{"id":25365,"date":"2025-03-13T11:54:10","date_gmt":"2025-03-13T04:54:10","guid":{"rendered":"https:\/\/interdata.vn\/blog\/?p=25365"},"modified":"2025-03-13T11:54:58","modified_gmt":"2025-03-13T04:54:58","slug":"thuat-toan-algorithm","status":"publish","type":"post","link":"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/","title":{"rendered":"Thu\u1eadt to\u00e1n (Algorithm) l\u00e0 g\u00ec? Vai tr\u00f2, \u1ee8ng d\u1ee5ng v\u00e0 Ph\u00e2n lo\u1ea1i"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">N\u1ed8I DUNG<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Algorithm-la-gi\" >Thu\u1eadt to\u00e1n (Algorithm) l\u00e0 g\u00ec?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Tam-quan-trong-cua-thuat-toan\" >T\u1ea7m quan tr\u1ecdng c\u1ee7a thu\u1eadt to\u00e1n<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Giai-quyet-van-de-hieu-qua\" >Gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 hi\u1ec7u qu\u1ea3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Toi-uu-hoa-hieu-suat\" >T\u1ed1i \u01b0u h\u00f3a hi\u1ec7u su\u1ea5t<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Nen-tang-cua-Tri-tue-nhan-tao-AI\" >N\u1ec1n t\u1ea3ng c\u1ee7a Tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o (AI)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Ung-dung-rong-rai-trong-moi-linh-vuc\" >\u1ee8ng d\u1ee5ng r\u1ed9ng r\u00e3i trong m\u1ecdi l\u0129nh v\u1ef1c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuc-day-su-doi-moi-va-sang-tao\" >Th\u00fac \u0111\u1ea9y s\u1ef1 \u0111\u1ed5i m\u1edbi v\u00e0 s\u00e1ng t\u1ea1o<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Nhung-ung-dung-thuc-tien-cua-thuat-toan\" >Nh\u1eefng \u1ee9ng d\u1ee5ng th\u1ef1c ti\u1ec5n c\u1ee7a thu\u1eadt to\u00e1n<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Cong-cu-tim-kiem-Search-Engines\" >C\u00f4ng c\u1ee5 t\u00ecm ki\u1ebfm (Search Engines)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Mang-xa-hoi-Social-Media\" >M\u1ea1ng x\u00e3 h\u1ed9i (Social Media)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuong-mai-dien-tu-E-commerce\" >Th\u01b0\u01a1ng m\u1ea1i \u0111i\u1ec7n t\u1eed (E-commerce)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Dan-duong-Navigation\" >D\u1eabn \u0111\u01b0\u1eddng (Navigation)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Tri-tue-nhan-tao-Artificial-Intelligence\" >Tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o (Artificial Intelligence)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Y-hoc-Medicine\" >Y h\u1ecdc (Medicine)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Tai-chinh-Finance\" >T\u00e0i ch\u00ednh (Finance)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Cac-linh-vuc-khac\" >C\u00e1c l\u0129nh v\u1ef1c kh\u00e1c<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Cac-loai-thuat-toan-pho-bien-hien-nay\" >C\u00e1c lo\u1ea1i thu\u1eadt to\u00e1n ph\u1ed5 bi\u1ebfn hi\u1ec7n nay<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Sap-xep-Sorting-Algorithms\" >Thu\u1eadt to\u00e1n S\u1eafp x\u1ebfp (Sorting Algorithms)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Tim-kiem-Searching-Algorithms\" >Thu\u1eadt to\u00e1n T\u00ecm ki\u1ebfm (Searching Algorithms)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Do-thi-Graph-Algorithms\" >Thu\u1eadt to\u00e1n \u0110\u1ed3 th\u1ecb (Graph Algorithms)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Chia-de-tri-Divide-and-Conquer\" >Thu\u1eadt to\u00e1n Chia \u0111\u1ec3 tr\u1ecb (Divide and Conquer)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Tham-lam-Greedy-Algorithms\" >Thu\u1eadt to\u00e1n Tham lam (Greedy Algorithms)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Quy-hoach-dong-Dynamic-Programming\" >Thu\u1eadt to\u00e1n Quy ho\u1ea1ch \u0111\u1ed9ng (Dynamic Programming)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Quay-lui-Backtracking\" >Thu\u1eadt to\u00e1n Quay lui (Backtracking)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/#Thuat-toan-Hoc-may-Machine-Learning-Algorithms\" >Thu\u1eadt to\u00e1n H\u1ecdc m\u00e1y (Machine Learning Algorithms)<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>B\u1ea1n c\u00f3 bao gi\u1edd t\u1ef1 h\u1ecfi Google t\u00ecm ki\u1ebfm nhanh \u0111\u1ebfn v\u1eady nh\u01b0 th\u1ebf n\u00e0o, hay Facebook g\u1ee3i \u00fd k\u1ebft b\u1ea1n &#8220;chu\u1ea9n&#8221; ra sao? T\u1ea5t c\u1ea3 \u0111\u1ec1u nh\u1edd v\u00e0o <strong>thu\u1eadt to\u00e1n<\/strong>. B\u00e0i vi\u1ebft n\u00e0y s\u1ebd gi\u1ea3i th\u00edch <a href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/\"><strong>thu\u1eadt to\u00e1n (algorithm) l\u00e0 g\u00ec<\/strong><\/a>, t\u1ea7m quan tr\u1ecdng, nh\u1eefng \u1ee9ng d\u1ee5ng th\u1ef1c ti\u1ec5n \u0111\u00e1ng kinh ng\u1ea1c v\u00e0 c\u00e1c lo\u1ea1i thu\u1eadt to\u00e1n ph\u1ed5 bi\u1ebfn m\u00e0 b\u1ea1n c\u1ea7n bi\u1ebft. T\u1eeb \u0111\u00f3, hi\u1ec3u \u0111\u01b0\u1ee3c c\u00e1ch thu\u1eadt to\u00e1n \u0111ang &#8220;v\u1eadn h\u00e0nh&#8221; th\u1ebf gi\u1edbi xung quanh ta.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Thuat-toan-Algorithm-la-gi\"><\/span>Thu\u1eadt to\u00e1n (Algorithm) l\u00e0 g\u00ec?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Thu\u1eadt to\u00e1n l\u00e0 m\u1ed9t chu\u1ed7i c\u00e1c h\u01b0\u1edbng d\u1eabn r\u00f5 r\u00e0ng, c\u00f3 th\u1ee9 t\u1ef1, \u0111\u1ec3 gi\u1ea3i quy\u1ebft m\u1ed9t v\u1ea5n \u0111\u1ec1 ho\u1eb7c th\u1ef1c hi\u1ec7n m\u1ed9t nhi\u1ec7m v\u1ee5 c\u1ee5 th\u1ec3.<\/strong> N\u00f3 gi\u1ed1ng nh\u01b0 m\u1ed9t c\u00f4ng th\u1ee9c n\u1ea5u \u0103n, ch\u1ec9 d\u1eabn t\u1eebng b\u01b0\u1edbc \u0111\u1ec3 \u0111\u1ea1t \u0111\u01b0\u1ee3c k\u1ebft qu\u1ea3 mong mu\u1ed1n.<\/p>\n<p>Th\u1ef1c ra, b\u1ea1n v\u1eabn th\u01b0\u1eddng xuy\u00ean s\u1eed d\u1ee5ng thu\u1eadt to\u00e1n trong cu\u1ed9c s\u1ed1ng h\u00e0ng ng\u00e0y m\u00e0 c\u00f3 th\u1ec3 kh\u00f4ng nh\u1eadn ra \u0111\u1ea5y! V\u00ed d\u1ee5, khi b\u1ea1n tra c\u1ee9u \u0111\u01b0\u1eddng \u0111i tr\u00ean Google Maps, \u1ee9ng d\u1ee5ng \u0111ang s\u1eed d\u1ee5ng m\u1ed9t <strong>thu\u1eadt to\u00e1n t\u00ecm \u0111\u01b0\u1eddng<\/strong> \u0111\u1ec3 t\u00ecm ra con \u0111\u01b0\u1eddng ng\u1eafn nh\u1ea5t ho\u1eb7c nhanh nh\u1ea5t t\u1eeb \u0111i\u1ec3m A \u0111\u1ebfn \u0111i\u1ec3m B. Hay khi b\u1ea1n t\u00ecm ki\u1ebfm tr\u00ean Google, h\u00e0ng t\u1ef7 <a href=\"https:\/\/interdata.vn\/blog\/page-la-gi\/\">trang web<\/a> \u0111\u01b0\u1ee3c qu\u00e9t v\u00e0 s\u1eafp x\u1ebfp ch\u1ec9 trong v\u00e0i gi\u00e2y, \u0111\u00f3 ch\u00ednh l\u00e0 nh\u1edd <strong>thu\u1eadt to\u00e1n t\u00ecm ki\u1ebfm<\/strong> c\u1ef1c k\u1ef3 ph\u1ee9c t\u1ea1p.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Thuat-toan-Algorithm.png\" alt=\"Thu\u1eadt to\u00e1n (Algorithm)\" width=\"800\" height=\"420\" class=\"aligncenter size-full wp-image-25370\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Thuat-toan-Algorithm.png 800w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Thuat-toan-Algorithm-300x158.png 300w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Thuat-toan-Algorithm-768x403.png 768w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Thuat-toan-Algorithm-750x394.png 750w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/p>\n<p>V\u1ec1 m\u1eb7t h\u1ecdc thu\u1eadt, thu\u1eadt to\u00e1n \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a l\u00e0 m\u1ed9t t\u1eadp h\u1ee3p <strong>h\u1eefu h\u1ea1n<\/strong> c\u00e1c <strong>h\u01b0\u1edbng d\u1eabn<\/strong> \u0111\u01b0\u1ee3c <strong>x\u00e1c \u0111\u1ecbnh r\u00f5 r\u00e0ng<\/strong>, c\u00f3 th\u1ec3 th\u1ef1c hi\u1ec7n \u0111\u01b0\u1ee3c b\u1eb1ng m\u00e1y t\u00ednh, th\u01b0\u1eddng \u0111\u1ec3 gi\u1ea3i quy\u1ebft m\u1ed9t<sup>\u00a0<\/sup>l\u1edbp b\u00e0i to\u00e1n ho\u1eb7c th\u1ef1c hi\u1ec7n m\u1ed9t ph\u00e9p t\u00ednh. Nghe c\u00f3 v\u1ebb &#8220;h\u00e0n l\u00e2m&#8221; qu\u00e1 ph\u1ea3i kh\u00f4ng? \u0110\u1eebng lo, ch\u00fang ta s\u1ebd &#8220;gi\u1ea3i m\u00e3&#8221; t\u1eebng ph\u1ea7n m\u1ed9t.<\/p>\n<ul>\n<li><strong>&#8220;H\u1eefu h\u1ea1n&#8221;<\/strong> c\u00f3 ngh\u0129a l\u00e0 thu\u1eadt to\u00e1n ph\u1ea3i k\u1ebft th\u00fac sau m\u1ed9t s\u1ed1 b\u01b0\u1edbc nh\u1ea5t \u0111\u1ecbnh, ch\u1ee9 kh\u00f4ng ch\u1ea1y m\u00e3i m\u00e3i.<\/li>\n<li><strong>&#8220;H\u01b0\u1edbng d\u1eabn&#8221;<\/strong> \u1edf \u0111\u00e2y l\u00e0 c\u00e1c ch\u1ec9 th\u1ecb, c\u00e1c b\u01b0\u1edbc th\u1ef1c hi\u1ec7n, gi\u1ed1ng nh\u01b0 c\u00e1c b\u01b0\u1edbc trong c\u00f4ng th\u1ee9c n\u1ea5u \u0103n v\u1eady.<\/li>\n<li><strong>&#8220;X\u00e1c \u0111\u1ecbnh r\u00f5 r\u00e0ng&#8221;<\/strong> ngh\u0129a l\u00e0 m\u1ed7i b\u01b0\u1edbc ph\u1ea3i th\u1eadt c\u1ee5 th\u1ec3, kh\u00f4ng m\u1eadp m\u1edd, kh\u00f4ng c\u00f3 ch\u1ed7 cho s\u1ef1 hi\u1ec3u l\u1ea7m. V\u00ed d\u1ee5: &#8220;th\u00eam m\u1ed9t ch\u00fat mu\u1ed1i&#8221; l\u00e0 kh\u00f4ng r\u00f5 r\u00e0ng, ph\u1ea3i l\u00e0 &#8220;th\u00eam 1\/2 th\u00eca c\u00e0 ph\u00ea mu\u1ed1i&#8221;.<\/li>\n<\/ul>\n<p>Ngo\u00e0i ra, thu\u1eadt to\u00e1n c\u00f2n c\u1ea7n \u0111\u1ea3m b\u1ea3o 2 y\u1ebfu t\u1ed1 Input v\u00e0 output:<\/p>\n<ul>\n<li><strong>\u0110\u1ea7u v\u00e0o (Input)<\/strong>: Thu\u1eadt to\u00e1n nh\u1eadn th\u00f4ng tin \u0111\u1ea7u v\u00e0o, v\u00ed d\u1ee5 nh\u01b0 danh s\u00e1ch c\u00e1c s\u1ed1 c\u1ea7n s\u1eafp x\u1ebfp, ho\u1eb7c \u0111i\u1ec3m xu\u1ea5t ph\u00e1t v\u00e0 \u0111i\u1ec3m \u0111\u1ebfn khi t\u00ecm \u0111\u01b0\u1eddng.<\/li>\n<li><strong>\u0110\u1ea7u ra (Output)<\/strong>: Sau khi th\u1ef1c hi\u1ec7n c\u00e1c b\u01b0\u1edbc, thu\u1eadt to\u00e1n s\u1ebd cho ra k\u1ebft qu\u1ea3, v\u00ed d\u1ee5 nh\u01b0 danh s\u00e1ch \u0111\u00e3 s\u1eafp x\u1ebfp, ho\u1eb7c l\u1ed9 tr\u00ecnh t\u1ed1i \u01b0u.<\/li>\n<\/ul>\n<p>\u0110\u1ec3 d\u1ec5 h\u00ecnh dung h\u01a1n, h\u00e3y x\u00e9t m\u1ed9t v\u00ed d\u1ee5 \u0111\u01a1n gi\u1ea3n: <strong>thu\u1eadt to\u00e1n pha m\u1ed9t t\u00e1ch c\u00e0 ph\u00ea.<\/strong><\/p>\n<ol>\n<li><strong>Input:<\/strong> N\u01b0\u1edbc n\u00f3ng, b\u1ed9t c\u00e0 ph\u00ea, \u0111\u01b0\u1eddng (t\u00f9y ch\u1ecdn), phin c\u00e0 ph\u00ea, t\u00e1ch.<\/li>\n<li><strong>C\u00e1c b\u01b0\u1edbc:<\/strong>\n<ul>\n<li>Tr\u00e1ng phin c\u00e0 ph\u00ea b\u1eb1ng n\u01b0\u1edbc n\u00f3ng.<\/li>\n<li>Cho b\u1ed9t c\u00e0 ph\u00ea v\u00e0o phin.<\/li>\n<li>\u0110\u1ed5 m\u1ed9t \u00edt n\u01b0\u1edbc n\u00f3ng v\u00e0o phin, ch\u1edd c\u00e0 ph\u00ea ng\u1ea5m.<\/li>\n<li>\u0110\u1ed5 th\u00eam n\u01b0\u1edbc n\u00f3ng v\u00e0o \u0111\u1ea7y phin.<\/li>\n<li>Ch\u1edd c\u00e0 ph\u00ea nh\u1ecf gi\u1ecdt h\u1ebft.<\/li>\n<li>Th\u00eam \u0111\u01b0\u1eddng (t\u00f9y ch\u1ecdn).<\/li>\n<\/ul>\n<\/li>\n<li><strong>Output:<\/strong> M\u1ed9t t\u00e1ch c\u00e0 ph\u00ea.<\/li>\n<\/ol>\n<h2><span class=\"ez-toc-section\" id=\"Tam-quan-trong-cua-thuat-toan\"><\/span>T\u1ea7m quan tr\u1ecdng c\u1ee7a thu\u1eadt to\u00e1n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Thu\u1eadt to\u00e1n l\u00e0 n\u1ec1n t\u1ea3ng c\u1ee7a khoa h\u1ecdc m\u00e1y t\u00ednh v\u00e0 c\u00f3 vai tr\u00f2 then ch\u1ed1t trong h\u1ea7u h\u1ebft m\u1ecdi l\u0129nh v\u1ef1c c\u00f4ng ngh\u1ec7 hi\u1ec7n \u0111\u1ea1i.<\/strong> N\u1ebfu kh\u00f4ng c\u00f3 thu\u1eadt to\u00e1n, m\u00e1y t\u00ednh ch\u1ec9 l\u00e0 nh\u1eefng c\u1ed7 m\u00e1y v\u00f4 d\u1ee5ng.<\/p>\n<p>Th\u1ef1c t\u1ebf, t\u1ea7m quan tr\u1ecdng c\u1ee7a thu\u1eadt to\u00e1n kh\u00f4ng ch\u1ec9 gi\u1edbi h\u1ea1n trong l\u0129nh v\u1ef1c c\u00f4ng ngh\u1ec7 th\u00f4ng tin. Ch\u00fang hi\u1ec7n di\u1ec7n v\u00e0 \u0111\u00f3ng vai tr\u00f2 quan tr\u1ecdng trong <i>h\u1ea7u h\u1ebft<\/i> m\u1ecdi kh\u00eda c\u1ea1nh c\u1ee7a cu\u1ed9c s\u1ed1ng hi\u1ec7n \u0111\u1ea1i, t\u1eeb nh\u1eefng vi\u1ec7c \u0111\u01a1n gi\u1ea3n h\u00e0ng ng\u00e0y \u0111\u1ebfn nh\u1eefng h\u1ec7 th\u1ed1ng ph\u1ee9c t\u1ea1p nh\u1ea5t.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Vai-tro-cua-Thuat-toan-Algorithm.jpg\" alt=\"Vai tr\u00f2 c\u1ee7a Thu\u1eadt to\u00e1n (Algorithm)\" width=\"800\" height=\"450\" class=\"aligncenter size-full wp-image-25372\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Vai-tro-cua-Thuat-toan-Algorithm.jpg 800w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Vai-tro-cua-Thuat-toan-Algorithm-300x169.jpg 300w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Vai-tro-cua-Thuat-toan-Algorithm-768x432.jpg 768w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Vai-tro-cua-Thuat-toan-Algorithm-750x422.jpg 750w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Giai-quyet-van-de-hieu-qua\"><\/span>Gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 hi\u1ec7u qu\u1ea3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Thu\u1eadt to\u00e1n cung c\u1ea5p <strong>c\u00e1ch ti\u1ebfp c\u1eadn c\u00f3 h\u1ec7 th\u1ed1ng<\/strong> \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1. Thay v\u00ec m\u00f2 m\u1eabm t\u00ecm gi\u1ea3i ph\u00e1p, thu\u1eadt to\u00e1n cho ph\u00e9p ch\u00fang ta chia nh\u1ecf v\u1ea5n \u0111\u1ec1 th\u00e0nh c\u00e1c b\u01b0\u1edbc nh\u1ecf h\u01a1n, d\u1ec5 qu\u1ea3n l\u00fd h\u01a1n, v\u00e0 gi\u1ea3i quy\u1ebft ch\u00fang m\u1ed9t c\u00e1ch tu\u1ea7n t\u1ef1, logic. \u0110i\u1ec1u n\u00e0y kh\u00f4ng ch\u1ec9 gi\u00fap ti\u1ebft ki\u1ec7m th\u1eddi gian v\u00e0 c\u00f4ng s\u1ee9c m\u00e0 c\u00f2n \u0111\u1ea3m b\u1ea3o t\u00ednh ch\u00ednh x\u00e1c v\u00e0 hi\u1ec7u qu\u1ea3 c\u1ee7a gi\u1ea3i ph\u00e1p.<\/p>\n<p>V\u00ed d\u1ee5, khi b\u1ea1n c\u1ea7n s\u1eafp x\u1ebfp m\u1ed9t ch\u1ed3ng s\u00e1ch theo th\u1ee9 t\u1ef1, b\u1ea1n c\u00f3 th\u1ec3 \u00e1p d\u1ee5ng c\u00e1c <strong>thu\u1eadt to\u00e1n s\u1eafp x\u1ebfp<\/strong> (nh\u01b0 s\u1eafp x\u1ebfp n\u1ed5i b\u1ecdt, s\u1eafp x\u1ebfp nhanh&#8230;) \u0111\u1ec3 th\u1ef1c hi\u1ec7n vi\u1ec7c n\u00e0y m\u1ed9t c\u00e1ch nhanh ch\u00f3ng v\u00e0 c\u00f3 h\u1ec7 th\u1ed1ng, thay v\u00ec th\u1eed t\u1eebng c\u00e1ch m\u1ed9t c\u00e1ch ng\u1eabu nhi\u00ean.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Toi-uu-hoa-hieu-suat\"><\/span>T\u1ed1i \u01b0u h\u00f3a hi\u1ec7u su\u1ea5t<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Trong th\u1ebf gi\u1edbi c\u00f4ng ngh\u1ec7, <strong>t\u1ed1c \u0111\u1ed9 v\u00e0 hi\u1ec7u su\u1ea5t<\/strong> l\u00e0 y\u1ebfu t\u1ed1 s\u1ed1ng c\u00f2n. Thu\u1eadt to\u00e1n gi\u00fap t\u1ed1i \u01b0u h\u00f3a c\u00e1c quy tr\u00ecnh, gi\u1ea3m thi\u1ec3u th\u1eddi gian x\u1eed l\u00fd v\u00e0 t\u00e0i nguy\u00ean s\u1eed d\u1ee5ng.<\/p>\n<p>V\u00ed d\u1ee5, c\u00e1c c\u00f4ng c\u1ee5 t\u00ecm ki\u1ebfm nh\u01b0 Google s\u1eed d\u1ee5ng c\u00e1c thu\u1eadt to\u00e1n c\u1ef1c k\u1ef3 tinh vi \u0111\u1ec3 x\u1eed l\u00fd h\u00e0ng t\u1ef7 trang web v\u00e0 tr\u1ea3 v\u1ec1 k\u1ebft qu\u1ea3 ph\u00f9 h\u1ee3p nh\u1ea5t ch\u1ec9 trong t\u00edch t\u1eafc. N\u1ebfu kh\u00f4ng c\u00f3 c\u00e1c thu\u1eadt to\u00e1n n\u00e0y, vi\u1ec7c t\u00ecm ki\u1ebfm th\u00f4ng tin tr\u00ean <a href=\"https:\/\/interdata.vn\/blog\/mang-internet\/\">Internet<\/a> s\u1ebd tr\u1edf n\u00ean v\u00f4 c\u00f9ng ch\u1eadm ch\u1ea1p v\u00e0 kh\u00f3 kh\u0103n.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Nen-tang-cua-Tri-tue-nhan-tao-AI\"><\/span>N\u1ec1n t\u1ea3ng c\u1ee7a Tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o (AI)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Thu\u1eadt to\u00e1n l\u00e0 &#8220;tr\u00e1i tim&#8221; c\u1ee7a <strong><a href=\"https:\/\/interdata.vn\/blog\/tri-tue-nhan-tao-ai\/\">Tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o<\/a> (AI)<\/strong> v\u00e0 <strong>h\u1ecdc m\u00e1y (Machine Learning)<\/strong>. C\u00e1c m\u00f4 h\u00ecnh AI hi\u1ec7n \u0111\u1ea1i, t\u1eeb nh\u1eadn d\u1ea1ng gi\u1ecdng n\u00f3i, d\u1ecbch m\u00e1y, \u0111\u1ebfn xe t\u1ef1 l\u00e1i, \u0111\u1ec1u d\u1ef1a tr\u00ean c\u00e1c thu\u1eadt to\u00e1n ph\u1ee9c t\u1ea1p \u0111\u1ec3 h\u1ecdc h\u1ecfi t\u1eeb d\u1eef li\u1ec7u v\u00e0 \u0111\u01b0a ra quy\u1ebft \u0111\u1ecbnh.<\/p>\n<p>V\u00ed d\u1ee5, khi b\u1ea1n s\u1eed d\u1ee5ng <a href=\"https:\/\/interdata.vn\/blog\/tro-ly-ao-la-gi\/\">tr\u1ee3 l\u00fd \u1ea3o<\/a> nh\u01b0 Siri ho\u1eb7c Google Assistant, c\u00e1c <strong>thu\u1eadt to\u00e1n x\u1eed l\u00fd ng\u00f4n ng\u1eef t\u1ef1 nhi\u00ean (NLP)<\/strong> gi\u00fap m\u00e1y t\u00ednh hi\u1ec3u \u0111\u01b0\u1ee3c y\u00eau c\u1ea7u c\u1ee7a b\u1ea1n v\u00e0 \u0111\u01b0a ra ph\u1ea3n h\u1ed3i ph\u00f9 h\u1ee3p.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Ung-dung-rong-rai-trong-moi-linh-vuc\"><\/span>\u1ee8ng d\u1ee5ng r\u1ed9ng r\u00e3i trong m\u1ecdi l\u0129nh v\u1ef1c<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Kh\u00f4ng ch\u1ec9 trong c\u00f4ng ngh\u1ec7, thu\u1eadt to\u00e1n c\u00f2n \u0111\u01b0\u1ee3c \u1ee9ng d\u1ee5ng r\u1ed9ng r\u00e3i trong nhi\u1ec1u l\u0129nh v\u1ef1c kh\u00e1c:<\/p>\n<ul>\n<li><strong>Y h\u1ecdc:<\/strong> Thu\u1eadt to\u00e1n gi\u00fap ch\u1ea9n \u0111o\u00e1n b\u1ec7nh, ph\u00e2n t\u00edch h\u00ecnh \u1ea3nh y khoa, ph\u00e1t tri\u1ec3n thu\u1ed1c m\u1edbi.<\/li>\n<li><strong>T\u00e0i ch\u00ednh:<\/strong> Thu\u1eadt to\u00e1n \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong giao d\u1ecbch ch\u1ee9ng kho\u00e1n, qu\u1ea3n l\u00fd r\u1ee7i ro, ph\u00e1t hi\u1ec7n gian l\u1eadn.<\/li>\n<li><strong>Logistics:<\/strong> Thu\u1eadt to\u00e1n gi\u00fap t\u1ed1i \u01b0u h\u00f3a chu\u1ed7i cung \u1ee9ng, qu\u1ea3n l\u00fd kho h\u00e0ng, l\u1eadp k\u1ebf ho\u1ea1ch v\u1eadn chuy\u1ec3n.<\/li>\n<li><strong>Khoa h\u1ecdc x\u00e3 h\u1ed9i:<\/strong> Thu\u1eadt to\u00e1n gi\u00fap ph\u00e2n t\u00edch d\u1eef li\u1ec7u, nghi\u00ean c\u1ee9u h\u00e0nh vi con ng\u01b0\u1eddi, d\u1ef1 b\u00e1o xu h\u01b0\u1edbng.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Thuc-day-su-doi-moi-va-sang-tao\"><\/span>Th\u00fac \u0111\u1ea9y s\u1ef1 \u0111\u1ed5i m\u1edbi v\u00e0 s\u00e1ng t\u1ea1o<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>B\u1eb1ng c\u00e1ch cung c\u1ea5p c\u00e1c c\u00f4ng c\u1ee5 v\u00e0 ph\u01b0\u01a1ng ph\u00e1p \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3, thu\u1eadt to\u00e1n th\u00fac \u0111\u1ea9y s\u1ef1 <strong>\u0111\u1ed5i m\u1edbi v\u00e0 s\u00e1ng t\u1ea1o<\/strong> trong m\u1ecdi l\u0129nh v\u1ef1c. Vi\u1ec7c ph\u00e1t tri\u1ec3n c\u00e1c thu\u1eadt to\u00e1n m\u1edbi, hi\u1ec7u qu\u1ea3 h\u01a1n lu\u00f4n l\u00e0 m\u1ed9t trong nh\u1eefng m\u1ee5c ti\u00eau h\u00e0ng \u0111\u1ea7u c\u1ee7a c\u00e1c nh\u00e0 khoa h\u1ecdc v\u00e0 k\u1ef9 s\u01b0.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Nhung-ung-dung-thuc-tien-cua-thuat-toan\"><\/span>Nh\u1eefng \u1ee9ng d\u1ee5ng th\u1ef1c ti\u1ec5n c\u1ee7a thu\u1eadt to\u00e1n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Thu\u1eadt to\u00e1n c\u00f3 m\u1eb7t \u1edf kh\u1eafp m\u1ecdi n\u01a1i, t\u1eeb chi\u1ebfc \u0111i\u1ec7n tho\u1ea1i b\u1ea1n \u0111ang c\u1ea7m, \u0111\u1ebfn c\u00e1c h\u1ec7 th\u1ed1ng t\u00e0i ch\u00ednh ph\u1ee9c t\u1ea1p v\u00e0 c\u00e1c nghi\u00ean c\u1ee9u khoa h\u1ecdc ti\u00ean ti\u1ebfn.<\/strong> Ch\u00fang l\u00e0 \u0111\u1ed9ng l\u1ef1c th\u1ea7m l\u1eb7ng \u0111\u1eb1ng sau v\u00f4 s\u1ed1 c\u00f4ng ngh\u1ec7 v\u00e0 \u1ee9ng d\u1ee5ng m\u00e0 ch\u00fang ta s\u1eed d\u1ee5ng h\u00e0ng ng\u00e0y.<\/p>\n<p>N\u00f3i m\u1ed9t c\u00e1ch kh\u00e1c, n\u1ebfu kh\u00f4ng c\u00f3 thu\u1eadt to\u00e1n, th\u1ebf gi\u1edbi hi\u1ec7n \u0111\u1ea1i nh\u01b0 ch\u00fang ta bi\u1ebft s\u1ebd kh\u00f4ng th\u1ec3 t\u1ed3n t\u1ea1i. Nghe c\u00f3 v\u1ebb &#8220;\u0111ao to b\u00faa l\u1edbn&#8221;, nh\u01b0ng h\u00e3y c\u00f9ng \u0111i\u1ec3m qua m\u1ed9t v\u00e0i v\u00ed d\u1ee5 c\u1ee5 th\u1ec3 \u0111\u1ec3 th\u1ea5y r\u00f5 \u0111i\u1ec1u n\u00e0y:<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cong-cu-tim-kiem-Search-Engines\"><\/span>C\u00f4ng c\u1ee5 t\u00ecm ki\u1ebfm (Search Engines)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>C\u00f3 l\u1ebd \u1ee9ng d\u1ee5ng thu\u1eadt to\u00e1n ph\u1ed5 bi\u1ebfn v\u00e0 quen thu\u1ed9c nh\u1ea5t ch\u00ednh l\u00e0 <strong>c\u00f4ng c\u1ee5 t\u00ecm ki\u1ebfm<\/strong> nh\u01b0 Google, Bing, C\u1ed1c C\u1ed1c&#8230; Khi b\u1ea1n nh\u1eadp m\u1ed9t truy v\u1ea5n, c\u00e1c thu\u1eadt to\u00e1n t\u00ecm ki\u1ebfm ph\u1ee9c t\u1ea1p s\u1ebd qu\u00e9t qua h\u00e0ng t\u1ef7 trang web, \u0111\u00e1nh gi\u00e1 m\u1ee9c \u0111\u1ed9 li\u00ean quan v\u00e0 ch\u1ea5t l\u01b0\u1ee3ng c\u1ee7a ch\u00fang, v\u00e0 tr\u1ea3 v\u1ec1 k\u1ebft qu\u1ea3 ph\u00f9 h\u1ee3p nh\u1ea5t ch\u1ec9 trong nh\u00e1y m\u1eaft.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> Thu\u1eadt to\u00e1n PageRank (\u0111\u00e3 t\u1eebng l\u00e0 m\u1ed9t ph\u1ea7n quan tr\u1ecdng c\u1ee7a Google Search) x\u1ebfp h\u1ea1ng c\u00e1c trang web d\u1ef1a tr\u00ean s\u1ed1 l\u01b0\u1ee3ng v\u00e0 ch\u1ea5t l\u01b0\u1ee3ng c\u00e1c li\u00ean k\u1ebft tr\u1ecf \u0111\u1ebfn ch\u00fang.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Mang-xa-hoi-Social-Media\"><\/span>M\u1ea1ng x\u00e3 h\u1ed9i (Social Media)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>C\u00e1c n\u1ec1n t\u1ea3ng m\u1ea1ng x\u00e3 h\u1ed9i nh\u01b0 Facebook, Instagram, TikTok&#8230; s\u1eed d\u1ee5ng thu\u1eadt to\u00e1n \u0111\u1ec3 c\u00e1 nh\u00e2n h\u00f3a <a href=\"https:\/\/interdata.vn\/blog\/ux-la-gi\/\">tr\u1ea3i nghi\u1ec7m ng\u01b0\u1eddi d\u00f9ng<\/a>. Ch\u00fang quy\u1ebft \u0111\u1ecbnh n\u1ed9i dung n\u00e0o s\u1ebd hi\u1ec3n th\u1ecb tr\u00ean b\u1ea3ng tin c\u1ee7a b\u1ea1n, qu\u1ea3ng c\u00e1o n\u00e0o b\u1ea1n s\u1ebd th\u1ea5y, v\u00e0 nh\u1eefng ng\u01b0\u1eddi b\u1ea1n n\u00e0o b\u1ea1n c\u00f3 th\u1ec3 mu\u1ed1n k\u1ebft n\u1ed1i.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> Thu\u1eadt to\u00e1n c\u1ee7a Facebook ph\u00e2n t\u00edch h\u00e0nh vi c\u1ee7a b\u1ea1n (like, share, comment&#8230;) \u0111\u1ec3 d\u1ef1 \u0111o\u00e1n nh\u1eefng n\u1ed9i dung b\u1ea1n s\u1ebd quan t\u00e2m v\u00e0 hi\u1ec3n th\u1ecb ch\u00fang nhi\u1ec1u h\u01a1n.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Ung-dung-cua-Thuat-toan-Algorithm.jpg\" alt=\"Nh\u1eefng \u1ee9ng d\u1ee5ng th\u1ef1c ti\u1ec5n c\u1ee7a thu\u1eadt to\u00e1n\" width=\"800\" height=\"450\" class=\"aligncenter size-full wp-image-25371\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Ung-dung-cua-Thuat-toan-Algorithm.jpg 800w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Ung-dung-cua-Thuat-toan-Algorithm-300x169.jpg 300w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Ung-dung-cua-Thuat-toan-Algorithm-768x432.jpg 768w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/03\/Ung-dung-cua-Thuat-toan-Algorithm-750x422.jpg 750w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Thuong-mai-dien-tu-E-commerce\"><\/span>Th\u01b0\u01a1ng m\u1ea1i \u0111i\u1ec7n t\u1eed (E-commerce)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>C\u00e1c trang web th\u01b0\u01a1ng m\u1ea1i \u0111i\u1ec7n t\u1eed nh\u01b0 Amazon, Shopee, Lazada&#8230; s\u1eed d\u1ee5ng thu\u1eadt to\u00e1n \u0111\u1ec3 g\u1ee3i \u00fd s\u1ea3n ph\u1ea9m, c\u00e1 nh\u00e2n h\u00f3a qu\u1ea3ng c\u00e1o, v\u00e0 ph\u00e1t hi\u1ec7n gian l\u1eadn.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> H\u1ec7 th\u1ed1ng g\u1ee3i \u00fd s\u1ea3n ph\u1ea9m c\u1ee7a Amazon ph\u00e2n t\u00edch l\u1ecbch s\u1eed mua h\u00e0ng v\u00e0 duy\u1ec7t web c\u1ee7a b\u1ea1n \u0111\u1ec3 \u0111\u1ec1 xu\u1ea5t nh\u1eefng s\u1ea3n ph\u1ea9m b\u1ea1n c\u00f3 th\u1ec3 quan t\u00e2m.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Dan-duong-Navigation\"><\/span>D\u1eabn \u0111\u01b0\u1eddng (Navigation)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>C\u00e1c \u1ee9ng d\u1ee5ng d\u1eabn \u0111\u01b0\u1eddng nh\u01b0 Google Maps, Apple Maps&#8230; s\u1eed d\u1ee5ng thu\u1eadt to\u00e1n \u0111\u1ec3 t\u00ecm \u0111\u01b0\u1eddng \u0111i ng\u1eafn nh\u1ea5t ho\u1eb7c nhanh nh\u1ea5t gi\u1eefa hai \u0111\u1ecba \u0111i\u1ec3m, c\u00f3 t\u00ednh \u0111\u1ebfn c\u00e1c y\u1ebfu t\u1ed1 nh\u01b0 giao th\u00f4ng, t\u1ed1c \u0111\u1ed9 gi\u1edbi h\u1ea1n, v\u00e0 c\u00e1c s\u1ef1 c\u1ed1 tr\u00ean \u0111\u01b0\u1eddng.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> Thu\u1eadt to\u00e1n Dijkstra \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng r\u1ed9ng r\u00e3i trong vi\u1ec7c t\u00ecm \u0111\u01b0\u1eddng \u0111i ng\u1eafn nh\u1ea5t.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tri-tue-nhan-tao-Artificial-Intelligence\"><\/span>Tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o (Artificial Intelligence)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Thu\u1eadt to\u00e1n l\u00e0 n\u1ec1n t\u1ea3ng c\u1ee7a <strong>Tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o (AI)<\/strong>. C\u00e1c \u1ee9ng d\u1ee5ng AI nh\u01b0 nh\u1eadn d\u1ea1ng gi\u1ecdng n\u00f3i, d\u1ecbch m\u00e1y, xe t\u1ef1 l\u00e1i&#8230; \u0111\u1ec1u d\u1ef1a tr\u00ean c\u00e1c thu\u1eadt to\u00e1n h\u1ecdc m\u00e1y ph\u1ee9c t\u1ea1p.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> M\u1ea1ng n\u01a1-ron s\u00e2u (Deep Neural Networks) l\u00e0 m\u1ed9t lo\u1ea1i thu\u1eadt to\u00e1n h\u1ecdc m\u00e1y \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong nhi\u1ec1u \u1ee9ng d\u1ee5ng AI, t\u1eeb nh\u1eadn d\u1ea1ng h\u00ecnh \u1ea3nh \u0111\u1ebfn x\u1eed l\u00fd ng\u00f4n ng\u1eef t\u1ef1 nhi\u00ean.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Y-hoc-Medicine\"><\/span>Y h\u1ecdc (Medicine)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Trong l\u0129nh v\u1ef1c y h\u1ecdc, thu\u1eadt to\u00e1n \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 ch\u1ea9n \u0111o\u00e1n b\u1ec7nh, ph\u00e2n t\u00edch h\u00ecnh \u1ea3nh y khoa, ph\u00e1t tri\u1ec3n thu\u1ed1c m\u1edbi, v\u00e0 c\u00e1 nh\u00e2n h\u00f3a ph\u00e1c \u0111\u1ed3 \u0111i\u1ec1u tr\u1ecb.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> Thu\u1eadt to\u00e1n h\u1ecdc m\u00e1y c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 ph\u00e2n t\u00edch h\u00ecnh \u1ea3nh X-quang ho\u1eb7c MRI \u0111\u1ec3 ph\u00e1t hi\u1ec7n c\u00e1c d\u1ea5u hi\u1ec7u s\u1edbm c\u1ee7a ung th\u01b0.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tai-chinh-Finance\"><\/span>T\u00e0i ch\u00ednh (Finance)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Trong l\u0129nh v\u1ef1c t\u00e0i ch\u00ednh, thu\u1eadt to\u00e1n \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong giao d\u1ecbch ch\u1ee9ng kho\u00e1n (high-frequency trading), qu\u1ea3n l\u00fd r\u1ee7i ro, ph\u00e1t hi\u1ec7n gian l\u1eadn, v\u00e0 ch\u1ea5m \u0111i\u1ec3m t\u00edn d\u1ee5ng.<\/p>\n<p><strong>V\u00ed d\u1ee5:<\/strong> C\u00e1c qu\u1ef9 \u0111\u1ea7u t\u01b0 \u0111\u1ecbnh l\u01b0\u1ee3ng (quantitative hedge funds) s\u1eed d\u1ee5ng c\u00e1c thu\u1eadt to\u00e1n ph\u1ee9c t\u1ea1p \u0111\u1ec3 t\u1ef1 \u0111\u1ed9ng th\u1ef1c hi\u1ec7n giao d\u1ecbch ch\u1ee9ng kho\u00e1n.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cac-linh-vuc-khac\"><\/span>C\u00e1c l\u0129nh v\u1ef1c kh\u00e1c<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Ngo\u00e0i ra, thu\u1eadt to\u00e1n c\u00f2n c\u00f3 v\u00f4 s\u1ed1 \u1ee9ng d\u1ee5ng trong c\u00e1c l\u0129nh v\u1ef1c kh\u00e1c nh\u01b0:<\/p>\n<ul>\n<li><strong>Logistics:<\/strong> T\u1ed1i \u01b0u h\u00f3a chu\u1ed7i cung \u1ee9ng, qu\u1ea3n l\u00fd kho h\u00e0ng.<\/li>\n<li><strong>S\u1ea3n xu\u1ea5t:<\/strong> L\u1eadp k\u1ebf ho\u1ea1ch s\u1ea3n xu\u1ea5t, \u0111i\u1ec1u khi\u1ec3n robot.<\/li>\n<li><strong>N\u00f4ng nghi\u1ec7p:<\/strong> D\u1ef1 b\u00e1o th\u1eddi ti\u1ebft, qu\u1ea3n l\u00fd t\u01b0\u1edbi ti\u00eau.<\/li>\n<li><strong>N\u0103ng l\u01b0\u1ee3ng:<\/strong> Qu\u1ea3n l\u00fd l\u01b0\u1edbi \u0111i\u1ec7n, d\u1ef1 b\u00e1o nhu c\u1ea7u n\u0103ng l\u01b0\u1ee3ng.<\/li>\n<li><strong>B\u1ea3o m\u1eadt:<\/strong> C\u00e1c thu\u1eadt to\u00e1n b\u1ea3o m\u1eadt v\u00e0 an ninh m\u1ea1ng, gi\u00fap b\u1ea3o v\u1ec7 h\u1ec7 th\u1ed1ng v\u00e0 ch\u1ed1ng l\u1ea1i x\u00e2m nh\u1eadp tr\u00e1i ph\u00e9p<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Cac-loai-thuat-toan-pho-bien-hien-nay\"><\/span>C\u00e1c lo\u1ea1i thu\u1eadt to\u00e1n ph\u1ed5 bi\u1ebfn hi\u1ec7n nay<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>C\u00f3 v\u00f4 s\u1ed1 thu\u1eadt to\u00e1n kh\u00e1c nhau, \u0111\u01b0\u1ee3c ph\u00e2n lo\u1ea1i theo nhi\u1ec1u c\u00e1ch: ch\u1ee9c n\u0103ng, c\u00e1ch ho\u1ea1t \u0111\u1ed9ng, \u0111\u1ed9 ph\u1ee9c t\u1ea1p, l\u0129nh v\u1ef1c \u1ee9ng d\u1ee5ng&#8230; Kh\u00f4ng c\u00f3 m\u1ed9t c\u00e1ch ph\u00e2n lo\u1ea1i duy nh\u1ea5t n\u00e0o l\u00e0 tuy\u1ec7t \u0111\u1ed1i c\u1ea3.<\/strong><\/p>\n<p>Tuy nhi\u00ean, \u0111\u1ec3 d\u1ec5 h\u00ecnh dung, ch\u00fang ta c\u00f3 th\u1ec3 nh\u00f3m c\u00e1c thu\u1eadt to\u00e1n ph\u1ed5 bi\u1ebfn d\u1ef1a tr\u00ean m\u1ee5c \u0111\u00edch s\u1eed d\u1ee5ng. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 lo\u1ea1i thu\u1eadt to\u00e1n quan tr\u1ecdng v\u00e0 th\u01b0\u1eddng g\u1eb7p, c\u00f9ng v\u1edbi v\u00ed d\u1ee5 minh h\u1ecda:<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Sap-xep-Sorting-Algorithms\"><\/span>Thu\u1eadt to\u00e1n S\u1eafp x\u1ebfp (Sorting Algorithms)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>S\u1eafp x\u1ebfp l\u00e0 m\u1ed9t trong nh\u1eefng b\u00e0i to\u00e1n c\u01a1 b\u1ea3n nh\u1ea5t trong khoa h\u1ecdc m\u00e1y t\u00ednh. C\u00e1c thu\u1eadt to\u00e1n s\u1eafp x\u1ebfp \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 b\u1ed1 tr\u00ed l\u1ea1i m\u1ed9t danh s\u00e1ch c\u00e1c ph\u1ea7n t\u1eed (v\u00ed d\u1ee5: s\u1ed1, t\u00ean&#8230;) theo m\u1ed9t th\u1ee9 t\u1ef1 nh\u1ea5t \u0111\u1ecbnh (t\u0103ng d\u1ea7n, gi\u1ea3m d\u1ea7n, theo b\u1ea3ng ch\u1eef c\u00e1i&#8230;).<\/p>\n<p>C\u00f3 nhi\u1ec1u thu\u1eadt to\u00e1n s\u1eafp x\u1ebfp kh\u00e1c nhau, m\u1ed7i thu\u1eadt to\u00e1n c\u00f3 \u01b0u v\u00e0 nh\u01b0\u1ee3c \u0111i\u1ec3m ri\u00eang:<\/p>\n<ul>\n<li><strong>S\u1eafp x\u1ebfp n\u1ed5i b\u1ecdt (Bubble Sort):<\/strong> Thu\u1eadt to\u00e1n n\u00e0y \u0111\u01a1n gi\u1ea3n, d\u1ec5 hi\u1ec3u: so s\u00e1nh v\u00e0 \u0111\u1ed5i ch\u1ed7 c\u00e1c c\u1eb7p ph\u1ea7n t\u1eed li\u1ec1n k\u1ec1 n\u1ebfu ch\u00fang kh\u00f4ng \u0111\u00fang th\u1ee9 t\u1ef1. Tuy nhi\u00ean, n\u00f3 th\u01b0\u1eddng kh\u00f4ng hi\u1ec7u qu\u1ea3 v\u1edbi <a href=\"https:\/\/interdata.vn\/blog\/big-data-la-gi\/\">d\u1eef li\u1ec7u l\u1edbn<\/a>.<\/li>\n<li><strong>S\u1eafp x\u1ebfp ch\u00e8n (Insertion Sort):<\/strong> Gi\u1ed1ng nh\u01b0 c\u00e1ch b\u1ea1n s\u1eafp x\u1ebfp b\u00e0i, thu\u1eadt to\u00e1n n\u00e0y ch\u00e8n t\u1eebng ph\u1ea7n t\u1eed v\u00e0o v\u1ecb tr\u00ed \u0111\u00fang trong danh s\u00e1ch \u0111\u00e3 s\u1eafp x\u1ebfp. N\u00f3 kh\u00e1 hi\u1ec7u qu\u1ea3 v\u1edbi d\u1eef li\u1ec7u g\u1ea7n nh\u01b0 \u0111\u00e3 \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp.<\/li>\n<li><strong>S\u1eafp x\u1ebfp ch\u1ecdn (Selection Sort):<\/strong> Li\u00ean t\u1ee5c t\u00ecm ph\u1ea7n t\u1eed nh\u1ecf nh\u1ea5t (ho\u1eb7c l\u1edbn nh\u1ea5t) v\u00e0 \u0111\u01b0a n\u00f3 v\u1ec1 v\u1ecb tr\u00ed \u0111\u1ea7u ti\u00ean (ho\u1eb7c cu\u1ed1i c\u00f9ng) c\u1ee7a danh s\u00e1ch ch\u01b0a s\u1eafp x\u1ebfp.<\/li>\n<li><strong>S\u1eafp x\u1ebfp nhanh (Quick Sort):<\/strong> M\u1ed9t thu\u1eadt to\u00e1n &#8220;chia \u0111\u1ec3 tr\u1ecb&#8221; r\u1ea5t ph\u1ed5 bi\u1ebfn. N\u00f3 ch\u1ecdn m\u1ed9t ph\u1ea7n t\u1eed l\u00e0m &#8220;ch\u1ed1t&#8221;, chia danh s\u00e1ch th\u00e0nh hai ph\u1ea7n (l\u1edbn h\u01a1n v\u00e0 nh\u1ecf h\u01a1n ch\u1ed1t), r\u1ed3i <a href=\"https:\/\/interdata.vn\/blog\/de-quy-la-gi\/\">\u0111\u1ec7 quy<\/a> s\u1eafp x\u1ebfp hai ph\u1ea7n n\u00e0y. Th\u01b0\u1eddng r\u1ea5t nhanh trong th\u1ef1c t\u1ebf.<\/li>\n<li><strong>S\u1eafp x\u1ebfp tr\u1ed9n (Merge Sort):<\/strong> C\u0169ng l\u00e0 m\u1ed9t thu\u1eadt to\u00e1n &#8220;chia \u0111\u1ec3 tr\u1ecb&#8221;. N\u00f3 chia danh s\u00e1ch th\u00e0nh c\u00e1c ph\u1ea7n nh\u1ecf, s\u1eafp x\u1ebfp t\u1eebng ph\u1ea7n, r\u1ed3i tr\u1ed9n ch\u00fang l\u1ea1i. \u1ed4n \u0111\u1ecbnh v\u00e0 hi\u1ec7u qu\u1ea3, nh\u01b0ng c\u1ea7n th\u00eam b\u1ed9 nh\u1edb.<\/li>\n<\/ul>\n<p>Vi\u1ec7c l\u1ef1a ch\u1ecdn thu\u1eadt to\u00e1n s\u1eafp x\u1ebfp n\u00e0o s\u1ebd t\u00f9y thu\u1ed9c v\u00e0o t\u00ecnh hu\u1ed1ng c\u1ee5 th\u1ec3: k\u00edch th\u01b0\u1edbc d\u1eef li\u1ec7u, m\u1ee9c \u0111\u1ed9 &#8220;l\u1ed9n x\u1ed9n&#8221; ban \u0111\u1ea7u, v\u00e0 y\u00eau c\u1ea7u v\u1ec1 hi\u1ec7u su\u1ea5t.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Tim-kiem-Searching-Algorithms\"><\/span>Thu\u1eadt to\u00e1n T\u00ecm ki\u1ebfm (Searching Algorithms)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>T\u00ecm ki\u1ebfm l\u00e0 m\u1ed9t thao t\u00e1c c\u01a1 b\u1ea3n kh\u00e1c. C\u00e1c thu\u1eadt to\u00e1n t\u00ecm ki\u1ebfm gi\u00fap ch\u00fang ta x\u00e1c \u0111\u1ecbnh v\u1ecb tr\u00ed c\u1ee7a m\u1ed9t ph\u1ea7n t\u1eed c\u1ee5 th\u1ec3 trong m\u1ed9t t\u1eadp h\u1ee3p d\u1eef li\u1ec7u.<\/p>\n<ul>\n<li><strong>T\u00ecm ki\u1ebfm tuy\u1ebfn t\u00ednh (Linear Search):<\/strong> C\u00e1ch \u0111\u01a1n gi\u1ea3n nh\u1ea5t: duy\u1ec7t qua t\u1eebng ph\u1ea7n t\u1eed c\u1ee7a danh s\u00e1ch cho \u0111\u1ebfn khi t\u00ecm th\u1ea5y ph\u1ea7n t\u1eed c\u1ea7n t\u00ecm. D\u1ec5 hi\u1ec3u, nh\u01b0ng kh\u00f4ng hi\u1ec7u qu\u1ea3 v\u1edbi d\u1eef li\u1ec7u l\u1edbn.<\/li>\n<li><strong>T\u00ecm ki\u1ebfm nh\u1ecb ph\u00e2n (Binary Search):<\/strong> M\u1ed9t &#8220;si\u00eau sao&#8221; t\u00ecm ki\u1ebfm! Nh\u01b0ng n\u00f3 ch\u1ec9 ho\u1ea1t \u0111\u1ed9ng tr\u00ean danh s\u00e1ch \u0111\u00e3 \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp. Thu\u1eadt to\u00e1n li\u00ean t\u1ee5c chia \u0111\u00f4i danh s\u00e1ch v\u00e0 so s\u00e1nh ph\u1ea7n t\u1eed c\u1ea7n t\u00ecm v\u1edbi ph\u1ea7n t\u1eed \u1edf gi\u1eefa. Nh\u1edd v\u1eady, n\u00f3 t\u00ecm ki\u1ebfm c\u1ef1c k\u1ef3 nhanh.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Do-thi-Graph-Algorithms\"><\/span>Thu\u1eadt to\u00e1n \u0110\u1ed3 th\u1ecb (Graph Algorithms)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0110\u1ed3 th\u1ecb l\u00e0 m\u1ed9t <a href=\"https:\/\/interdata.vn\/blog\/cau-truc-du-lieu\/\">c\u1ea5u tr\u00fac d\u1eef li\u1ec7u<\/a> bi\u1ec3u di\u1ec5n m\u1ed1i quan h\u1ec7 gi\u1eefa c\u00e1c \u0111\u1ed1i t\u01b0\u1ee3ng (v\u00ed d\u1ee5: m\u1ea1ng x\u00e3 h\u1ed9i, b\u1ea3n \u0111\u1ed3 \u0111\u01b0\u1eddng \u0111i&#8230;). C\u00e1c thu\u1eadt to\u00e1n \u0111\u1ed3 th\u1ecb gi\u1ea3i quy\u1ebft c\u00e1c b\u00e0i to\u00e1n li\u00ean quan \u0111\u1ebfn m\u1ea1ng l\u01b0\u1edbi n\u00e0y.<\/p>\n<ul>\n<li><strong>T\u00ecm \u0111\u01b0\u1eddng \u0111i ng\u1eafn nh\u1ea5t:<\/strong> \u0110\u00e2y l\u00e0 m\u1ed9t b\u00e0i to\u00e1n kinh \u0111i\u1ec3n!\n<ol>\n<li><strong>Thu\u1eadt to\u00e1n Dijkstra:<\/strong> T\u00ecm \u0111\u01b0\u1eddng \u0111i ng\u1eafn nh\u1ea5t t\u1eeb m\u1ed9t \u0111\u1ec9nh \u0111\u1ebfn t\u1ea5t c\u1ea3 c\u00e1c \u0111\u1ec9nh kh\u00e1c trong \u0111\u1ed3 th\u1ecb c\u00f3 tr\u1ecdng s\u1ed1 kh\u00f4ng \u00e2m.<\/li>\n<li><strong>Thu\u1eadt to\u00e1n Bellman-Ford:<\/strong> T\u01b0\u01a1ng t\u1ef1 nh\u01b0 Dijkstra, nh\u01b0ng x\u1eed l\u00fd \u0111\u01b0\u1ee3c c\u1ea3 tr\u1ecdng s\u1ed1 \u00e2m (nh\u01b0ng ch\u1eadm h\u01a1n).<\/li>\n<li><strong>Thu\u1eadt to\u00e1n Floyd-Warshall:<\/strong> T\u00ecm \u0111\u01b0\u1eddng \u0111i ng\u1eafn nh\u1ea5t gi\u1eefa <i>t\u1ea5t c\u1ea3<\/i> c\u00e1c c\u1eb7p \u0111\u1ec9nh trong \u0111\u1ed3 th\u1ecb.<\/li>\n<\/ol>\n<\/li>\n<li><strong>T\u00ecm ki\u1ebfm theo chi\u1ec1u r\u1ed9ng (BFS) v\u00e0 T\u00ecm ki\u1ebfm theo chi\u1ec1u s\u00e2u (DFS):<\/strong> \u0110\u00e2y l\u00e0 hai thu\u1eadt to\u00e1n c\u01a1 b\u1ea3n \u0111\u1ec3 &#8220;duy\u1ec7t&#8221; \u0111\u1ed3 th\u1ecb. BFS duy\u1ec7t theo t\u1eebng &#8220;l\u1edbp&#8221;, c\u00f2n DFS \u0111i s\u00e2u v\u00e0o t\u1eebng &#8220;nh\u00e1nh&#8221;.<\/li>\n<li><strong>C\u00e2y bao tr\u00f9m t\u1ed1i thi\u1ec3u (Minimum Spanning <a href=\"https:\/\/interdata.vn\/blog\/cau-truc-du-lieu-tree-cay\/\">Tree<\/a>):<\/strong> T\u00ecm m\u1ed9t t\u1eadp h\u1ee3p c\u00e1c c\u1ea1nh n\u1ed1i t\u1ea5t c\u1ea3 c\u00e1c \u0111\u1ec9nh c\u1ee7a \u0111\u1ed3 th\u1ecb v\u1edbi t\u1ed5ng tr\u1ecdng s\u1ed1 nh\u1ecf nh\u1ea5t (v\u00ed d\u1ee5: thi\u1ebft k\u1ebf m\u1ea1ng l\u01b0\u1edbi \u0111i\u1ec7n).<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Chia-de-tri-Divide-and-Conquer\"><\/span>Thu\u1eadt to\u00e1n Chia \u0111\u1ec3 tr\u1ecb (Divide and Conquer)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0110\u00e2y kh\u00f4ng ph\u1ea3i l\u00e0 m\u1ed9t lo\u1ea1i thu\u1eadt to\u00e1n c\u1ee5 th\u1ec3, m\u00e0 l\u00e0 m\u1ed9t <i>chi\u1ebfn l\u01b0\u1ee3c<\/i> thi\u1ebft k\u1ebf thu\u1eadt to\u00e1n. \u00dd t\u01b0\u1edfng l\u00e0 chia b\u00e0i to\u00e1n l\u1edbn th\u00e0nh c\u00e1c b\u00e0i to\u00e1n con nh\u1ecf h\u01a1n, gi\u1ea3i quy\u1ebft c\u00e1c b\u00e0i to\u00e1n con n\u00e0y (th\u01b0\u1eddng l\u00e0 \u0111\u1ec7 quy), v\u00e0 sau \u0111\u00f3 k\u1ebft h\u1ee3p c\u00e1c k\u1ebft qu\u1ea3 l\u1ea1i.<\/p>\n<ul>\n<li><strong>S\u1eafp x\u1ebfp tr\u1ed9n (Merge Sort)<\/strong> v\u00e0 <strong>S\u1eafp x\u1ebfp nhanh (Quick Sort)<\/strong> (\u0111\u00e3 n\u00f3i \u1edf tr\u00ean) l\u00e0 nh\u1eefng v\u00ed d\u1ee5 \u0111i\u1ec3n h\u00ecnh.<\/li>\n<li><strong>T\u00ecm ki\u1ebfm nh\u1ecb ph\u00e2n (Binary Search)<\/strong> c\u0169ng c\u00f3 th\u1ec3 xem l\u00e0 1 tr\u01b0\u1eddng h\u1ee3p c\u1ee7a chia \u0111\u1ec3 tr\u1ecb.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Tham-lam-Greedy-Algorithms\"><\/span>Thu\u1eadt to\u00e1n Tham lam (Greedy Algorithms)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Thu\u1eadt to\u00e1n tham lam \u0111\u01b0a ra quy\u1ebft \u0111\u1ecbnh &#8220;t\u1ed1t nh\u1ea5t c\u00f3 th\u1ec3&#8221; <i>t\u1ea1i m\u1ed7i b\u01b0\u1edbc<\/i>, v\u1edbi hy v\u1ecdng r\u1eb1ng chu\u1ed7i c\u00e1c quy\u1ebft \u0111\u1ecbnh c\u1ee5c b\u1ed9 t\u1ed1i \u01b0u n\u00e0y s\u1ebd d\u1eabn \u0111\u1ebfn l\u1eddi gi\u1ea3i to\u00e0n c\u1ee5c t\u1ed1i \u01b0u. \u0110\u00f4i khi n\u00f3 hi\u1ec7u qu\u1ea3, \u0111\u00f4i khi kh\u00f4ng!<\/p>\n<ul>\n<li><strong>B\u00e0i to\u00e1n c\u00e1i t\u00fai (Knapsack Problem):<\/strong> L\u00e0m sao \u0111\u1ec3 ch\u1ecdn c\u00e1c v\u1eadt ph\u1ea9m c\u00f3 gi\u00e1 tr\u1ecb cao nh\u1ea5t \u0111\u1ec3 cho v\u00e0o t\u00fai m\u00e0 kh\u00f4ng v\u01b0\u1ee3t qu\u00e1 tr\u1ecdng l\u01b0\u1ee3ng cho ph\u00e9p?<\/li>\n<li><strong>Thu\u1eadt to\u00e1n Kruskal v\u00e0 Prim<\/strong> (t\u00ecm c\u00e2y bao tr\u00f9m t\u1ed1i thi\u1ec3u) c\u0169ng l\u00e0 v\u00ed d\u1ee5 v\u1ec1 thu\u1eadt to\u00e1n tham lam.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Quy-hoach-dong-Dynamic-Programming\"><\/span>Thu\u1eadt to\u00e1n Quy ho\u1ea1ch \u0111\u1ed9ng (Dynamic Programming)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Quy ho\u1ea1ch \u0111\u1ed9ng c\u0169ng chia b\u00e0i to\u00e1n th\u00e0nh c\u00e1c b\u00e0i to\u00e1n con, nh\u01b0ng \u0111i\u1ec3m kh\u00e1c bi\u1ec7t l\u00e0 n\u00f3 ghi nh\u1edb (l\u01b0u tr\u1eef) k\u1ebft qu\u1ea3 c\u1ee7a c\u00e1c b\u00e0i to\u00e1n con \u0111\u00e3 gi\u1ea3i \u0111\u1ec3 tr\u00e1nh ph\u1ea3i t\u00ednh to\u00e1n l\u1ea1i. \u0110i\u1ec1u n\u00e0y gi\u00fap c\u1ea3i thi\u1ec7n \u0111\u00e1ng k\u1ec3 hi\u1ec7u su\u1ea5t trong nhi\u1ec1u tr\u01b0\u1eddng h\u1ee3p.<\/p>\n<ul>\n<li><strong>D\u00e3y Fibonacci:<\/strong> T\u00ednh s\u1ed1 Fibonacci th\u1ee9 n b\u1eb1ng c\u00e1ch s\u1eed d\u1ee5ng k\u1ebft qu\u1ea3 c\u1ee7a c\u00e1c s\u1ed1 Fibonacci tr\u01b0\u1edbc \u0111\u00f3.<\/li>\n<li><strong>B\u00e0i to\u00e1n c\u00e1i t\u00fai<\/strong> (phi\u00ean b\u1ea3n quy ho\u1ea1ch \u0111\u1ed9ng).<\/li>\n<li><strong>Thu\u1eadt to\u00e1n Floyd-Warshall<\/strong> (t\u00ecm \u0111\u01b0\u1eddng \u0111i ng\u1eafn nh\u1ea5t gi\u1eefa m\u1ecdi c\u1eb7p \u0111\u1ec9nh) c\u0169ng l\u00e0 m\u1ed9t v\u00ed d\u1ee5.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Quay-lui-Backtracking\"><\/span>Thu\u1eadt to\u00e1n Quay lui (Backtracking)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Thu\u1eadt to\u00e1n quay lui th\u1eed t\u1ea5t c\u1ea3 c\u00e1c kh\u1ea3 n\u0103ng c\u00f3 th\u1ec3. N\u1ebfu m\u1ed9t kh\u1ea3 n\u0103ng kh\u00f4ng d\u1eabn \u0111\u1ebfn l\u1eddi gi\u1ea3i, thu\u1eadt to\u00e1n s\u1ebd &#8220;quay lui&#8221; v\u00e0 th\u1eed kh\u1ea3 n\u0103ng kh\u00e1c. N\u00f3 gi\u1ed1ng nh\u01b0 vi\u1ec7c \u0111i v\u00e0o m\u1ed9t m\u00ea cung v\u00e0 th\u1eed t\u1eebng \u0111\u01b0\u1eddng \u0111i, n\u1ebfu g\u1eb7p ng\u00f5 c\u1ee5t th\u00ec quay l\u1ea1i v\u00e0 th\u1eed \u0111\u01b0\u1eddng kh\u00e1c.<\/p>\n<p><strong>B\u00e0i to\u00e1n m\u00e3 \u0111i tu\u1ea7n<\/strong>, <strong>b\u00e0i to\u00e1n x\u1ebfp h\u1eadu<\/strong> l\u00e0 c\u00e1c v\u00ed d\u1ee5 kinh \u0111i\u1ec3n.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Thuat-toan-Hoc-may-Machine-Learning-Algorithms\"><\/span>Thu\u1eadt to\u00e1n H\u1ecdc m\u00e1y (Machine Learning Algorithms)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0110\u00e2y l\u00e0 m\u1ed9t l\u0129nh v\u1ef1c &#8220;hot&#8221; hi\u1ec7n nay! C\u00e1c thu\u1eadt to\u00e1n h\u1ecdc m\u00e1y cho ph\u00e9p m\u00e1y t\u00ednh &#8220;h\u1ecdc&#8221; t\u1eeb d\u1eef li\u1ec7u m\u00e0 kh\u00f4ng c\u1ea7n \u0111\u01b0\u1ee3c <a href=\"https:\/\/interdata.vn\/blog\/lap-trinh-la-gi\/\">l\u1eadp tr\u00ecnh<\/a> c\u1ee5 th\u1ec3 cho t\u1eebng tr\u01b0\u1eddng h\u1ee3p.<\/p>\n<p><strong>H\u1ed3i quy tuy\u1ebfn t\u00ednh\/Logistic, <a href=\"https:\/\/interdata.vn\/blog\/decision-tree-la-gi\/\">C\u00e2y quy\u1ebft \u0111\u1ecbnh<\/a>, M\u00e1y h\u1ecdc v\u00e9c-t\u01a1 h\u1ed7 tr\u1ee3 (<a href=\"https:\/\/interdata.vn\/blog\/support-vector-machine-la-gi\/\">SVM<\/a>), M\u1ea1ng n\u01a1-ron, K-means <a href=\"https:\/\/interdata.vn\/blog\/clustering-la-gi\/\">clustering<\/a><\/strong>&#8230; v\u00e0 r\u1ea5t nhi\u1ec1u thu\u1eadt to\u00e1n kh\u00e1c, m\u1ed7i thu\u1eadt to\u00e1n c\u00f3 \u0111i\u1ec3m m\u1ea1nh v\u00e0 \u1ee9ng d\u1ee5ng ri\u00eang.<\/p>\n<hr \/>\n<p>Nh\u01b0 v\u1eady, ch\u00fang ta \u0111\u00e3 c\u00f9ng nhau t\u00ecm hi\u1ec3u thu\u1eadt to\u00e1n (algorithm) l\u00e0 g\u00ec, vai tr\u00f2 then ch\u1ed1t c\u1ee7a n\u00f3 trong m\u1ecdi m\u1eb7t \u0111\u1eddi s\u1ed1ng, nh\u1eefng \u1ee9ng d\u1ee5ng r\u1ed9ng r\u00e3i v\u00e0 c\u00e1c lo\u1ea1i thu\u1eadt to\u00e1n ph\u1ed5 bi\u1ebfn. Hy v\u1ecdng b\u00e0i vi\u1ebft n\u00e0y \u0111\u00e3 gi\u00fap b\u1ea1n c\u00f3 c\u00e1i nh\u00ecn t\u1ed5ng quan v\u00e0 s\u00e2u s\u1eafc h\u01a1n v\u1ec1 m\u1ed9t kh\u00e1i ni\u1ec7m t\u01b0\u1edfng ch\u1eebng &#8220;kh\u00f3 nh\u1eb1n&#8221; nh\u01b0ng l\u1ea1i v\u00f4 c\u00f9ng th\u00fa v\u1ecb n\u00e0y. N\u1ebfu b\u1ea1n mu\u1ed1n t\u00ecm hi\u1ec3u s\u00e2u h\u01a1n v\u1ec1 l\u1eadp tr\u00ecnh v\u00e0 khoa h\u1ecdc m\u00e1y t\u00ednh, \u0111\u1eebng ng\u1ea7n ng\u1ea1i kh\u00e1m ph\u00e1 th\u00eam c\u00e1c b\u00e0i vi\u1ebft kh\u00e1c tr\u00ean <a target=\"_blank\" rel=\"noopener noreferrer\" href=\"https:\/\/interdata.vn\/\">InterData<\/a> nh\u00e9!<\/p>\n<p><strong>INTERDATA<\/strong><\/p>\n<ul>\n<li><strong>Website:<\/strong><span>\u00a0<\/span>Interdata.vn<\/li>\n<li><strong>Hotline:<\/strong><span>\u00a0<\/span>1900-636822<\/li>\n<li><strong>Email:<\/strong><span>\u00a0<\/span>Info@interdata.vn<\/li>\n<li><strong>VP\u0110D:<\/strong><span>\u00a0<\/span>240 Nguy\u1ec5n \u0110\u00ecnh Ch\u00ednh, P.11. Q. Ph\u00fa Nhu\u1eadn, TP. Ho\u0302\u0300 Ch\u00ed Minh<\/li>\n<li><strong>VPGD:<\/strong><span>\u00a0<\/span>S\u1ed1 211 \u0110\u01b0\u1eddng s\u1ed1 5, K\u0110T Lakeview City, P. An Ph\u00fa, TP. Th\u1ee7 \u0110\u1ee9c, TP. H\u1ed3 Ch\u00ed Minh<\/li>\n<\/ul>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul><\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>B\u1ea1n c\u00f3 bao gi\u1edd t\u1ef1 h\u1ecfi Google t\u00ecm ki\u1ebfm nhanh \u0111\u1ebfn v\u1eady nh\u01b0 th\u1ebf n\u00e0o, hay Facebook g\u1ee3i \u00fd k\u1ebft b\u1ea1n &#8220;chu\u1ea9n&#8221; ra sao? T\u1ea5t c\u1ea3 \u0111\u1ec1u nh\u1edd v\u00e0o thu\u1eadt to\u00e1n. B\u00e0i vi\u1ebft n\u00e0y s\u1ebd gi\u1ea3i th\u00edch thu\u1eadt to\u00e1n (algorithm) l\u00e0 g\u00ec, t\u1ea7m quan tr\u1ecdng, nh\u1eefng \u1ee9ng d\u1ee5ng th\u1ef1c ti\u1ec5n \u0111\u00e1ng kinh ng\u1ea1c v\u00e0 c\u00e1c<\/p>\n","protected":false},"author":2,"featured_media":25370,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[108],"tags":[],"class_list":["post-25365","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ai"],"_links":{"self":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts\/25365","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/comments?post=25365"}],"version-history":[{"count":0,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts\/25365\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/media\/25370"}],"wp:attachment":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/media?parent=25365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/categories?post=25365"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/tags?post=25365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}