{"id":324269,"date":"2016-11-18T14:48:14","date_gmt":"2016-11-18T22:48:14","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=324269"},"modified":"2018-10-16T20:37:30","modified_gmt":"2018-10-17T03:37:30","slug":"optimal-lower-bound-monotonicity-testing-hypergrids","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/optimal-lower-bound-monotonicity-testing-hypergrids\/","title":{"rendered":"An Optimal Lower Bound for Monotonicity Testing Over Hypergrids"},"content":{"rendered":"<p class=\"Para\">For positive integers <em class=\"EmphasisTypeItalic \">n<\/em>, <em class=\"EmphasisTypeItalic \">d<\/em>, consider the hypergrid [<em class=\"EmphasisTypeItalic \">n<\/em>]<sup> <em class=\"EmphasisTypeItalic \">d<\/em> <\/sup>with the coordinate-wise product partial ordering denoted by \u227a. A function <em class=\"EmphasisTypeItalic \">f<\/em>: [<em class=\"EmphasisTypeItalic \">n<\/em>]<sup> <em class=\"EmphasisTypeItalic \">d<\/em> <\/sup>\u2009\u2192\u2009\u2115 is monotone if \u2200\u2009<em class=\"EmphasisTypeItalic \">x<\/em>\u2009\u227a\u2009<em class=\"EmphasisTypeItalic \">y<\/em>, <em class=\"EmphasisTypeItalic \">f<\/em>(<em class=\"EmphasisTypeItalic \">x<\/em>)\u2009\u2264\u2009<em class=\"EmphasisTypeItalic \">f<\/em>(<em class=\"EmphasisTypeItalic \">y<\/em>). A function <em class=\"EmphasisTypeItalic \">f<\/em> is <em class=\"EmphasisTypeItalic \">\u03b5<\/em>-far from monotone if at least an <em class=\"EmphasisTypeItalic \">\u03b5<\/em>-fraction of values must be changed to make <em class=\"EmphasisTypeItalic \">f<\/em> monotone. Given a parameter <em class=\"EmphasisTypeItalic \">\u03b5<\/em>, a <em class=\"EmphasisTypeItalic \">monotonicity tester<\/em> must distinguish with high probability a monotone function from one that is <em class=\"EmphasisTypeItalic \">\u03b5<\/em>-far.<\/p>\n<p class=\"Para\">We prove that any (adaptive, two-sided) monotonicity tester for functions <em class=\"EmphasisTypeItalic \">f<\/em>:[<em class=\"EmphasisTypeItalic \">n<\/em>]<sup> <em class=\"EmphasisTypeItalic \">d<\/em> <\/sup>\u2009\u2192\u2009\u2115 must make \u03a9(<em class=\"EmphasisTypeItalic \">\u03b5<\/em> <sup>\u2212\u20091<\/sup> <em class=\"EmphasisTypeItalic \">d<\/em>log<em class=\"EmphasisTypeItalic \">n<\/em>\u2009\u2212\u2009<em class=\"EmphasisTypeItalic \">\u03b5<\/em> <sup>\u2212\u20091<\/sup>log<em class=\"EmphasisTypeItalic \">\u03b5<\/em> <sup>\u2212\u20091<\/sup>) queries. Recent upper bounds show the existence of <em class=\"EmphasisTypeItalic \">O<\/em>(<em class=\"EmphasisTypeItalic \">\u03b5<\/em> <sup>\u2212\u20091<\/sup> <em class=\"EmphasisTypeItalic \">d<\/em> log<em class=\"EmphasisTypeItalic \">n<\/em>) query monotonicity testers for hypergrids. This closes the question of monotonicity testing for hypergrids over arbitrary ranges. The previous best lower bound for general hypergrids was a non-adaptive bound of \u03a9(<em class=\"EmphasisTypeItalic \">d<\/em> log<em class=\"EmphasisTypeItalic \">n<\/em>).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For positive integers n, d, consider the hypergrid [n] d with the coordinate-wise product partial ordering denoted by \u227a. A function f: [n] d \u2009\u2192\u2009\u2115 is monotone if \u2200\u2009x\u2009\u227a\u2009y, f(x)\u2009\u2264\u2009f(y). A function f is \u03b5-far from monotone if at least an \u03b5-fraction of values must be changed to make f monotone. Given a parameter \u03b5, [&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":"Springer Berlin Heidelberg","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA. Proceedings","msr_editors":"","msr_how_published":"","msr_isbn":"978-3-642-40327-9","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"425-435","msr_page_range_start":"425","msr_page_range_end":"435","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA. 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