{"id":324281,"date":"2016-11-18T14:58:28","date_gmt":"2016-11-18T22:58:28","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=324281"},"modified":"2018-10-16T20:38:16","modified_gmt":"2018-10-17T03:38:16","slug":"monotonicity-tester-boolean-functions-hypercube","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/monotonicity-tester-boolean-functions-hypercube\/","title":{"rendered":"A o(n) Monotonicity Tester for Boolean Functions on the Hypercube"},"content":{"rendered":"<p>Given oracle access to a Boolean function <span class=\"typeset\"><span class=\"scale\"><span class=\"icmmi10\">f<\/span><span class=\"icmr10\">:<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmsy10\/alpha\/120\/char66.png\" \/><span class=\"icmr10\">0<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/120\/char3B.png\" \/><span class=\"icmr10\">1<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmsy10\/alpha\/120\/char67.png\" \/><span class=\"size2\"><span class=\"icmmi10\">n<\/span><\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmsy10\/alpha\/120\/char37.png\" \/><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmsy10\/alpha\/120\/char21.png\" \/><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmsy10\/alpha\/120\/char66.png\" \/><span class=\"icmr10\">0<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/120\/char3B.png\" \/><span class=\"icmr10\">1<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmsy10\/alpha\/120\/char67.png\" \/>\u00a0<\/span><\/span>, we design a randomized tester that takes as input a parameter <span class=\"typeset\"><span class=\"scale\"><span class=\"unknown\">\\eps<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/120\/char3E.png\" \/><span class=\"icmr10\">0<\/span><\/span><\/span>, and outputs {\\sf Yes} if the function is monotone, and outputs {\\sf No} with probability <span class=\"typeset\"><span class=\"scale\"><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/120\/char3E.png\" \/><span class=\"icmr10\">2<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/120\/char3D.png\" \/><span class=\"icmr10\">3<\/span>\u00a0<\/span><\/span>, if the function is <span class=\"typeset\"><span class=\"scale\"><span class=\"unknown\">\\eps<\/span><\/span><\/span>-far from monotone. That is, <span class=\"typeset\"><span class=\"scale\"><span class=\"icmmi10\">f<\/span><\/span><\/span> needs to be modified at <span class=\"typeset\"><span class=\"scale\"><span class=\"unknown\">\\eps<\/span><\/span><\/span>-fraction of the points to make it monotone. Our non-adaptive, one-sided tester makes <span class=\"typeset\"><span class=\"scale\"><span class=\"icmmi10\">O<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmex10\/alpha\/120\/char65.png\" \/><span class=\"icmr10\">(<\/span><span class=\"icmmi10\">n<\/span><span class=\"size2\"><span class=\"icmr10\">7<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/85\/char3D.png\" \/><span class=\"icmr10\">8<\/span><\/span><span class=\"unknown\">\\eps<\/span><span class=\"size2\"><span class=\"icmsy10\">\u2212<\/span><span class=\"icmr10\">3<\/span><img decoding=\"async\" src=\"http:\/\/eccc.hpi-web.de\/resources\/jsMath\/fonts\/cmmi10\/alpha\/85\/char3D.png\" \/><span class=\"icmr10\">2<\/span><\/span><span class=\"icmr10\">)<\/span>\u00a0<\/span><\/span> queries to the oracle.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given oracle access to a Boolean function f:01n01\u00a0, we design a randomized tester that takes as input a parameter \\eps0, and outputs {\\sf Yes} if the function is monotone, and outputs {\\sf No} with probability 23\u00a0, if the function is \\eps-far from monotone. That is, f needs to be modified at \\eps-fraction of the points [&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":"ACM New York, NY, USA","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"STOC '13 Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, California, USA","msr_editors":"","msr_how_published":"","msr_isbn":"978-1-4503-2029-0","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"411-418","msr_page_range_start":"411","msr_page_range_end":"418","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"STOC '13 Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, California, USA","msr_doi":"10.1145\/2488608.2488660","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":"2013-06-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488660","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"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-324281","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":"ACM New York, NY, USA","msr_edition":"STOC '13 Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, California, USA","msr_affiliation":"","msr_published_date":"2013-06-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"411-418","msr_chapter":"","msr_isbn":"978-1-4503-2029-0","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:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488660","msr_doi":"10.1145\/2488608.2488660","msr_publication_uploader":[{"type":"url","title":"http:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488660","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1145\/2488608.2488660","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:\/\/dl.acm.org\/citation.cfm?doid=2488608.2488660"}],"msr-author-ordering":[{"type":"user_nicename","value":"dechakr","user_id":31593,"rest_url":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=dechakr"},{"type":"text","value":"C. 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