{"id":5090,"date":"2022-08-26T14:21:49","date_gmt":"2022-08-26T08:51:49","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-how-do-i-prove-or-disprove-that-this-function-is-%cf%89n1-5\/"},"modified":"2022-08-26T14:21:49","modified_gmt":"2022-08-26T08:51:49","slug":"solved-how-do-i-prove-or-disprove-that-this-function-is-%cf%89n1-5","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-how-do-i-prove-or-disprove-that-this-function-is-%cf%89n1-5\/","title":{"rendered":"[Solved] How do I prove or disprove that this function is \u03a9(n^1.5)?"},"content":{"rendered":"<p> [ad_1]<br \/>\n<\/p>\n<div id=\"answer-45543103\" class=\"answer js-answer accepted-answer js-accepted-answer\" data-answerid=\"45543103\" data-parentid=\"45535248\" data-score=\"2\" data-position-on-page=\"1\" data-highest-scored=\"1\" data-question-has-accepted-highest-score=\"1\" itemprop=\"acceptedAnswer\" itemscope itemtype=\"https:\/\/schema.org\/Answer\">\n<div class=\"post-layout\">\n<div class=\"votecell post-layout--left\"><\/div>\n<div class=\"answercell post-layout--right\">\n<div class=\"s-prose js-post-body\" itemprop=\"text\">\n<p>Assuming that <code>T(n)<\/code> doesn&#8217;t suddenly become negative at some value of <code>n<\/code>, we can give a lower bound for the left hand side if we neglect the first term:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>We define a new function <code>S(n)<\/code> such that:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503906_371_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503906_371_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>We can immediately see that it has <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503906_133_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503906_133_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a> terms (ignoring off-by-one etc.). Thus if we keep expanding:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503907_880_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503907_880_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>At this stage, since we know that <code>log(n) &lt;&lt; n<\/code> for any large <code>n<\/code>, we can apply a Taylor expansion to the third term in the recursive call to <code>S(n)<\/code>:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503907_124_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503907_124_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>We can realistically ignore the 2nd term too. Applying this approximation to every <code>S(n)<\/code> call:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503907_718_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503907_718_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>Now, we know that:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503908_258_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503908_258_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p><code>b<\/code> obviously can be 1.5; therefore:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503908_826_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503908_826_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<hr>\n<p>EDIT: some numerical tests to confirm this result &#8211;<\/p>\n<p>Code:<\/p>\n<pre><code>uint64_t T(int n) {\n  return n &lt;= 1 ? 0 : T(n - log(n)) + T(log(n)) + (uint64_t)n;\n}\n<\/code><\/pre>\n<p>Results:<\/p>\n<pre><code>N           T(N)\n--------------------------\n2           2\n4           6\n8           18\n16          60\n32          181\n64          578\n128         1911\n256         6331\n512         22011\n1024        79304\n2048        279719\n4096        1016217\n8192        3814210\n16384       13902832\n32768       51540129\n65536       195366441\n131072      732435510\n262144      2744988819\n524288      10457580914\n1048576     39910628826\n<\/code><\/pre>\n<p>Plot of <code>N^2\/log(N)<\/code> against <code>T(N)<\/code>:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/Solved-How-do-I-prove-or-disprove-that-this-function.png\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/Solved-How-do-I-prove-or-disprove-that-this-function.png\" alt=\"enter image description here\"><\/a><\/p>\n<p>The relationship is linear, meaning that<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503909_337_Solved-How-do-I-prove-or-disprove-that-this-function.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/08\/1661503909_337_Solved-How-do-I-prove-or-disprove-that-this-function.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>&#8230; confirming the given result.<\/p>\n<\/p><\/div>\n<div class=\"mt24\"><\/div>\n<\/div>\n<p>            <span class=\"d-none\" itemprop=\"commentCount\">5<\/span> <\/p><\/div>\n<\/div>\n<p>[ad_2]<\/p>\n<p>solved How do I prove or disprove that this function is \u03a9(n^1.5)? <\/p>\n","protected":false},"excerpt":{"rendered":"<p>[ad_1] Assuming that T(n) doesn&#8217;t suddenly become negative at some value of n, we can give a lower bound for the left hand side if we neglect the first term: We define a new function S(n) such that: We can immediately see that it has terms (ignoring off-by-one etc.). Thus if we keep expanding: At &#8230; <a title=\"[Solved] How do I prove or disprove that this function is \u03a9(n^1.5)?\" class=\"read-more\" href=\"https:\/\/jassweb.com\/solved\/solved-how-do-i-prove-or-disprove-that-this-function-is-%cf%89n1-5\/\" aria-label=\"More on [Solved] How do I prove or disprove that this function is \u03a9(n^1.5)?\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[612,859,1238],"class_list":["post-5090","post","type-post","status-publish","format-standard","hentry","category-solved","tag-big-o","tag-data-structures","tag-time-complexity"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>[Solved] How do I prove or disprove that this function is \u03a9(n^1.5)? - JassWeb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jassweb.com\/solved\/solved-how-do-i-prove-or-disprove-that-this-function-is-\u03c9n1-5\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[Solved] How do I prove or disprove that this function is \u03a9(n^1.5)? - JassWeb\" \/>\n<meta property=\"og:description\" content=\"[ad_1] Assuming that T(n) doesn&#8217;t suddenly become negative at some value of n, we can give a lower bound for the left hand side if we neglect the first term: We define a new function S(n) such that: We can immediately see that it has terms (ignoring off-by-one etc.). 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