{"id":478658,"date":"2023-08-09T09:36:38","date_gmt":"2023-08-09T09:36:38","guid":{"rendered":""},"modified":"2023-09-05T11:17:18","modified_gmt":"2023-09-05T11:17:18","slug":"recursion","status":"publish","type":"wiki","link":"https:\/\/oneproxy.pro\/vn\/wiki\/recursion\/","title":{"rendered":"\u0111\u1ec7 quy"},"content":{"rendered":"<p>\u0110\u1ec7 quy l\u00e0 m\u1ed9t k\u1ef9 thu\u1eadt t\u00ednh to\u00e1n ho\u1eb7c to\u00e1n h\u1ecdc trong \u0111\u00f3 m\u1ed9t h\u00e0m g\u1ecdi ch\u00ednh n\u00f3 tr\u1ef1c ti\u1ebfp ho\u1eb7c gi\u00e1n ti\u1ebfp \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1. \u0110\u00f3 l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m thi\u1ebft y\u1ebfu trong khoa h\u1ecdc m\u00e1y t\u00ednh v\u00e0 to\u00e1n h\u1ecdc, cho ph\u00e9p \u0111\u01b0a ra c\u00e1c gi\u1ea3i ph\u00e1p tinh t\u1ebf cho m\u1ed9t s\u1ed1 v\u1ea5n \u0111\u1ec1 nh\u1ea5t \u0111\u1ecbnh, nh\u01b0ng n\u00f3 c\u0169ng c\u00f3 th\u1ec3 d\u1eabn \u0111\u1ebfn nh\u1eefng r\u1eafc r\u1ed1i n\u1ebfu kh\u00f4ng \u0111\u01b0\u1ee3c tri\u1ec3n khai \u0111\u00fang c\u00e1ch.<\/p>\n<h2>L\u1ecbch s\u1eed ngu\u1ed3n g\u1ed1c c\u1ee7a \u0111\u1ec7 quy v\u00e0 s\u1ef1 \u0111\u1ec1 c\u1eadp \u0111\u1ea7u ti\u00ean v\u1ec1 n\u00f3<\/h2>\n<p>Ngu\u1ed3n g\u1ed1c c\u1ee7a \u0111\u1ec7 quy c\u00f3 th\u1ec3 b\u1eaft ngu\u1ed3n t\u1eeb to\u00e1n h\u1ecdc v\u00e0 tri\u1ebft h\u1ecdc c\u1ed5 \u0111\u1ea1i. Ngh\u1ecbch l\u00fd v\u1ec1 s\u1ef1 t\u1ef1 quy chi\u1ebfu, ch\u1eb3ng h\u1ea1n nh\u01b0 \u201cngh\u1ecbch l\u00fd k\u1ebb n\u00f3i d\u1ed1i\u201d, l\u00e0 m\u1ed9t v\u00ed d\u1ee5 ban \u0111\u1ea7u v\u1ec1 s\u1ef1 \u0111\u1ec7 quy trong t\u01b0 duy logic.<\/p>\n<p>Trong to\u00e1n h\u1ecdc, c\u00e1c c\u00f4ng th\u1ee9c \u0111\u1ec7 quy s\u1edbm nh\u1ea5t \u0111\u01b0\u1ee3c t\u00ecm th\u1ea5y trong c\u00e1c t\u00e1c ph\u1ea9m c\u1ee7a c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc \u1ea4n \u0110\u1ed9 v\u00e0o th\u1ebf k\u1ef7 th\u1ee9 6. Trong khoa h\u1ecdc m\u00e1y t\u00ednh, \u0111\u1ec7 quy tr\u1edf n\u00ean ph\u1ed5 bi\u1ebfn h\u01a1n v\u1edbi s\u1ef1 ra \u0111\u1eddi c\u1ee7a c\u00e1c ng\u00f4n ng\u1eef l\u1eadp tr\u00ecnh h\u00e0m v\u00e0o gi\u1eefa th\u1ebf k\u1ef7 20.<\/p>\n<h2>Th\u00f4ng tin chi ti\u1ebft v\u1ec1 \u0111\u1ec7 quy: M\u1edf r\u1ed9ng ch\u1ee7 \u0111\u1ec1 \u0111\u1ec7 quy<\/h2>\n<p>\u0110\u1ec7 quy c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c xem nh\u01b0 m\u1ed9t qu\u00e1 tr\u00ecnh \u00e1p d\u1ee5ng l\u1eb7p \u0111i l\u1eb7p l\u1ea1i c\u00f9ng m\u1ed9t h\u00e0m ho\u1eb7c m\u1ed9t t\u1eadp h\u1ee3p h\u00e0m \u0111\u1ec3 gi\u1ea3m \u0111\u1ed9 ph\u1ee9c t\u1ea1p c\u1ee7a m\u1ed9t b\u00e0i to\u00e1n. N\u00f3 \u0111\u1eb7c bi\u1ec7t h\u1eefu \u00edch khi m\u1ed9t v\u1ea5n \u0111\u1ec1 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c chia th\u00e0nh c\u00e1c tr\u01b0\u1eddng h\u1ee3p nh\u1ecf h\u01a1n c\u1ee7a c\u00f9ng m\u1ed9t v\u1ea5n \u0111\u1ec1.<\/p>\n<h3>C\u00e1c lo\u1ea1i \u0111\u1ec7 quy<\/h3>\n<ol>\n<li><strong>\u0110\u1ec7 quy tr\u1ef1c ti\u1ebfp<\/strong>: Khi m\u1ed9t h\u00e0m g\u1ecdi tr\u1ef1c ti\u1ebfp ch\u00ednh n\u00f3.<\/li>\n<li><strong>\u0110\u1ec7 quy gi\u00e1n ti\u1ebfp<\/strong>: Khi m\u1ed9t h\u00e0m g\u1ecdi h\u00e0m kh\u00e1c v\u00e0 h\u00e0m \u0111\u00f3 g\u1ecdi h\u00e0m g\u1ed1c.<\/li>\n<\/ol>\n<h3>V\u00ed d\u1ee5 to\u00e1n h\u1ecdc<\/h3>\n<ul>\n<li>H\u00e0m giai th\u1eeba<\/li>\n<li>Chu\u1ed7i Fibonacci<\/li>\n<\/ul>\n<h3>\u1ee8ng d\u1ee5ng l\u1eadp tr\u00ecnh<\/h3>\n<ul>\n<li>Thu\u1eadt to\u00e1n s\u1eafp x\u1ebfp (S\u1eafp x\u1ebfp nhanh, S\u1eafp x\u1ebfp h\u1ee3p nh\u1ea5t)<\/li>\n<li>Duy\u1ec7t c\u00e2y<\/li>\n<\/ul>\n<h2>C\u1ea5u tr\u00fac b\u00ean trong c\u1ee7a \u0111\u1ec7 quy: C\u00e1ch th\u1ee9c ho\u1ea1t \u0111\u1ed9ng c\u1ee7a \u0111\u1ec7 quy<\/h2>\n<p>H\u00e0m \u0111\u1ec7 quy th\u01b0\u1eddng c\u00f3 hai th\u00e0nh ph\u1ea7n ch\u00ednh:<\/p>\n<ol>\n<li><strong>(C\u00e1c) tr\u01b0\u1eddng h\u1ee3p c\u01a1 s\u1edf<\/strong>: \u0110i\u1ec1u ki\u1ec7n \u0111\u1ec3 qu\u00e1 tr\u00ecnh \u0111\u1ec7 quy d\u1eebng l\u1ea1i.<\/li>\n<li><strong>Cu\u1ed9c g\u1ecdi \u0111\u1ec7 quy<\/strong>: Ph\u1ea7n m\u00e0 h\u00e0m g\u1ecdi ch\u00ednh n\u00f3, th\u01b0\u1eddng c\u00f3 c\u00e1c tham s\u1ed1 \u0111\u01b0\u1ee3c s\u1eeda \u0111\u1ed5i.<\/li>\n<\/ol>\n<p>H\u00e0m ti\u1ebfp t\u1ee5c g\u1ecdi ch\u00ednh n\u00f3 cho \u0111\u1ebfn khi \u0111\u1ea1t \u0111\u01b0\u1ee3c tr\u01b0\u1eddng h\u1ee3p c\u01a1 s\u1edf v\u00e0 sau \u0111\u00f3 n\u00f3 b\u1eaft \u0111\u1ea7u quay tr\u1edf l\u1ea1i, l\u00e0m s\u00e1ng t\u1ecf c\u00e1c l\u1ec7nh g\u1ecdi \u0111\u1ec7 quy.<\/p>\n<h2>Ph\u00e2n t\u00edch c\u00e1c t\u00ednh n\u0103ng ch\u00ednh c\u1ee7a \u0111\u1ec7 quy<\/h2>\n<ul>\n<li><strong>S\u1ef1 \u0111\u01a1n gi\u1ea3n<\/strong>: Th\u01b0\u1eddng d\u1eabn \u0111\u1ebfn m\u00e3 s\u1ea1ch h\u01a1n, d\u1ec5 \u0111\u1ecdc h\u01a1n.<\/li>\n<li><strong>Ti\u00eau th\u1ee5 b\u1ed9 nh\u1edb<\/strong>: C\u00f3 th\u1ec3 d\u1eabn \u0111\u1ebfn m\u1ee9c s\u1eed d\u1ee5ng b\u1ed9 nh\u1edb cao n\u1ebfu kh\u00f4ng \u0111\u01b0\u1ee3c x\u1eed l\u00fd \u0111\u00fang c\u00e1ch.<\/li>\n<li><strong>G\u1ee1 l\u1ed7i<\/strong>: Vi\u1ec7c g\u1ee1 l\u1ed7i c\u00f3 th\u1ec3 kh\u00f3 kh\u0103n h\u01a1n.<\/li>\n<li><strong>Hi\u1ec7u su\u1ea5t<\/strong>: C\u00f3 th\u1ec3 k\u00e9m hi\u1ec7u qu\u1ea3 h\u01a1n so v\u1edbi c\u00e1c gi\u1ea3i ph\u00e1p l\u1eb7p l\u1ea1i \u0111\u1ed1i v\u1edbi m\u1ed9t s\u1ed1 v\u1ea5n \u0111\u1ec1.<\/li>\n<\/ul>\n<h2>C\u00e1c lo\u1ea1i \u0111\u1ec7 quy: S\u1eed d\u1ee5ng b\u1ea3ng v\u00e0 danh s\u00e1ch \u0111\u1ec3 vi\u1ebft<\/h2>\n<table>\n<thead>\n<tr>\n<th>Ki\u1ec3u<\/th>\n<th>S\u1ef1 mi\u00eau t\u1ea3<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Tr\u1ef1c ti\u1ebfp<\/td>\n<td>H\u00e0m n\u00e0y g\u1ecdi tr\u1ef1c ti\u1ebfp ch\u00ednh n\u00f3.<\/td>\n<\/tr>\n<tr>\n<td>gi\u00e1n ti\u1ebfp<\/td>\n<td>H\u00e0m n\u00e0y g\u1ecdi h\u00e0m kh\u00e1c, h\u00e0m n\u00e0y g\u1ecdi h\u00e0m g\u1ed1c.<\/td>\n<\/tr>\n<tr>\n<td>\u0110u\u00f4i<\/td>\n<td>Tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t trong \u0111\u00f3 l\u1ec7nh g\u1ecdi \u0111\u1ec7 quy l\u00e0 thao t\u00e1c cu\u1ed1i c\u00f9ng trong h\u00e0m.<\/td>\n<\/tr>\n<tr>\n<td>Qua l\u1ea1i<\/td>\n<td>Hai ho\u1eb7c nhi\u1ec1u h\u00e0m g\u1ecdi nhau m\u1ed9t c\u00e1ch \u0111\u1ec7 quy.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>C\u00e1ch s\u1eed d\u1ee5ng \u0111\u1ec7 quy, v\u1ea5n \u0111\u1ec1 v\u00e0 gi\u1ea3i ph\u00e1p li\u00ean quan \u0111\u1ebfn vi\u1ec7c s\u1eed d\u1ee5ng<\/h2>\n<ul>\n<li><strong>S\u1eed d\u1ee5ng trong thu\u1eadt to\u00e1n<\/strong>: Ph\u1ed5 bi\u1ebfn trong c\u00e1c thu\u1eadt to\u00e1n chia \u0111\u1ec3 tr\u1ecb.<\/li>\n<li><strong>v\u1ea5n \u0111\u1ec1 ti\u1ec1m \u1ea9n<\/strong>: Tr\u00e0n ng\u0103n x\u1ebfp, d\u01b0 th\u1eeba, k\u00e9m hi\u1ec7u qu\u1ea3.<\/li>\n<li><strong>C\u00e1c gi\u1ea3i ph\u00e1p<\/strong>: S\u1eed d\u1ee5ng \u0111\u1ec7 quy \u0111u\u00f4i, ghi nh\u1edb ho\u1eb7c c\u00e1c l\u1ef1a ch\u1ecdn thay th\u1ebf l\u1eb7p l\u1ea1i.<\/li>\n<\/ul>\n<h2>C\u00e1c \u0111\u1eb7c \u0111i\u1ec3m ch\u00ednh v\u00e0 nh\u1eefng so s\u00e1nh kh\u00e1c v\u1edbi c\u00e1c thu\u1eadt ng\u1eef t\u01b0\u01a1ng t\u1ef1<\/h2>\n<table>\n<thead>\n<tr>\n<th>Thu\u1eadt ng\u1eef<\/th>\n<th>\u0111\u1ec7 quy<\/th>\n<th>L\u1eb7p l\u1ea1i<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>S\u1ef1 \u0111\u1ecbnh ngh\u0129a<\/td>\n<td>H\u00e0m g\u1ecdi ch\u00ednh n\u00f3 \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1.<\/td>\n<td>Th\u1ef1c thi l\u1eb7p \u0111i l\u1eb7p l\u1ea1i m\u00e3 b\u1eb1ng c\u00e1ch s\u1eed d\u1ee5ng c\u00e1c v\u00f2ng l\u1eb7p.<\/td>\n<\/tr>\n<tr>\n<td>Hi\u1ec7u qu\u1ea3<\/td>\n<td>C\u00f3 th\u1ec3 k\u00e9m hi\u1ec7u qu\u1ea3 h\u01a1n trong m\u1ed9t s\u1ed1 tr\u01b0\u1eddng h\u1ee3p.<\/td>\n<td>Th\u01b0\u1eddng hi\u1ec7u qu\u1ea3 h\u01a1n.<\/td>\n<\/tr>\n<tr>\n<td>\u0110\u1ed9 ph\u1ee9c t\u1ea1p<\/td>\n<td>C\u00f3 th\u1ec3 d\u1eabn \u0111\u1ebfn m\u00e3 s\u1ea1ch h\u01a1n.<\/td>\n<td>C\u00f3 th\u1ec3 ph\u1ee9c t\u1ea1p h\u01a1n trong m\u1ed9t s\u1ed1 tr\u01b0\u1eddng h\u1ee3p.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Quan \u0111i\u1ec3m v\u00e0 c\u00f4ng ngh\u1ec7 c\u1ee7a t\u01b0\u01a1ng lai li\u00ean quan \u0111\u1ebfn \u0111\u1ec7 quy<\/h2>\n<p>\u0110\u1ec7 quy ti\u1ebfp t\u1ee5c l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m quan tr\u1ecdng trong khoa h\u1ecdc m\u00e1y t\u00ednh, v\u1edbi nghi\u00ean c\u1ee9u \u0111ang di\u1ec5n ra trong vi\u1ec7c t\u1ed1i \u01b0u h\u00f3a c\u00e1c thu\u1eadt to\u00e1n \u0111\u1ec7 quy. C\u00e1c c\u00f4ng ngh\u1ec7 trong t\u01b0\u01a1ng lai c\u00f3 th\u1ec3 th\u00fac \u0111\u1ea9y \u0111\u1ec7 quy theo nh\u1eefng c\u00e1ch ph\u1ee9c t\u1ea1p h\u01a1n, bao g\u1ed3m c\u1ea3 \u0111i\u1ec7n to\u00e1n l\u01b0\u1ee3ng t\u1eed v\u00e0 tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o.<\/p>\n<h2>C\u00e1ch s\u1eed d\u1ee5ng ho\u1eb7c li\u00ean k\u1ebft m\u00e1y ch\u1ee7 proxy v\u1edbi \u0111\u1ec7 quy<\/h2>\n<p>M\u00e1y ch\u1ee7 proxy c\u00f3 th\u1ec3 s\u1eed d\u1ee5ng thu\u1eadt to\u00e1n \u0111\u1ec7 quy \u0111\u1ec3 x\u1eed l\u00fd c\u00e1c t\u00e1c v\u1ee5 nh\u01b0 \u0111\u1ecbnh tuy\u1ebfn, c\u00e2n b\u1eb1ng t\u1ea3i v\u00e0 l\u1ecdc d\u1eef li\u1ec7u. B\u1eb1ng c\u00e1ch t\u1eadn d\u1ee5ng \u0111\u1ec7 quy, c\u00e1c t\u00e1c v\u1ee5 n\u00e0y c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c t\u1ed1i \u01b0u h\u00f3a \u0111\u1ec3 cung c\u1ea5p c\u00e1c d\u1ecbch v\u1ee5 hi\u1ec7u qu\u1ea3 v\u00e0 linh ho\u1ea1t. \u0110\u1ed1i v\u1edbi m\u1ed9t nh\u00e0 cung c\u1ea5p nh\u01b0 OneProxy, vi\u1ec7c hi\u1ec3u \u0111\u1ec7 quy c\u00f3 th\u1ec3 gi\u00fap qu\u1ea3n l\u00fd v\u00e0 c\u1ea5u h\u00ecnh m\u00e1y ch\u1ee7 proxy t\u1ed1t h\u01a1n.<\/p>\n<h2>Li\u00ean k\u1ebft li\u00ean quan<\/h2>\n<ul>\n<li><a href=\"https:\/\/web.stanford.edu\/class\/cs97si\/02-recursion.pdf\" target=\"_new\" rel=\"noopener nofollow\">Stanford: Gi\u1edbi thi\u1ec7u v\u1ec1 \u0111\u1ec7 quy<\/a><\/li>\n<li><a href=\"https:\/\/ocw.mit.edu\/courses\/electrical-engineering-and-computer-science\/6-042j-mathematics-for-computer-science-fall-2005\/readings\/r01.pdf\" target=\"_new\" rel=\"noopener nofollow\">MIT OpenCourseWare: \u0110\u1ec7 quy v\u00e0 l\u1eb7p l\u1ea1i<\/a><\/li>\n<li><a href=\"https:\/\/oneproxy.pro\/vn\/recursion-in-proxy-services\/\" target=\"_new\" rel=\"noopener\">OneProxy: C\u00e1ch ch\u00fang t\u00f4i s\u1eed d\u1ee5ng \u0111\u1ec7 quy trong c\u00e1c d\u1ecbch v\u1ee5 proxy c\u1ee7a m\u00ecnh<\/a><\/li>\n<\/ul>","protected":false},"featured_media":469333,"menu_order":0,"template":"","meta":{"_acf_changed":false,"content-type":"","inline_featured_image":false,"footnotes":""},"class_list":["post-478658","wiki","type-wiki","status-publish","has-post-thumbnail","hentry"],"acf":{"faq_title":"Frequently Asked Questions about <mark>Recursion<\/mark>","faq_items":[{"question":"What is Recursion?","answer":"<p>Recursion is a technique in mathematics and computer science where a function calls itself directly or indirectly to solve a problem. It can simplify complex problems by breaking them down into smaller, more manageable instances of the same problem.<\/p>"},{"question":"What are the Different Types of Recursion?","answer":"<p>There are several types of recursion, including Direct, Indirect, Tail, and Mutual recursion. Direct recursion occurs when a function calls itself directly, while Indirect recursion involves a function calling another that in turn calls the original. Tail recursion is a special case where the recursive call is the last operation, and Mutual recursion involves two or more functions calling each other recursively.<\/p>"},{"question":"How Does Recursion Work?","answer":"<p>A recursive function generally consists of two parts: the base case(s) and the recursive call. The function continues to call itself with modified parameters until the base case is reached, at which point it begins to return and unravel the recursive calls.<\/p>"},{"question":"What are the Key Features of Recursion?","answer":"<p>Recursion offers simplicity and often leads to cleaner code. However, it can consume more memory, be challenging to debug, and may be less efficient than iterative solutions for some problems.<\/p>"},{"question":"What are the Problems Associated with Recursion, and How Can They be Solved?","answer":"<p>Problems with recursion include the potential for stack overflow, redundancy, and inefficiency. Solutions include using tail recursion, memoization, or switching to iterative alternatives.<\/p>"},{"question":"How are Recursion and Iteration Different?","answer":"<p>While recursion involves a function calling itself to solve a problem, iteration involves the repeated execution of code using loops. Recursion can lead to cleaner but possibly less efficient code, while iteration may be more efficient but potentially more complex.<\/p>"},{"question":"How are Proxy Servers Associated with Recursion?","answer":"<p>Proxy servers like those provided by OneProxy can leverage recursive algorithms for tasks like routing, load balancing, and data filtering. Understanding recursion can lead to better proxy server configuration and management.<\/p>"},{"question":"What are the Future Perspectives of Recursion?","answer":"<p>Recursion continues to be a vital concept with ongoing research in optimizing recursive algorithms. Future technologies may leverage recursion in more complex ways, including applications in quantum computing and artificial intelligence.<\/p>"}]},"_links":{"self":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/478658","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki"}],"about":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/types\/wiki"}],"version-history":[{"count":0,"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/478658\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media\/469333"}],"wp:attachment":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media?parent=478658"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}