type":[193716],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-162033","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"IEEE","msr_edition":"Symposium on Foundations of Computer Science (FOCS)","msr_affiliation":"","msr_published_date":"2011-01-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"http:\/\/arxiv.org\/abs\/1011.1263","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"http:\/\/arxiv.org\/abs\/1011.1263","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"http:\/\/arxiv.org\/abs\/1011.1263"}],"msr-author-ordering":[{"type":"user_nicename","value":"andoni","user_id":31012,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=andoni"},{"type":"text","value":"Robert Krauthgamer","user_id":0,"rest_url":false},{"type":"text","value":"Krzysztof Onak","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"inproceedings","related_content":[],"_links":{"self":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/162033","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/162033\/revisions"}],"predecessor-version":[{"id":522668,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/162033\/revisions\/522668"}],"wp:attachment":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media?parent=162033"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=162033"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=162033"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=162033"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=162033"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=162033"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=162033"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=162033"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=162033"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=162033"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=162033"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=162033"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=162033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}