{"id":518,"date":"2024-03-10T15:32:47","date_gmt":"2024-03-10T15:32:47","guid":{"rendered":"https:\/\/cyberenlightener.com\/?page_id=518"},"modified":"2025-03-23T07:48:13","modified_gmt":"2025-03-23T07:48:13","slug":"randomized-algorithms","status":"publish","type":"page","link":"https:\/\/cyberenlightener.com\/?page_id=518","title":{"rendered":"Randomized Algorithms"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li><strong>Randomized Algorithms<\/strong><\/li>\n\n\n\n<li><strong>QuickSort&#8217;s Time Complexities : Average case, Best Case, Worst Case<\/strong><\/li>\n\n\n\n<li><strong>Partitioning technique : Balanced and Imbalanced<\/strong><\/li>\n\n\n\n<li><strong>Randomized QuickSort<\/strong><\/li>\n\n\n\n<li><strong>Worst and average case time complexities of Randomized QuickSort<\/strong><\/li>\n\n\n\n<li><strong>Finding expected running time of Randomized Quick Sort<\/strong>.<\/li>\n\n\n\n<li><strong>Randomized Primality Testing<\/strong><\/li>\n\n\n\n<li><strong>Fermat&#8217;s  primality testing<\/strong><\/li>\n\n\n\n<li><strong>Miller\u2013Rabin primality testing(Randomized version)<\/strong><\/li>\n\n\n\n<li><strong>Freivalds\u2019 Algorithm for Matrix Multiplication : randomized algorithm for verification of matrix multiplication<\/strong><\/li>\n\n\n\n<li><strong>Randomized algorithm solution  for Hiring Problem<\/strong><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Randomized Algorithm<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Randomized Algorithm &amp; QuickSort  : Average Case, Worst Case, &amp; Imbalance partition\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZtlOOq7q2Mk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Expected Running time of Randomized QuickSort<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"\ud835\udc11\ud835\udc1a\ud835\udc27\ud835\udc1d\ud835\udc28\ud835\udc26\ud835\udc22\ud835\udc33\ud835\udc1e\ud835\udc1d \ud835\udc10\ud835\udc2e\ud835\udc22\ud835\udc1c\ud835\udc24 \ud835\udc12\ud835\udc28\ud835\udc2b\ud835\udc2d: \ud835\udc04\ud835\udc31\ud835\udc29\ud835\udc25\ud835\udc28\ud835\udc2b\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc04\ud835\udc31\ud835\udc29\ud835\udc1e\ud835\udc1c\ud835\udc2d\ud835\udc1e\ud835\udc1d \ud835\udc11\ud835\udc2e\ud835\udc27\ud835\udc27\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc13\ud835\udc22\ud835\udc26\ud835\udc1e \ud835\udc22.\ud835\udc1e \ud835\udc04(\ud835\udc31)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WTljb7oKnkY?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What is Randomized Algorithms?<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure.<\/li>\n\n\n\n<li>The algorithm typically uses <strong>uniformly random bits<\/strong> as an auxiliary input to guide its behavior, in the hope of achieving good performance in the <strong>&#8220;average case&#8221;<\/strong> over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"598\" height=\"535\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized.png\" alt=\"\" class=\"wp-image-519\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized.png 598w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized-300x268.png 300w\" sizes=\"auto, (max-width: 598px) 100vw, 598px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Applications of Randomized Algorithms<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Determining outcomes in games<\/li>\n\n\n\n<li>Preventing communications on a shared channel from interfering with each other<\/li>\n\n\n\n<li>Initializing passwords<\/li>\n\n\n\n<li>Performing statistical (e.g. &#8220;Monte Carlo&#8221;) analysis on a system<\/li>\n\n\n\n<li>Simulating real-world behaviors and processes<\/li>\n\n\n\n<li>Emergent system generation using genetic hybridization<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Types of Randomized Algorithms<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"865\" height=\"273\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized2.png\" alt=\"\" class=\"wp-image-520\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized2.png 865w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized2-300x95.png 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized2-768x242.png 768w\" sizes=\"auto, (max-width: 865px) 100vw, 865px\" \/><\/figure>\n\n\n\n<p>A <strong>Las Vegas <\/strong>algorithm always produces the correct result, but its running time is based on a random value. A <strong>Monte Carlo algorithm<\/strong> has a deterministic running time and produces an answer that has a probability of\u2009 \u2265\u20091\/3 of being correct. <\/p>\n\n\n\n<p>The<strong> Las Vegas<\/strong> method of randomized algorithms never gives incorrect outputs, making the time constraint as the random variable. For example, in string matching algorithms, Las Vegas algorithms start from the beginning once they encounter an error. This increases the probability of correctness. Eg., Randomized Quick Sort Algorithm.<\/p>\n\n\n\n<p>The <strong>Monte Carlo method <\/strong>of randomized algorithms focuses on finishing the execution within the given time constraint. Therefore, the running time of this method is <strong>deterministic.<\/strong> For example, in string matching, if <strong>monte carlo<\/strong> encounters an error, it restarts the algorithm from the same point. Thus, saving time. Eg., Karger\u2019s Minimum Cut Algorithm.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\"><strong>Randomized Quick Sort<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"808\" height=\"347\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized3.png\" alt=\"\" class=\"wp-image-522\" style=\"width:1202px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized3.png 808w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized3-300x129.png 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized3-768x330.png 768w\" sizes=\"auto, (max-width: 808px) 100vw, 808px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"811\" height=\"481\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized4.png\" alt=\"\" class=\"wp-image-523\" style=\"width:1205px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized4.png 811w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized4-300x178.png 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized4-768x455.png 768w\" sizes=\"auto, (max-width: 811px) 100vw, 811px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"776\" height=\"228\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized5.png\" alt=\"\" class=\"wp-image-524\" style=\"width:1197px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized5.png 776w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized5-300x88.png 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized5-768x226.png 768w\" sizes=\"auto, (max-width: 776px) 100vw, 776px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"133\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized6.png\" alt=\"\" class=\"wp-image-525\" style=\"width:1202px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized6.png 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized6-300x52.png 300w\" sizes=\"auto, (max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"761\" height=\"514\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized7.png\" alt=\"\" class=\"wp-image-526\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized7.png 761w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized7-300x203.png 300w\" sizes=\"auto, (max-width: 761px) 100vw, 761px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"608\" height=\"486\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized8.png\" alt=\"\" class=\"wp-image-527\" style=\"width:1204px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized8.png 608w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized8-300x240.png 300w\" sizes=\"auto, (max-width: 608px) 100vw, 608px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"663\" height=\"225\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized9.png\" alt=\"\" class=\"wp-image-528\" style=\"width:1202px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized9.png 663w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized9-300x102.png 300w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"615\" height=\"454\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized10.png\" alt=\"\" class=\"wp-image-529\" style=\"width:1200px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized10.png 615w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized10-300x221.png 300w\" sizes=\"auto, (max-width: 615px) 100vw, 615px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"643\" height=\"361\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized11.png\" alt=\"\" class=\"wp-image-530\" style=\"width:1202px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized11.png 643w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized11-300x168.png 300w\" sizes=\"auto, (max-width: 643px) 100vw, 643px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"652\" height=\"462\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized12.png\" alt=\"\" class=\"wp-image-531\" style=\"width:1197px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized12.png 652w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized12-300x213.png 300w\" sizes=\"auto, (max-width: 652px) 100vw, 652px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"641\" height=\"186\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized13.png\" alt=\"\" class=\"wp-image-532\" style=\"width:1194px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized13.png 641w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized13-300x87.png 300w\" sizes=\"auto, (max-width: 641px) 100vw, 641px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"660\" height=\"402\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized14.png\" alt=\"\" class=\"wp-image-533\" style=\"width:1203px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized14.png 660w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Randomized14-300x183.png 300w\" sizes=\"auto, (max-width: 660px) 100vw, 660px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-724x1024.jpg\" alt=\"\" class=\"wp-image-536\" style=\"width:1201px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0010-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-724x1024.jpg\" alt=\"\" class=\"wp-image-537\" style=\"width:1198px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0011-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-724x1024.jpg\" alt=\"\" class=\"wp-image-538\" style=\"width:1200px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0012-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-724x1024.jpg\" alt=\"\" class=\"wp-image-539\" style=\"width:1202px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0013-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-724x1024.jpg\" alt=\"\" class=\"wp-image-540\" style=\"width:1198px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0014-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-724x1024.jpg\" alt=\"\" class=\"wp-image-541\" style=\"width:1201px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0015-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-724x1024.jpg\" alt=\"\" class=\"wp-image-542\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-724x1024.jpg 724w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-212x300.jpg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-768x1086.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-1086x1536.jpg 1086w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-1448x2048.jpg 1448w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/0016-scaled.jpg 1810w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h1 class=\"wp-block-heading\"><strong>Primality testing<\/strong><\/h1>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"717\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/1-3-717x1024.jpeg\" alt=\"\" class=\"wp-image-598\" style=\"width:1201px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/1-3-717x1024.jpeg 717w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/1-3-210x300.jpeg 210w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/1-3-768x1096.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/1-3-1076x1536.jpeg 1076w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/1-3.jpeg 1121w\" sizes=\"auto, (max-width: 717px) 100vw, 717px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"725\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/2-1-725x1024.jpeg\" alt=\"\" class=\"wp-image-575\" style=\"width:1064px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/2-1-725x1024.jpeg 725w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/2-1-212x300.jpeg 212w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/2-1-768x1085.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/2-1-1088x1536.jpeg 1088w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/2-1.jpeg 1133w\" sizes=\"auto, (max-width: 725px) 100vw, 725px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"719\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/3-2-719x1024.jpeg\" alt=\"\" class=\"wp-image-585\" style=\"width:1202px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/3-2-719x1024.jpeg 719w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/3-2-211x300.jpeg 211w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/3-2-768x1094.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/3-2-1078x1536.jpeg 1078w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/3-2.jpeg 1123w\" sizes=\"auto, (max-width: 719px) 100vw, 719px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"727\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/4-2-727x1024.jpeg\" alt=\"\" class=\"wp-image-586\" style=\"width:1201px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/4-2-727x1024.jpeg 727w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/4-2-213x300.jpeg 213w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/4-2-768x1082.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/4-2-1091x1536.jpeg 1091w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/4-2.jpeg 1136w\" sizes=\"auto, (max-width: 727px) 100vw, 727px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"716\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/5-1-716x1024.jpeg\" alt=\"\" class=\"wp-image-587\" style=\"width:1201px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/5-1-716x1024.jpeg 716w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/5-1-210x300.jpeg 210w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/5-1-768x1098.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/5-1-1074x1536.jpeg 1074w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/5-1.jpeg 1119w\" sizes=\"auto, (max-width: 716px) 100vw, 716px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"745\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/6-745x1024.jpeg\" alt=\"\" class=\"wp-image-579\" style=\"width:1068px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/6-745x1024.jpeg 745w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/6-218x300.jpeg 218w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/6-768x1056.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/6-1117x1536.jpeg 1117w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/6.jpeg 1163w\" sizes=\"auto, (max-width: 745px) 100vw, 745px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"742\" height=\"1024\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/7-1-742x1024.jpeg\" alt=\"\" class=\"wp-image-589\" style=\"width:1200px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/7-1-742x1024.jpeg 742w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/7-1-217x300.jpeg 217w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/7-1-768x1060.jpeg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/7-1-1113x1536.jpeg 1113w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/7-1.jpeg 1159w\" sizes=\"auto, (max-width: 742px) 100vw, 742px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Miller-Rabbin_Primality-test-2-1024x768.jpg\" alt=\"\" class=\"wp-image-614\" style=\"width:1201px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Miller-Rabbin_Primality-test-2-1024x768.jpg 1024w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Miller-Rabbin_Primality-test-2-300x225.jpg 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Miller-Rabbin_Primality-test-2-768x576.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Miller-Rabbin_Primality-test-2-1536x1152.jpg 1536w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Miller-Rabbin_Primality-test-2-2048x1536.jpg 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Miller Rabin Primality Test\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yT35yO_Untc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><figcaption class=\"wp-element-caption\"><strong>Miller Rabin Primality Test<\/strong><\/figcaption><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Freivalds\u2019 Algorithm for Matrix Multiplication : randomized algorithm for verification of matrix multiplication<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Freivalds\u2019 Algorithm   : a randomized algorithm to  verify matrix multiplication || Monte Carlo\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yh73ChFj5ik?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-1-1024x768.jpg\" alt=\"\" class=\"wp-image-601\" style=\"width:1203px;height:auto\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-1-1024x768.jpg 1024w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-1-300x225.jpg 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-1-768x576.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-1-1536x1152.jpg 1536w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-1-2048x1536.jpg 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-2-1024x768.jpg\" alt=\"\" class=\"wp-image-602\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-2-1024x768.jpg 1024w, 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height=\"768\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-8-1024x768.jpg\" alt=\"\" class=\"wp-image-608\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-8-1024x768.jpg 1024w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-8-300x225.jpg 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-8-768x576.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-8-1536x1152.jpg 1536w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-8-2048x1536.jpg 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-9-1024x768.jpg\" alt=\"\" class=\"wp-image-609\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-9-1024x768.jpg 1024w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-9-300x225.jpg 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-9-768x576.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-9-1536x1152.jpg 1536w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-9-2048x1536.jpg 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-10-1024x768.jpg\" alt=\"\" class=\"wp-image-610\" srcset=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-10-1024x768.jpg 1024w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-10-300x225.jpg 300w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-10-768x576.jpg 768w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-10-1536x1152.jpg 1536w, https:\/\/cyberenlightener.com\/wp-content\/uploads\/2024\/03\/Freivalds-Algorithm-10-2048x1536.jpg 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/03\/Randomized-Algorithm-for-the-Hiring-Problem_-Detailed-Numerical-Explanation-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Randomized Algorithm for the Hiring Problem_ Detailed Numerical Explanation (1).\"><\/object><a id=\"wp-block-file--media-0a692c1b-e42a-4341-9fd6-dbb9ccd8768b\" href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/03\/Randomized-Algorithm-for-the-Hiring-Problem_-Detailed-Numerical-Explanation-1.pdf\">Randomized Algorithm for the Hiring Problem_ Detailed Numerical Explanation (1)<\/a><a href=\"https:\/\/cyberenlightener.com\/wp-content\/uploads\/2025\/03\/Randomized-Algorithm-for-the-Hiring-Problem_-Detailed-Numerical-Explanation-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-0a692c1b-e42a-4341-9fd6-dbb9ccd8768b\">Download<\/a><\/div>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"blob:https:\/\/cyberenlightener.com\/c7d9e3ff-7998-48ff-b365-3cc5f738bda6\" alt=\"\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Randomized Algorithm Expected Running time of Randomized QuickSort What is Randomized Algorithms? Applications of Randomized Algorithms Types of Randomized Algorithms A Las Vegas algorithm always produces the correct result, but its running time is based on a random value. A Monte Carlo algorithm has a deterministic running time and produces an answer that has a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-518","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages\/518","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=518"}],"version-history":[{"count":13,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages\/518\/revisions"}],"predecessor-version":[{"id":1779,"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=\/wp\/v2\/pages\/518\/revisions\/1779"}],"wp:attachment":[{"href":"https:\/\/cyberenlightener.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}