{"id":475961,"date":"2023-08-09T07:24:43","date_gmt":"2023-08-09T07:24:43","guid":{"rendered":""},"modified":"2023-09-05T11:11:42","modified_gmt":"2023-09-05T11:11:42","slug":"backtracking","status":"publish","type":"wiki","link":"https:\/\/oneproxy.pro\/vn\/wiki\/backtracking\/","title":{"rendered":"Quay l\u1ea1i"},"content":{"rendered":"<p>Quay lui l\u00e0 m\u1ed9t k\u1ef9 thu\u1eadt thu\u1eadt to\u00e1n m\u1ea1nh m\u1ebd \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 gi\u1ea3i quy\u1ebft c\u00e1c v\u1ea5n \u0111\u1ec1 t\u1ed5 h\u1ee3p m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3. \u0110\u00f3 l\u00e0 m\u1ed9t c\u00e1ch c\u00f3 h\u1ec7 th\u1ed1ng \u0111\u1ec3 t\u00ecm gi\u1ea3i ph\u00e1p b\u1eb1ng c\u00e1ch kh\u00e1m ph\u00e1 t\u1ea5t c\u1ea3 c\u00e1c con \u0111\u01b0\u1eddng c\u00f3 th\u1ec3 v\u00e0 quay l\u1ea1i b\u1ea5t c\u1ee9 khi n\u00e0o g\u1eb7p ph\u1ea3i ng\u00f5 c\u1ee5t. K\u1ef9 thu\u1eadt n\u00e0y \u0111\u1eb7c bi\u1ec7t h\u1eefu \u00edch cho c\u00e1c b\u00e0i to\u00e1n c\u00f3 kh\u00f4ng gian t\u00ecm ki\u1ebfm l\u1edbn v\u1edbi nhi\u1ec1u gi\u1ea3i ph\u00e1p ti\u1ec1m n\u0103ng.<\/p>\n<h2>L\u1ecbch s\u1eed v\u1ec1 ngu\u1ed3n g\u1ed1c c\u1ee7a Backtracking v\u00e0 l\u1ea7n \u0111\u1ea7u ti\u00ean \u0111\u1ec1 c\u1eadp \u0111\u1ebfn n\u00f3<\/h2>\n<p>Kh\u00e1i ni\u1ec7m quay lui c\u00f3 t\u1eeb \u0111\u1ea7u nh\u1eefng n\u0103m 1970 khi c\u00e1c nh\u00e0 khoa h\u1ecdc m\u00e1y t\u00ednh v\u00e0 to\u00e1n h\u1ecdc \u0111ang kh\u00e1m ph\u00e1 nhi\u1ec1u c\u00e1ch ti\u1ebfp c\u1eadn kh\u00e1c nhau \u0111\u1ec3 gi\u1ea3i quy\u1ebft c\u00e1c v\u1ea5n \u0111\u1ec1 ph\u1ee9c t\u1ea1p. L\u1ea7n \u0111\u1ea7u ti\u00ean \u0111\u1ec1 c\u1eadp \u0111\u1ebfn vi\u1ec7c quay lui c\u00f3 th\u1ec3 b\u1eaft ngu\u1ed3n t\u1eeb t\u00e1c ph\u1ea9m c\u00f3 \u1ea3nh h\u01b0\u1edfng l\u1edbn c\u1ee7a Donald Knuth \u201cNgh\u1ec7 thu\u1eadt l\u1eadp tr\u00ecnh m\u00e1y t\u00ednh\u201d xu\u1ea5t b\u1ea3n n\u0103m 1968. Trong T\u1eadp 1 c\u1ee7a b\u1ed9 s\u00e1ch c\u1ee7a m\u00ecnh, Knuth \u0111\u00e3 gi\u1edbi thi\u1ec7u \u00fd t\u01b0\u1edfng v\u1ec1 \u201cThu\u1eadt to\u00e1n X\u201d, \u0111\u01b0\u1ee3c d\u00f9ng l\u00e0m n\u1ec1n t\u1ea3ng cho nhi\u1ec1u thu\u1eadt to\u00e1n. thu\u1eadt to\u00e1n quay lui.<\/p>\n<h2>Th\u00f4ng tin chi ti\u1ebft v\u1ec1 Quay l\u1ea1i. M\u1edf r\u1ed9ng ch\u1ee7 \u0111\u1ec1 Quay l\u1ea1i.<\/h2>\n<p>Quay l\u1ea1i d\u1ef1a tr\u00ean \u00fd t\u01b0\u1edfng x\u00e2y d\u1ef1ng d\u1ea7n d\u1ea7n m\u1ed9t gi\u1ea3i ph\u00e1p v\u00e0 t\u1eeb b\u1ecf n\u00f3 khi n\u00f3 kh\u00f4ng \u0111\u00e1p \u1ee9ng \u0111\u01b0\u1ee3c m\u1ed9t s\u1ed1 \u0111i\u1ec1u ki\u1ec7n nh\u1ea5t \u0111\u1ecbnh. Thu\u1eadt to\u00e1n kh\u00e1m ph\u00e1 kh\u00f4ng gian gi\u1ea3i ph\u00e1p th\u00f4ng qua chi\u1ebfn l\u01b0\u1ee3c t\u00ecm ki\u1ebfm theo chi\u1ec1u s\u00e2u v\u00e0 c\u1eaft t\u1ec9a c\u00e1c nh\u00e1nh \u0111\u01b0\u1ee3c \u0111\u1ea3m b\u1ea3o d\u1eabn \u0111\u1ebfn gi\u1ea3i ph\u00e1p kh\u00f4ng ch\u00ednh x\u00e1c, do \u0111\u00f3 gi\u1ea3m \u0111\u00e1ng k\u1ec3 g\u00e1nh n\u1eb7ng t\u00ednh to\u00e1n.<\/p>\n<p>\u0110\u1ec3 th\u1ef1c hi\u1ec7n quay lui, thu\u1eadt to\u00e1n tu\u00e2n theo c\u00e1c b\u01b0\u1edbc chung sau:<\/p>\n<ol>\n<li>\n<p><strong>Ch\u1ecdn<\/strong>: \u0110\u01b0a ra quy\u1ebft \u0111\u1ecbnh v\u00e0 ch\u1ecdn m\u1ed9t ph\u01b0\u01a1ng \u00e1n trong s\u1ed1 c\u00e1c l\u1ef1a ch\u1ecdn c\u00f3 s\u1eb5n.<\/p>\n<\/li>\n<li>\n<p><strong>Kh\u00e1m ph\u00e1<\/strong>: Ti\u1ebfn v\u1ec1 ph\u00eda tr\u01b0\u1edbc v\u00e0 kh\u00e1m ph\u00e1 h\u1eadu qu\u1ea3 c\u1ee7a ph\u01b0\u01a1ng \u00e1n \u0111\u00e3 ch\u1ecdn.<\/p>\n<\/li>\n<li>\n<p><strong>Ki\u1ec3m tra<\/strong>: Ki\u1ec3m tra xem ph\u01b0\u01a1ng \u00e1n \u0111\u00e3 ch\u1ecdn c\u00f3 d\u1eabn \u0111\u1ebfn gi\u1ea3i ph\u00e1p h\u1ee3p l\u1ec7 hay kh\u00f4ng.<\/p>\n<\/li>\n<li>\n<p><strong>Quay l\u1ea1i<\/strong>: N\u1ebfu t\u00f9y ch\u1ecdn \u0111\u00e3 ch\u1ecdn kh\u00f4ng d\u1eabn \u0111\u1ebfn gi\u1ea3i ph\u00e1p h\u1ee3p l\u1ec7, h\u00e3y quay l\u1ea1i tr\u1ea1ng th\u00e1i tr\u01b0\u1edbc \u0111\u00f3 v\u00e0 kh\u00e1m ph\u00e1 c\u00e1c t\u00f9y ch\u1ecdn kh\u00e1c.<\/p>\n<\/li>\n<\/ol>\n<p>Qu\u00e1 tr\u00ecnh ti\u1ebfp t\u1ee5c cho \u0111\u1ebfn khi t\u1ea5t c\u1ea3 c\u00e1c k\u1ebft h\u1ee3p c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c kh\u00e1m ph\u00e1 ho\u1eb7c t\u00ecm th\u1ea5y gi\u1ea3i ph\u00e1p h\u1ee3p l\u1ec7.<\/p>\n<h2>C\u1ea5u tr\u00fac b\u00ean trong c\u1ee7a Backtracking. C\u00e1ch th\u1ee9c ho\u1ea1t \u0111\u1ed9ng c\u1ee7a Backtracking.<\/h2>\n<p>V\u1ec1 c\u1ed1t l\u00f5i, quay lui l\u00e0 m\u1ed9t thu\u1eadt to\u00e1n \u0111\u1ec7 quy s\u1eed d\u1ee5ng ng\u0103n x\u1ebfp cu\u1ed9c g\u1ecdi \u0111\u1ec3 qu\u1ea3n l\u00fd qu\u00e1 tr\u00ecnh th\u0103m d\u00f2 v\u00e0 quay lui. Khi thu\u1eadt to\u00e1n ch\u1ecdn m\u1ed9t t\u00f9y ch\u1ecdn, thu\u1eadt to\u00e1n s\u1ebd th\u1ef1c hi\u1ec7n l\u1ec7nh g\u1ecdi \u0111\u1ec7 quy \u0111\u1ec3 kh\u00e1m ph\u00e1 s\u00e2u h\u01a1n, t\u00ecm hi\u1ec3u s\u00e2u h\u01a1n v\u1ec1 kh\u00f4ng gian gi\u1ea3i ph\u00e1p. Tuy nhi\u00ean, n\u1ebfu n\u00f3 g\u1eb7p ng\u00f5 c\u1ee5t (t\u1ee9c l\u00e0 tr\u1ea1ng th\u00e1i kh\u00f4ng h\u1ee3p l\u1ec7 ho\u1eb7c \u0111i\u1ec1u ki\u1ec7n vi ph\u1ea1m c\u00e1c r\u00e0ng bu\u1ed9c c\u1ee7a b\u00e0i to\u00e1n), n\u00f3 s\u1ebd quay l\u1ea1i b\u1eb1ng c\u00e1ch quay l\u1ea1i \u0111i\u1ec3m quy\u1ebft \u0111\u1ecbnh tr\u01b0\u1edbc \u0111\u00f3 v\u00e0 th\u1eed c\u00e1c l\u1ef1a ch\u1ecdn thay th\u1ebf.<\/p>\n<p>S\u1ef1 th\u00e0nh c\u00f4ng c\u1ee7a thu\u1eadt to\u00e1n quay lui ph\u1ee5 thu\u1ed9c r\u1ea5t nhi\u1ec1u v\u00e0o vi\u1ec7c x\u1eed l\u00fd hi\u1ec7u qu\u1ea3 h\u1ec7 s\u1ed1 ph\u00e2n nh\u00e1nh v\u00e0 \u0111\u1ed9 s\u00e2u c\u1ee7a c\u00e2y t\u00ecm ki\u1ebfm. Trong tr\u01b0\u1eddng h\u1ee3p h\u1ec7 s\u1ed1 ph\u00e2n nh\u00e1nh cao ho\u1eb7c \u0111\u1ed9 s\u00e2u c\u1ee7a c\u00e2y t\u00ecm ki\u1ebfm r\u1ed9ng, hi\u1ec7u su\u1ea5t c\u1ee7a thu\u1eadt to\u00e1n c\u00f3 th\u1ec3 gi\u1ea3m.<\/p>\n<h2>Ph\u00e2n t\u00edch c\u00e1c t\u00ednh n\u0103ng ch\u00ednh c\u1ee7a Backtracking<\/h2>\n<p>Quay lui cung c\u1ea5p m\u1ed9t s\u1ed1 t\u00ednh n\u0103ng ch\u00ednh khi\u1ebfn n\u00f3 tr\u1edf th\u00e0nh m\u1ed9t k\u1ef9 thu\u1eadt thu\u1eadt to\u00e1n c\u00f3 gi\u00e1 tr\u1ecb:<\/p>\n<ol>\n<li>\n<p><strong>T\u00ednh \u0111\u1ea7y \u0111\u1ee7<\/strong>: Quay lui \u0111\u1ea3m b\u1ea3o t\u00ecm ra t\u1ea5t c\u1ea3 c\u00e1c gi\u1ea3i ph\u00e1p kh\u1ea3 thi b\u1eb1ng c\u00e1ch kh\u00e1m ph\u00e1 to\u00e0n b\u1ed9 kh\u00f4ng gian gi\u1ea3i ph\u00e1p.<\/p>\n<\/li>\n<li>\n<p><strong>S\u1ef1 t\u1ed1i \u01b0u<\/strong>: Trong m\u1ed9t s\u1ed1 v\u1ea5n \u0111\u1ec1 nh\u1ea5t \u0111\u1ecbnh, vi\u1ec7c quay lui c\u00f3 th\u1ec3 x\u00e1c \u0111\u1ecbnh gi\u1ea3i ph\u00e1p t\u1ed1i \u01b0u b\u1eb1ng c\u00e1ch kh\u00e1m ph\u00e1 kh\u00f4ng gian gi\u1ea3i ph\u00e1p m\u1ed9t c\u00e1ch c\u00f3 h\u1ec7 th\u1ed1ng.<\/p>\n<\/li>\n<li>\n<p><strong>Uy\u1ec3n chuy\u1ec3n<\/strong>: Thu\u1eadt to\u00e1n quay lui c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c \u0111i\u1ec1u ch\u1ec9nh \u0111\u1ec3 ph\u00f9 h\u1ee3p v\u1edbi nhi\u1ec1u mi\u1ec1n v\u1ea5n \u0111\u1ec1 kh\u00e1c nhau, l\u00e0m cho n\u00f3 tr\u1edf th\u00e0nh m\u1ed9t k\u1ef9 thu\u1eadt linh ho\u1ea1t.<\/p>\n<\/li>\n<li>\n<p><strong>Hi\u1ec7u qu\u1ea3 b\u1ed9 nh\u1edb<\/strong>: C\u00e1c thu\u1eadt to\u00e1n quay lui th\u01b0\u1eddng ti\u00eau t\u1ed1n \u00edt b\u1ed9 nh\u1edb h\u01a1n v\u00ec ch\u00fang kh\u00e1m ph\u00e1 c\u00e1c gi\u1ea3i ph\u00e1p t\u0103ng d\u1ea7n m\u00e0 kh\u00f4ng l\u01b0u tr\u1eef to\u00e0n b\u1ed9 c\u00e2y t\u00ecm ki\u1ebfm.<\/p>\n<\/li>\n<li>\n<p><strong>C\u1eaft t\u1ec9a<\/strong>: Kh\u1ea3 n\u0103ng c\u1eaft t\u1ec9a c\u00e1c nh\u00e1nh c\u00f3 kh\u1ea3 n\u0103ng d\u1eabn \u0111\u1ebfn gi\u1ea3i ph\u00e1p kh\u00f4ng ch\u00ednh x\u00e1c cho ph\u00e9p quay lui \u0111\u1ec3 kh\u00e1m ph\u00e1 kh\u00f4ng gian gi\u1ea3i ph\u00e1p l\u1edbn m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3.<\/p>\n<\/li>\n<\/ol>\n<h2>C\u00e1c lo\u1ea1i quay l\u1ea1i<\/h2>\n<p>K\u1ef9 thu\u1eadt quay lui c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ph\u00e2n lo\u1ea1i th\u00e0nh c\u00e1c lo\u1ea1i kh\u00e1c nhau d\u1ef1a tr\u00ean mi\u1ec1n \u1ee9ng d\u1ee5ng c\u1ee5 th\u1ec3 c\u1ee7a ch\u00fang. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 ki\u1ec3u quay lui ph\u1ed5 bi\u1ebfn:<\/p>\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><strong>Quay lui \u0111\u1ec7 quy<\/strong><\/td>\n<td>C\u00e1ch ti\u1ebfp c\u1eadn quay lui ti\u00eau chu\u1ea9n s\u1eed d\u1ee5ng c\u00e1c l\u1ec7nh g\u1ecdi h\u00e0m \u0111\u1ec7 quy.<\/td>\n<\/tr>\n<tr>\n<td><strong>Quay l\u1ea1i l\u1eb7p \u0111i l\u1eb7p l\u1ea1i<\/strong><\/td>\n<td>M\u1ed9t bi\u1ebfn th\u1ec3 s\u1eed d\u1ee5ng ph\u01b0\u01a1ng ph\u00e1p l\u1eb7p l\u1ea1i, th\u01b0\u1eddng c\u00f3 ng\u0103n x\u1ebfp.<\/td>\n<\/tr>\n<tr>\n<td><strong>R\u00e0ng bu\u1ed9c quay l\u1ea1i<\/strong><\/td>\n<td>T\u1eadp trung v\u00e0o c\u00e1c v\u1ea5n \u0111\u1ec1 th\u1ecfa m\u00e3n h\u1ea1n ch\u1ebf nh\u01b0 Sudoku.<\/td>\n<\/tr>\n<tr>\n<td><strong>\u0110\u01b0\u1eddng \u0111i Hamilton<\/strong><\/td>\n<td>T\u00ecm \u0111\u01b0\u1eddng \u0111i qua m\u1ed7i \u0111\u1ec9nh c\u1ee7a \u0111\u1ed3 th\u1ecb \u0111\u00fang m\u1ed9t l\u1ea7n.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>C\u00e1c c\u00e1ch s\u1eed d\u1ee5ng Quay lui, c\u00e1c v\u1ea5n \u0111\u1ec1 v\u00e0 gi\u1ea3i ph\u00e1p li\u00ean quan \u0111\u1ebfn vi\u1ec7c s\u1eed d\u1ee5ng.<\/h2>\n<p>Quay lui t\u00ecm th\u1ea5y \u1ee9ng d\u1ee5ng trong c\u00e1c l\u0129nh v\u1ef1c kh\u00e1c nhau, bao g\u1ed3m:<\/p>\n<ol>\n<li>\n<p><strong>Gi\u1ea3i c\u00e2u \u0111\u1ed1<\/strong>: Thu\u1eadt to\u00e1n quay lui c\u00f3 th\u1ec3 gi\u1ea3i c\u00e1c c\u00e2u \u0111\u1ed1 c\u1ed5 \u0111i\u1ec3n nh\u01b0 b\u00e0i to\u00e1n N-Queens, Sudoku v\u00e0 C\u00e2u \u0111\u1ed1 t\u00e1m qu\u00e2n h\u1eadu.<\/p>\n<\/li>\n<li>\n<p><strong>T\u1ed1i \u01b0u h\u00f3a t\u1ed5 h\u1ee3p<\/strong>: C\u00e1c b\u00e0i to\u00e1n nh\u01b0 B\u00e0i to\u00e1n ng\u01b0\u1eddi b\u00e1n h\u00e0ng du l\u1ecbch (TSP) v\u00e0 B\u00e0i to\u00e1n t\u1ed5ng t\u1eadp h\u1ee3p con c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c gi\u1ea3i quy\u1ebft m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3 b\u1eb1ng c\u00e1ch quay lui.<\/p>\n<\/li>\n<li>\n<p><strong>V\u1ea5n \u0111\u1ec1 v\u1ec1 \u0111\u1ed3 th\u1ecb<\/strong>: Quay lui c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng cho c\u00e1c b\u00e0i to\u00e1n truy\u1ec1n t\u1ea3i \u0111\u1ed3 th\u1ecb nh\u01b0 t\u00ecm \u0111\u01b0\u1eddng \u0111i ho\u1eb7c chu tr\u00ecnh Hamilton.<\/p>\n<\/li>\n<li>\n<p><strong>Chi\u1ebfn l\u01b0\u1ee3c tr\u00f2 ch\u01a1i<\/strong>: C\u00e1c thu\u1eadt to\u00e1n ch\u01a1i tr\u00f2 ch\u01a1i, ch\u1eb3ng h\u1ea1n nh\u01b0 c\u1edd vua v\u00e0 tic-tac-toe, th\u01b0\u1eddng s\u1eed d\u1ee5ng t\u00ednh n\u0103ng quay lui \u0111\u1ec3 t\u00ecm ki\u1ebfm n\u01b0\u1edbc \u0111i t\u1ed1t nh\u1ea5t.<\/p>\n<\/li>\n<\/ol>\n<p>M\u1eb7c d\u00f9 t\u00ednh linh ho\u1ea1t c\u1ee7a n\u00f3, vi\u1ec7c quay lui c\u00f3 m\u1ed9t s\u1ed1 th\u00e1ch th\u1ee9c:<\/p>\n<ul>\n<li>\n<p><strong>\u0110\u1ed9 ph\u1ee9c t\u1ea1p th\u1eddi gian h\u00e0m m\u0169<\/strong>: Trong tr\u01b0\u1eddng h\u1ee3p x\u1ea5u nh\u1ea5t, vi\u1ec7c quay lui c\u00f3 th\u1ec3 c\u00f3 \u0111\u1ed9 ph\u1ee9c t\u1ea1p v\u1ec1 th\u1eddi gian theo c\u1ea5p s\u1ed1 nh\u00e2n, khi\u1ebfn n\u00f3 kh\u00f4ng hi\u1ec7u qu\u1ea3 \u0111\u1ed1i v\u1edbi m\u1ed9t s\u1ed1 v\u1ea5n \u0111\u1ec1.<\/p>\n<\/li>\n<li>\n<p><strong>Kh\u00f3 kh\u0103n trong vi\u1ec7c c\u1eaft t\u1ec9a<\/strong>: Vi\u1ec7c x\u00e1c \u0111\u1ecbnh c\u00e1c chi\u1ebfn l\u01b0\u1ee3c c\u1eaft t\u1ec9a hi\u1ec7u qu\u1ea3 c\u00f3 th\u1ec3 l\u00e0 m\u1ed9t th\u00e1ch th\u1ee9c, \u1ea3nh h\u01b0\u1edfng \u0111\u1ebfn hi\u1ec7u su\u1ea5t c\u1ee7a thu\u1eadt to\u00e1n.<\/p>\n<\/li>\n<\/ul>\n<p>\u0110\u1ec3 gi\u1ea3i quy\u1ebft nh\u1eefng th\u00e1ch th\u1ee9c n\u00e0y, c\u00e1c nh\u00e0 nghi\u00ean c\u1ee9u \u0111\u00e3 kh\u00e1m ph\u00e1 c\u00e1c k\u1ef9 thu\u1eadt t\u1ed1i \u01b0u h\u00f3a v\u00e0 ch\u1ea9n \u0111o\u00e1n \u0111\u1ec3 c\u1ea3i thi\u1ec7n hi\u1ec7u qu\u1ea3 c\u1ee7a thu\u1eadt to\u00e1n quay lui.<\/p>\n<h2>C\u00e1c \u0111\u1eb7c \u0111i\u1ec3m ch\u00ednh v\u00e0 so s\u00e1nh kh\u00e1c v\u1edbi c\u00e1c thu\u1eadt ng\u1eef t\u01b0\u01a1ng t\u1ef1<\/h2>\n<p>D\u01b0\u1edbi \u0111\u00e2y l\u00e0 so s\u00e1nh vi\u1ec7c quay lui v\u1edbi c\u00e1c k\u1ef9 thu\u1eadt thu\u1eadt to\u00e1n kh\u00e1c:<\/p>\n<table>\n<thead>\n<tr>\n<th>K\u1ef9 thu\u1eadt<\/th>\n<th>\u0110\u1eb7c tr\u01b0ng<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Quay l\u1ea1i<\/strong><\/td>\n<td>T\u00ecm ki\u1ebfm to\u00e0n di\u1ec7n, t\u00ecm ra m\u1ecdi gi\u1ea3i ph\u00e1p, \u0111\u1ec7 quy.<\/td>\n<\/tr>\n<tr>\n<td><strong>L\u1ef1c l\u01b0\u1ee3ng v\u0169 phu<\/strong><\/td>\n<td>T\u00ecm ki\u1ebfm to\u00e0n di\u1ec7n, c\u00f3 th\u1ec3 kh\u00f4ng \u0111\u1ec7 quy.<\/td>\n<\/tr>\n<tr>\n<td><strong>L\u1eadp tr\u00ecnh n\u0103ng \u0111\u1ed9ng<\/strong><\/td>\n<td>Ghi nh\u1edb c\u00e1c gi\u1ea3i ph\u00e1p, c\u1ea5u tr\u00fac t\u1ed1i \u01b0u.<\/td>\n<\/tr>\n<tr>\n<td><strong>Ph\u00e2n chia v\u00e0 chinh ph\u1ee5c<\/strong><\/td>\n<td>\u0110\u1ec7 quy, chia b\u00e0i to\u00e1n th\u00e0nh c\u00e1c b\u00e0i to\u00e1n con nh\u1ecf h\u01a1n.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Trong khi vi\u1ec7c quay lui v\u00e0 d\u00f9ng v\u0169 l\u1ef1c \u0111\u1ec1u li\u00ean quan \u0111\u1ebfn vi\u1ec7c t\u00ecm ki\u1ebfm to\u00e0n di\u1ec7n, th\u00ec vi\u1ec7c quay l\u1ea1i bao g\u1ed3m kh\u1ea3 n\u0103ng quay l\u1ea1i v\u00e0 t\u1eeb b\u1ecf nh\u1eefng con \u0111\u01b0\u1eddng kh\u00f4ng h\u1ee9a h\u1eb9n, l\u00e0m cho n\u00f3 hi\u1ec7u qu\u1ea3 h\u01a1n so v\u1edbi d\u00f9ng v\u0169 l\u1ef1c thu\u1ea7n t\u00fay.<\/p>\n<h2>C\u00e1c quan \u0111i\u1ec3m v\u00e0 c\u00f4ng ngh\u1ec7 c\u1ee7a t\u01b0\u01a1ng lai li\u00ean quan \u0111\u1ebfn Quay lui<\/h2>\n<p>C\u00e1c thu\u1eadt to\u00e1n quay lui s\u1ebd ti\u1ebfp t\u1ee5c \u0111\u00f3ng m\u1ed9t vai tr\u00f2 quan tr\u1ecdng trong vi\u1ec7c gi\u1ea3i c\u00e1c b\u00e0i to\u00e1n t\u1ed5 h\u1ee3p ph\u1ee9c t\u1ea1p. V\u1edbi nh\u1eefng ti\u1ebfn b\u1ed9 v\u1ec1 s\u1ee9c m\u1ea1nh t\u00ednh to\u00e1n v\u00e0 k\u1ef9 thu\u1eadt t\u1ed1i \u01b0u h\u00f3a, c\u00e1c nh\u00e0 nghi\u00ean c\u1ee9u c\u00f3 th\u1ec3 s\u1ebd \u0111\u01b0a ra c\u00e1c chi\u1ebfn l\u01b0\u1ee3c quay lui hi\u1ec7u qu\u1ea3 h\u01a1n. Ngo\u00e0i ra, vi\u1ec7c t\u00edch h\u1ee3p tr\u00ed tu\u1ec7 nh\u00e2n t\u1ea1o v\u00e0 h\u1ecdc m\u00e1y v\u00e0o c\u00e1c thu\u1eadt to\u00e1n quay lui c\u00f3 th\u1ec3 d\u1eabn \u0111\u1ebfn c\u00e1c gi\u1ea3i ph\u00e1p th\u00f4ng minh v\u00e0 t\u1ed1i \u01b0u h\u01a1n n\u1eefa.<\/p>\n<h2>C\u00e1ch s\u1eed d\u1ee5ng ho\u1eb7c li\u00ean k\u1ebft m\u00e1y ch\u1ee7 proxy v\u1edbi Quay lui<\/h2>\n<p>M\u00e1y ch\u1ee7 proxy v\u00e0 quay lui c\u00f3 th\u1ec3 t\u00ecm th\u1ea5y s\u1ef1 li\u00ean quan trong c\u00e1c t\u00ecnh hu\u1ed1ng c\u1ea7n ti\u1ebfn h\u00e0nh nhi\u1ec1u t\u00ednh to\u00e1n song song ho\u1eb7c khi mi\u1ec1n v\u1ea5n \u0111\u1ec1 y\u00eau c\u1ea7u \u1ea9n danh ho\u1eb7c ph\u00e2n b\u1ed5 \u0111\u1ecba l\u00fd. C\u00e1c m\u00e1y ch\u1ee7 proxy c\u00f3 th\u1ec3 t\u1ea1o \u0111i\u1ec1u ki\u1ec7n thu\u1eadn l\u1ee3i cho vi\u1ec7c ph\u00e2n ph\u1ed1i c\u00e1c t\u00e1c v\u1ee5 quay lui tr\u00ean c\u00e1c n\u00fat kh\u00e1c nhau, gi\u1ea3m t\u1ea3i t\u00ednh to\u00e1n tr\u00ean c\u00e1c h\u1ec7 th\u1ed1ng ri\u00eang l\u1ebb v\u00e0 \u0111\u1ea3m b\u1ea3o kh\u00e1m ph\u00e1 kh\u00f4ng gian gi\u1ea3i ph\u00e1p hi\u1ec7u qu\u1ea3 h\u01a1n.<\/p>\n<h2>Li\u00ean k\u1ebft li\u00ean quan<\/h2>\n<p>\u0110\u1ec3 bi\u1ebft th\u00eam th\u00f4ng tin v\u1ec1 Quay l\u1ea1i, b\u1ea1n c\u00f3 th\u1ec3 tham kh\u1ea3o c\u00e1c t\u00e0i nguy\u00ean sau:<\/p>\n<ul>\n<li><a href=\"https:\/\/www-cs-faculty.stanford.edu\/~uno\/taocp.html\" target=\"_new\" rel=\"noopener nofollow\">\u201cNgh\u1ec7 thu\u1eadt l\u1eadp tr\u00ecnh m\u00e1y t\u00ednh\u201d c\u1ee7a Donald Knuth<\/a><\/li>\n<li><a href=\"https:\/\/www.geeksforgeeks.org\/backtracking-algorithms\/\" target=\"_new\" rel=\"noopener nofollow\">Gi\u1ea3i th\u00edch v\u1ec1 thu\u1eadt to\u00e1n quay lui<\/a><\/li>\n<li><a href=\"https:\/\/en.wikipedia.org\/wiki\/Backtracking\" target=\"_new\" rel=\"noopener nofollow\">Quay l\u1ea1i tr\u00ean Wikipedia<\/a><\/li>\n<\/ul>","protected":false},"featured_media":0,"menu_order":0,"template":"","meta":{"_acf_changed":false,"content-type":"","inline_featured_image":false,"footnotes":""},"class_list":["post-475961","wiki","type-wiki","status-publish","hentry"],"acf":{"faq_title":"Frequently Asked Questions about <mark>Backtracking: A Comprehensive Guide<\/mark>","faq_items":[{"question":"What is Backtracking?","answer":"<p>Backtracking is a powerful algorithmic technique used to efficiently solve combinatorial problems. It involves exploring all possible paths and backtracking whenever a dead end is encountered.<\/p>"},{"question":"Who introduced Backtracking and when was it first mentioned?","answer":"<p>Backtracking was introduced by Donald Knuth and was first mentioned in his book \"The Art of Computer Programming,\" published in 1968.<\/p>"},{"question":"How does Backtracking work?","answer":"<p>Backtracking is based on a recursive approach where decisions are made, consequences are explored, and validity is checked. If the chosen option leads to an invalid solution, the algorithm backtracks and explores alternative choices.<\/p>"},{"question":"What are the key features of Backtracking?","answer":"<p>The key features of Backtracking include its completeness, optimality, flexibility, memory efficiency, and the ability to prune branches leading to incorrect solutions.<\/p>"},{"question":"What types of Backtracking exist?","answer":"<p>Backtracking techniques can be classified into various types, including Recursive Backtracking, Iterative Backtracking, Constraint Backtracking, and Hamiltonian Path.<\/p>"},{"question":"In which domains is Backtracking commonly used?","answer":"<p>Backtracking finds application in puzzle solving, combinatorial optimization, graph problems, and game strategies.<\/p>"},{"question":"What challenges does Backtracking face?","answer":"<p>Backtracking may have exponential time complexity in some cases, and identifying effective pruning strategies can be challenging.<\/p>"},{"question":"How does Backtracking compare with other algorithms?","answer":"<p>Backtracking involves exhaustive search with backtracking capabilities, making it more efficient than pure brute force. It also differs from Dynamic Programming and Divide and Conquer.<\/p>"},{"question":"What can we expect for the future of Backtracking?","answer":"<p>With advancements in computing power and optimization techniques, researchers may devise more efficient backtracking strategies. Integrating AI and machine learning may lead to even more intelligent solutions.<\/p>"},{"question":"How is Backtracking associated with proxy servers?","answer":"<p>Proxy servers can be used to distribute backtracking tasks across different nodes, optimizing the exploration of the solution space.<\/p>"}]},"_links":{"self":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/475961","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\/475961\/revisions"}],"wp:attachment":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media?parent=475961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}