{"id":404,"date":"2026-03-29T19:48:20","date_gmt":"2026-03-29T11:48:20","guid":{"rendered":"https:\/\/jiangqvweihuan.cn\/?p=404"},"modified":"2026-03-30T18:37:17","modified_gmt":"2026-03-30T10:37:17","slug":"dp","status":"publish","type":"post","link":"https:\/\/jiangqvweihuan.cn\/index.php\/2026\/03\/29\/dp\/","title":{"rendered":"\u7b97\u6cd5\u5fc3\u4f20\u00b7DP"},"content":{"rendered":"\n<p>\u52a8\u6001\u89c4\u5212\uff0c\u6c5f\u6e56\u4eba\u79f0\u201cDP\u201d\uff0c\u662f\u7b97\u6cd5\u9762\u8bd5\u4e2d\u7684\u201c\u62e6\u8def\u864e\u201d\uff0c\u4e5f\u662f\u7ade\u8d5b\u9009\u624b\u7684\u201c\u770b\u5bb6\u672c\u9886\u201d\u3002\u5f88\u591a\u4eba\u89c9\u5f97DP\u96be\uff0c\u662f\u56e0\u4e3a\u5b83\u4e0d\u50cf\u6392\u5e8f\u3001\u641c\u7d22\u90a3\u6837\u6709\u56fa\u5b9a\u7684\u6a21\u677f\uff0c\u66f4\u50cf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u601d\u60f3\u201d\u2014\u2014\u4f60\u9700\u8981\u5bf9\u95ee\u9898\u8fdb\u884c\u62bd\u8c61\u3001\u5b9a\u4e49\u72b6\u6001\u3001\u5bfb\u627e\u8f6c\u79fb\u3002<\/p>\n\n\n\n<p>\u672c\u6587\u5c06\u5e26\u4f60\u4ece\u96f6\u5f00\u59cb\uff0c\u7cfb\u7edf\u68b3\u7406\u52a8\u6001\u89c4\u5212\u7684\u6838\u5fc3\u601d\u60f3\uff0c\u901a\u8fc7\u5927\u91cf\u7ecf\u5178\u6848\u4f8b\uff0c\u5e2e\u4f60\u5efa\u7acbDP\u7684\u89e3\u9898\u601d\u7ef4\u6846\u67b6\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e00\u3001\u4ec0\u4e48\u662f\u52a8\u6001\u89c4\u5212<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u4e00\u4e2a\u6734\u7d20\u7684\u7406\u89e3<\/h3>\n\n\n\n<p>\u52a8\u6001\u89c4\u5212\u7684\u6838\u5fc3\u601d\u60f3\u662f\uff1a<strong>\u5c06\u4e00\u4e2a\u590d\u6742\u7684\u95ee\u9898\u5206\u89e3\u6210\u82e5\u5e72\u4e2a\u7b80\u5355\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u4e14\u4fdd\u5b58\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u9ad8\u6548\u5730\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002<\/strong><br>\u5b83\u901a\u5e38\u7528\u4e8e\u6c42\u89e3<strong>\u6700\u4f18\u5316\u95ee\u9898<\/strong>\uff08\u6700\u5927\u503c\u3001\u6700\u5c0f\u503c\u3001\u65b9\u6848\u6570\u7b49\uff09\uff0c\u8fd9\u4e9b\u95ee\u9898\u7684\u5171\u540c\u7279\u5f81\u662f\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5177\u6709<strong>\u91cd\u53e0\u5b50\u95ee\u9898<\/strong>\uff1a\u5b50\u95ee\u9898\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u88ab\u53cd\u590d\u7528\u5230\u3002<\/li>\n\n\n\n<li>\u5177\u6709<strong>\u6700\u4f18\u5b50\u7ed3\u6784<\/strong>\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u52a8\u6001\u89c4\u5212\u4e0e\u5206\u6cbb\u548c\u8d2a\u5fc3<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>\u7b97\u6cd5<\/th><th>\u6838\u5fc3\u7279\u70b9<\/th><th>\u5178\u578b\u4ee3\u8868<\/th><\/tr><\/thead><tbody><tr><td>\u5206\u6cbb<\/td><td>\u5b50\u95ee\u9898\u76f8\u4e92\u72ec\u7acb\uff0c\u65e0\u91cd\u53e0<\/td><td>\u5f52\u5e76\u6392\u5e8f<\/td><\/tr><tr><td>\u8d2a\u5fc3<\/td><td>\u6bcf\u4e00\u6b65\u53d6\u5c40\u90e8\u6700\u4f18\uff0c\u4e0d\u53ef\u56de\u9000<\/td><td>\u6700\u5c0f\u751f\u6210\u6811\u3001Dijkstra<\/td><\/tr><tr><td>\u52a8\u6001\u89c4\u5212<\/td><td>\u5b50\u95ee\u9898\u91cd\u53e0\uff0c\u8bb0\u5f55\u72b6\u6001\uff0c\u5168\u5c40\u6700\u4f18<\/td><td>\u80cc\u5305\u3001LCS\u3001\u6700\u77ed\u8def\u5f84<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>DP\u672c\u8d28\u4e0a\u662f\u201c\u5e26\u5907\u5fd8\u5f55\u7684\u66b4\u529b\u641c\u7d22\u201d\u2014\u2014\u5b83\u679a\u4e3e\u4e86\u6240\u6709\u53ef\u80fd\u7684\u72b6\u6001\uff0c\u4f46\u901a\u8fc7\u72b6\u6001\u8f6c\u79fb\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e8c\u3001\u52a8\u6001\u89c4\u5212\u7684\u6838\u5fc3\u8981\u7d20<\/h2>\n\n\n\n<p>\u4efb\u4f55\u4e00\u4e2aDP\u95ee\u9898\uff0c\u90fd\u53ef\u4ee5\u7528\u4ee5\u4e0b\u56db\u4e2a\u6b65\u9aa4\u6765\u62c6\u89e3\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5b9a\u4e49\u72b6\u6001\uff08State\uff09<\/strong><br>\u7528\u6570\u7ec4<code>dp[i]<\/code>\u6216<code>dp[i][j]<\/code>\u8868\u793a\u67d0\u4e2a\u9636\u6bb5\u7684\u6700\u4f18\u89e3\u6216\u65b9\u6848\u6570\u3002\u72b6\u6001\u5b9a\u4e49\u662fDP\u7684\u7075\u9b42\uff0c\u5b9a\u4e49\u5f97\u5de7\uff0c\u95ee\u9898\u5c31\u89e3\u51b3\u4e86\u4e00\u534a\u3002<\/li>\n\n\n\n<li><strong>\u786e\u5b9a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff08Transition\uff09<\/strong><br>\u63cf\u8ff0\u5f53\u524d\u72b6\u6001\u5982\u4f55\u7531\u4e4b\u524d\u7684\u72b6\u6001\u63a8\u5bfc\u51fa\u6765\u3002\u8fd9\u662fDP\u7684\u201c\u9012\u63a8\u516c\u5f0f\u201d\u3002<\/li>\n\n\n\n<li><strong>\u521d\u59cb\u5316\uff08Initialization\uff09<\/strong><br>\u786e\u5b9a\u8fb9\u754c\u72b6\u6001\u7684\u503c\uff0c\u901a\u5e38\u662f\u9012\u63a8\u7684\u8d77\u70b9\u3002<\/li>\n\n\n\n<li><strong>\u786e\u5b9a\u904d\u5386\u987a\u5e8f\uff08Order\uff09<\/strong><br>\u6839\u636e\u4f9d\u8d56\u5173\u7cfb\uff0c\u51b3\u5b9a\u5faa\u73af\u7684\u65b9\u5411\uff08\u6b63\u5e8f\u3001\u5012\u5e8f\u3001\u6309\u5c42\u7b49\uff09\u3002<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e09\u3001\u4e00\u7ef4DP<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u6590\u6ce2\u90a3\u5951\u6570\u5217<\/h3>\n\n\n\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/B2064\">B2064 \u6590\u6ce2\u90a3\u5951\u6570\u5217 &#8211; \u6d1b\u8c37<\/a><br><strong>\u95ee\u9898<\/strong>\uff1a\u6c42\u7b2cn\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\uff0cF(0)=0, F(1)=1, F(n)=F(n-1)+F(n-2)\u3002<br>\u8fd9\u662f\u6700\u76f4\u89c2\u7684DP\u6a21\u578b\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a<code>dp[i]<\/code>\u8868\u793a\u7b2ci\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u3002<br><strong>\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a<code>dp[i] = dp[i-1] + dp[i-2]<\/code><br><strong>\u521d\u59cb\u5316<\/strong>\uff1a<code>dp[0] = 0, dp[1] = 1<\/code><\/p>\n\n\n\n<p>\u8fd9\u4e00\u6bb5\u6bd4\u8f83\u7b80\u5355\uff0c\u76f4\u63a5\u7ed9\u51fa\u4ee3\u7801\u4f9b\u8bfb\u8005\u53c2\u8003\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>long long fib(int n) {\n    if (n &lt; 2) return n;\n    long long a = 0, b = 1;\n    for (int i = 2; i &lt;= n; ++i) {\n        int c = a + b;\n        a = b;\n        b = c;\n    }\n    return b;\n}<\/code><\/pre>\n\n\n\n<p><strong>\u5173\u952e\u601d\u60f3<\/strong>\uff1aDP\u7684\u672c\u8d28\u5c31\u662f\u7528\u7a7a\u95f4\u6362\u65f6\u95f4\uff0c\u8fd9\u91cc\u7528\u4e24\u4e2a\u53d8\u91cf\u5c31\u5b9e\u73b0\u4e86\u72b6\u6001\u6eda\u52a8\uff0c\u590d\u6742\u5ea6O(n)\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.\u722c\u697c\u68af\u95ee\u9898<\/h3>\n\n\n\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1255\">P1255 \u6570\u697c\u68af &#8211; \u6d1b\u8c37<\/a><br><strong>\u95ee\u9898<\/strong>\uff1a\u6bcf\u6b21\u53ef\u4ee5\u722c1\u7ea7\u62162\u7ea7\u53f0\u9636\uff0c\u6c42\u722c\u5230\u7b2cn\u7ea7\u6709\u591a\u5c11\u79cd\u4e0d\u540c\u65b9\u6cd5\u3002<br>\u8fd9\u662f\u6590\u6ce2\u90a3\u5951\u7684\u201c\u5e94\u7528\u9898\u201d\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a<code>dp[i]<\/code>\u8868\u793a\u5230\u8fbe\u7b2ci\u7ea7\u53f0\u9636\u7684\u65b9\u6cd5\u6570\u3002<br><strong>\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a<code>dp[i] = dp[i-1] + dp[i-2]<\/code>\uff08\u56e0\u4e3a\u6700\u540e\u4e00\u6b65\u8981\u4e48\u4ecei-1\u8de81\u6b65\uff0c\u8981\u4e48\u4ecei-2\u8de82\u6b65\uff09<br><strong>\u521d\u59cb\u5316<\/strong>\uff1a<code>dp[1] = 1, dp[2] = 2<\/code><\/p>\n\n\n\n<p>\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>long long climbStairs(int n) {\n    if (n &lt;= 2) return n;\n    vector&lt;long long&gt; dp(n + 1);\n    dp&#91;1] = 1;\n    dp&#91;2] = 2;\n    for (int i = 3; i &lt;= n; ++i) {\n        dp&#91;i] = dp&#91;i-1] + dp&#91;i-2];\n    }\n    return dp&#91;n];\n}<\/code><\/pre>\n\n\n\n<p><strong>\u603b\u7ed3<\/strong>\uff1a\u4e00\u7ef4DP\u901a\u5e38\u72b6\u6001\u7b80\u5355\uff0c\u9012\u63a8\u76f4\u89c2\uff0c\u9002\u5408\u4f5c\u4e3aDP\u7684\u542f\u8499\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u56db\u3001\u4e8c\u7ef4DP\u548c\u8def\u5f84\u95ee\u9898<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u4e0d\u540c\u8def\u5f84\uff08\u673a\u5668\u4eba\u8d70\u683c\u5b50\uff09<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1am\u00d7n\u7f51\u683c\uff0c\u673a\u5668\u4eba\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u5411\u53f3\uff0c\u6c42\u4e0d\u540c\u8def\u5f84\u6570\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a<code>dp[i][j]<\/code>\u8868\u793a\u4ece\u8d77\u70b9\u8d70\u5230(i,j)\u7684\u8def\u5f84\u6570\u3002<br><strong>\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a<code>dp[i][j] = dp[i-1][j] + dp[i][j-1]<\/code>\uff08\u53ea\u80fd\u4ece\u4e0a\u8fb9\u6216\u5de6\u8fb9\u6765\uff09<br><strong>\u521d\u59cb\u5316<\/strong>\uff1a\u7b2c\u4e00\u884c\u548c\u7b2c\u4e00\u5217\u5747\u4e3a1\uff08\u53ea\u6709\u4e00\u6761\u8def\u5f84\uff09<\/p>\n\n\n\n<p>\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int uniquePaths(int m, int n) {\n    vector&lt;vector&lt;int>> dp(m, vector&lt;int>(n, 1));\n    for (int i = 1; i &lt; m; ++i) {\n        for (int j = 1; j &lt; n; ++j) {\n            dp&#91;i]&#91;j] = dp&#91;i-1]&#91;j] + dp&#91;i]&#91;j-1];\n        }\n    }\n    return dp&#91;m-1]&#91;n-1];\n}<\/code><\/pre>\n\n\n\n<p><strong>\u7a7a\u95f4\u4f18\u5316<\/strong>\uff1a\u53ef\u4ee5\u7528\u4e00\u7ef4\u6570\u7ec4\u6eda\u52a8\uff0c\u56e0\u4e3a<code>dp[i][j]<\/code>\u53ea\u4f9d\u8d56\u4e0a\u4e00\u884c\u548c\u5f53\u524d\u884c\u5de6\u8fb9\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u6700\u5c0f\u8def\u5f84\u548c<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1a\u7ed9\u5b9a\u5e26\u6743\u503c\u7684\u7f51\u683c\uff0c\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\uff0c\u8def\u5f84\u548c\u6700\u5c0f\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a<code>dp[i][j]<\/code>\u8868\u793a\u4ece\u8d77\u70b9\u5230(i,j)\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002<br><strong>\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a<code>dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1])<\/code><br><strong>\u521d\u59cb\u5316<\/strong>\uff1a<code>dp[0][0] = grid[0][0]<\/code>\uff0c\u7b2c\u4e00\u884c\u548c\u7b2c\u4e00\u5217\u7d2f\u52a0\u3002<\/p>\n\n\n\n<p>\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int minPathSum(vector&lt;vector&lt;int>>&amp; grid) {\n    int m = grid.size(), n = grid&#91;0].size();\n    vector&lt;vector&lt;int>> dp(m, vector&lt;int>(n));\n    dp&#91;0]&#91;0] = grid&#91;0]&#91;0];\n    for (int i = 1; i &lt; m; ++i) dp&#91;i]&#91;0] = dp&#91;i-1]&#91;0] + grid&#91;i]&#91;0];\n    for (int j = 1; j &lt; n; ++j) dp&#91;0]&#91;j] = dp&#91;0]&#91;j-1] + grid&#91;0]&#91;j];\n    for (int i = 1; i &lt; m; ++i) {\n        for (int j = 1; j &lt; n; ++j) {\n            dp&#91;i]&#91;j] = grid&#91;i]&#91;j] + min(dp&#91;i-1]&#91;j], dp&#91;i]&#91;j-1]);\n        }\n    }\n    return dp&#91;m-1]&#91;n-1];\n}<\/code><\/pre>\n\n\n\n<p><strong>\u603b\u7ed3<\/strong>\uff1a\u4e8c\u7ef4DP\u5e38\u89c1\u4e8e\u7f51\u683c\u3001\u77e9\u9635\u7c7b\u95ee\u9898\uff0c\u5173\u952e\u5728\u4e8e\u7406\u6e05\u4f9d\u8d56\u65b9\u5411\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e94\u3001\u80cc\u5305\u95ee\u9898<\/h2>\n\n\n\n<p>\u80cc\u5305\u95ee\u9898\u662fDP\u4e2d\u975e\u5e38\u7ecf\u5178\u7684\u4e00\u7c7b\uff0c\u4e5f\u662f\u9762\u8bd5\u9ad8\u9891\u3002\u6211\u4eec\u91cd\u70b9\u4ecb\u7ecd<strong>01\u80cc\u5305<\/strong>\u548c<strong>\u5b8c\u5168\u80cc\u5305<\/strong>\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. 01\u80cc\u5305<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1a\u6709N\u4ef6\u7269\u54c1\uff0c\u6bcf\u4ef6\u91cd\u91cfw[i]\uff0c\u4ef7\u503cv[i]\uff0c\u80cc\u5305\u5bb9\u91cfW\uff0c\u6bcf\u4ef6\u7269\u54c1\u6700\u591a\u9009\u4e00\u6b21\uff0c\u6c42\u80fd\u88c5\u7684\u6700\u5927\u4ef7\u503c\u3002<br><strong>\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u4e8c\u7ef4\uff1a<code>dp[i][j] = max(dp[i-1][j], dp[i-1][j-w[i]] + v[i])<\/code><\/li>\n\n\n\n<li>\u4e00\u7ef4\uff1a<code>dp[j] = max(dp[j], dp[j - w[i]] + v[i])<\/code>\uff0c<strong>\u6ce8\u610fj\u5fc5\u987b\u5012\u5e8f\u904d\u5386<\/strong>\uff0c\u786e\u4fdd\u6bcf\u4e2a\u7269\u54c1\u53ea\u53d6\u4e00\u6b21\u3002<br><\/li>\n<\/ul>\n\n\n\n<p>\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int knapsack_01(vector&lt;int>&amp; weights, vector&lt;int>&amp; values, int W) {\n    vector&lt;int> dp(W + 1, 0);\n    int n = weights.size();\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = W; j >= weights&#91;i]; --j) {\n            dp&#91;j] = max(dp&#91;j], dp&#91;j - weights&#91;i]] + values&#91;i]);\n        }\n    }\n    return dp&#91;W];\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u5b8c\u5168\u80cc\u5305<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1a\u6bcf\u4ef6\u7269\u54c1\u53ef\u4ee5\u9009\u65e0\u9650\u6b21\u3002<br><strong>\u533a\u522b<\/strong>\uff1a\u53ea\u9700\u5c06\u5185\u5c42\u5faa\u73af\u6539\u4e3a<strong>\u6b63\u5e8f<\/strong>\uff0c\u56e0\u4e3a\u53ef\u4ee5\u91cd\u590d\u4f7f\u7528\u540c\u4e00\u7269\u54c1\u3002<\/p>\n\n\n\n<p>\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int knapsack_complete(vector&lt;int>&amp; weights, vector&lt;int>&amp; values, int W) {\n    vector&lt;int> dp(W + 1, 0);\n    int n = weights.size();\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = weights&#91;i]; j &lt;= W; ++j) {\n            dp&#91;j] = max(dp&#91;j], dp&#91;j - weights&#91;i]] + values&#91;i]);\n        }\n    }\n    return dp&#91;W];\n}<\/code><\/pre>\n\n\n\n<p><strong>\u603b\u7ed3<\/strong>\uff1a\u80cc\u5305\u95ee\u9898\u7684\u6838\u5fc3\u5728\u4e8e\u201c\u9009\u4e0e\u4e0d\u9009\u201d\u7684\u51b3\u7b56\uff0c\u4ee5\u53ca\u904d\u5386\u987a\u5e8f\u5bf9\u7269\u54c1\u4f7f\u7528\u6b21\u6570\u7684\u63a7\u5236\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u516d\u3001\u533a\u95f4DP<\/h2>\n\n\n\n<p>\u533a\u95f4DP\u662f\u5728\u533a\u95f4\u4e0a\u8fdb\u884c\u52a8\u6001\u89c4\u5212\uff0c\u901a\u5e38\u7528\u4e8e\u6c42\u89e3\u4e00\u4e2a\u533a\u95f4\u5185\u7684\u6700\u4f18\u503c\uff08\u5982\u5408\u5e76\u77f3\u5b50\u3001\u56de\u6587\u5206\u5272\u7b49\uff09\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u77f3\u5b50\u5408\u5e76\uff08\u7ecf\u5178\u533a\u95f4DP\uff09<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1aN\u5806\u77f3\u5b50\u6392\u6210\u4e00\u6761\u7ebf\uff0c\u6bcf\u6b21\u5408\u5e76\u76f8\u90bb\u4e24\u5806\uff0c\u4ee3\u4ef7\u4e3a\u4e24\u5806\u77f3\u5b50\u6570\u4e4b\u548c\uff0c\u6c42\u5408\u5e76\u6210\u4e00\u5806\u7684\u6700\u5c0f\u603b\u4ee3\u4ef7\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a<code>dp[i][j]<\/code>\u8868\u793a\u5408\u5e76\u7b2ci\u5806\u5230\u7b2cj\u5806\u7684\u6700\u5c0f\u4ee3\u4ef7\u3002<br><strong>\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a<code>dp[i][j] = min(dp[i][k] + dp[k+1][j] + sum(i,j))<\/code>\uff0c\u5176\u4e2d<code>i \u2264 k &lt; j<\/code>\u3002<br><strong>\u679a\u4e3e\u987a\u5e8f<\/strong>\uff1a\u6309\u533a\u95f4\u957f\u5ea6\u4ece\u5c0f\u5230\u5927\u679a\u4e3e\u3002<\/p>\n\n\n\n<p>\u4ee3\u7801\u6846\u67b6\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int stoneMerge(vector&lt;int>&amp; stones) {\n    int n = stones.size();\n    vector&lt;int> prefix(n + 1, 0);\n    for (int i = 1; i &lt;= n; ++i) {\n        prefix&#91;i] = prefix&#91;i-1] + stones&#91;i-1];\n    }\n    \n    vector&lt;vector&lt;int>> dp(n, vector&lt;int>(n, 0));\n    \/\/ \u533a\u95f4\u957f\u5ea6\u4ece2\u5230n\n    for (int len = 2; len &lt;= n; ++len) {\n        for (int i = 0; i + len &lt;= n; ++i) {\n            int j = i + len - 1;\n            dp&#91;i]&#91;j] = INT_MAX;\n            for (int k = i; k &lt; j; ++k) {\n                int cost = dp&#91;i]&#91;k] + dp&#91;k+1]&#91;j] + prefix&#91;j+1] - prefix&#91;i];\n                dp&#91;i]&#91;j] = min(dp&#91;i]&#91;j], cost);\n            }\n        }\n    }\n    return dp&#91;0]&#91;n-1];\n}<\/code><\/pre>\n\n\n\n<p><strong>\u603b\u7ed3<\/strong>\uff1a\u533a\u95f4DP\u7684\u7279\u5f81\u662f\u72b6\u6001\u7531\u5c0f\u533a\u95f4\u5408\u5e76\u800c\u6765\uff0c\u5faa\u73af\u987a\u5e8f\u5e38\u6309\u957f\u5ea6\u9012\u589e\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e03\u3001\u6811\u5f62DP<\/h2>\n\n\n\n<p>\u6811\u5f62DP\u662f\u5728\u6811\u7ed3\u6784\u4e0a\u505aDP\uff0c\u901a\u5e38\u7528DFS\u540e\u5e8f\u904d\u5386\u5b9e\u73b0\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. 7.1 \u6811\u7684\u6700\u5927\u72ec\u7acb\u96c6\uff08\u6253\u5bb6\u52ab\u820dIII\uff09<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1a\u4e8c\u53c9\u6811\uff0c\u76f8\u90bb\u8282\u70b9\u4e0d\u80fd\u540c\u65f6\u9009\uff0c\u6c42\u80fd\u9009\u7684\u6700\u5927\u6743\u503c\u548c\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a\u6bcf\u4e2a\u8282\u70b9\u8fd4\u56de\u4e24\u4e2a\u503c\uff1a<code>[\u9009\u5f53\u524d\u8282\u70b9\uff0c\u4e0d\u9009\u5f53\u524d\u8282\u70b9]<\/code>\u7684\u6700\u5927\u6536\u76ca\u3002<br><strong>\u8f6c\u79fb<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u9009\u5f53\u524d\u8282\u70b9\uff1a\u5de6\u53f3\u5b69\u5b50\u90fd\u4e0d\u80fd\u9009 \u2192\u00a0<code>val + left[1] + right[1]<\/code><\/li>\n\n\n\n<li>\u4e0d\u9009\u5f53\u524d\u8282\u70b9\uff1a\u5de6\u53f3\u5b69\u5b50\u53ef\u9009\u53ef\u4e0d\u9009 \u2192\u00a0<code>max(left[0], left[1]) + max(right[0], right[1])<\/code><\/li>\n<\/ul>\n\n\n\n<p>\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>struct TreeNode {\n    int val;\n    TreeNode *left;\n    TreeNode *right;\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n\npair&lt;int, int> dfs(TreeNode* node) {\n    if (!node) return {0, 0};\n    auto left = dfs(node->left);\n    auto right = dfs(node->right);\n    int rob_cur = node->val + left.second + right.second;\n    int not_rob = max(left.first, left.second) + max(right.first, right.second);\n    return {rob_cur, not_rob};\n}\n\nint rob(TreeNode* root) {\n    auto res = dfs(root);\n    return max(res.first, res.second);\n}<\/code><\/pre>\n\n\n\n<p><strong>\u603b\u7ed3<\/strong>\uff1a\u6811\u5f62DP\u7684\u5173\u952e\u662f\u540e\u5e8f\u6536\u96c6\u5b50\u8282\u70b9\u4fe1\u606f\uff0c\u518d\u5411\u4e0a\u5408\u5e76\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u516b\u3001\u72b6\u6001\u538b\u7f29DP<\/h2>\n\n\n\n<p>\u5f53\u72b6\u6001\u7ef4\u5ea6\u5f88\u591a\uff08\u6bd4\u5982\u96c6\u5408\u3001\u6392\u5217\uff09\u65f6\uff0c\u53ef\u4ee5\u7528\u4e8c\u8fdb\u5236\u4f4d\u8868\u793a\u72b6\u6001\uff0c\u5e38\u89c1\u4e8eNP\u96be\u95ee\u9898\u7684\u5c0f\u89c4\u6a21\u6c42\u89e3\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u65c5\u884c\u5546\u95ee\u9898\uff08TSP\uff09<\/h3>\n\n\n\n<p><strong>\u95ee\u9898<\/strong>\uff1a\u7ed9\u5b9a\u57ce\u5e02\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u4ece0\u51fa\u53d1\uff0c\u7ecf\u8fc7\u6240\u6709\u57ce\u5e02\u4e00\u6b21\uff0c\u56de\u5230\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002<br><strong>\u72b6\u6001\u5b9a\u4e49<\/strong>\uff1a<code>dp[mask][i]<\/code>\u8868\u793a\u5df2\u7ecf\u8bbf\u95ee\u8fc7\u7684\u57ce\u5e02\u96c6\u5408\u4e3amask\uff08\u4e8c\u8fdb\u5236\uff09\uff0c\u5f53\u524d\u5728\u57ce\u5e02i\u7684\u6700\u77ed\u8def\u5f84\u3002<br><strong>\u8f6c\u79fb<\/strong>\uff1a<code>dp[mask][i] = min(dp[mask ^ (1&lt;&lt;i)][j] + dist[j][i])<\/code><\/p>\n\n\n\n<p>\u6838\u5fc3\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int tsp(vector&lt;vector&lt;int>>&amp; dist) {\n    int n = dist.size();\n    int fullMask = (1 &lt;&lt; n) - 1;\n    const int INF = 1e9;\n    vector&lt;vector&lt;int>> dp(1 &lt;&lt; n, vector&lt;int>(n, INF));\n    dp&#91;1]&#91;0] = 0;  \/\/ \u4ece0\u51fa\u53d1\uff0c\u53ea\u8bbf\u95ee\u4e860\n    for (int mask = 0; mask &lt; (1 &lt;&lt; n); ++mask) {\n        for (int i = 0; i &lt; n; ++i) {\n            if (dp&#91;mask]&#91;i] == INF) continue;\n            for (int j = 0; j &lt; n; ++j) {\n                if (!(mask >> j &amp; 1)) {  \/\/ j\u672a\u8bbf\u95ee\n                    int newMask = mask | (1 &lt;&lt; j);\n                    dp&#91;newMask]&#91;j] = min(dp&#91;newMask]&#91;j], dp&#91;mask]&#91;i] + dist&#91;i]&#91;j]);\n                }\n            }\n        }\n    }\n    int ans = INF;\n    for (int i = 0; i &lt; n; ++i) {\n        ans = min(ans, dp&#91;fullMask]&#91;i] + dist&#91;i]&#91;0]);\n    }\n    return ans;\n}<\/code><\/pre>\n\n\n\n<p><strong>\u603b\u7ed3<\/strong>\uff1a\u72b6\u6001\u538b\u7f29DP\u5e38\u7528\u4e8e\u201c\u96c6\u5408\u8986\u76d6\u201d\u201c\u54c8\u5bc6\u987f\u8def\u5f84\u201d\u7b49\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0cn\u901a\u5e38\u226420\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e5d\u3001DP\u4f18\u5316\u6280\u5de7<\/h2>\n\n\n\n<p>\u5f53DP\u7684\u590d\u6742\u5ea6\u4e0d\u6ee1\u8db3\u8981\u6c42\u65f6\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u4ee5\u4e0b\u4f18\u5316\uff1a<\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li><strong>\u7a7a\u95f4\u4f18\u5316<\/strong>\uff1a\u6eda\u52a8\u6570\u7ec4\uff08\u5982\u80cc\u5305\u3001\u8def\u5f84\u95ee\u9898\uff09\u3002<\/li>\n\n\n\n<li><strong>\u5355\u8c03\u961f\u5217\u4f18\u5316<\/strong>\uff1a\u7528\u4e8e\u6ed1\u52a8\u7a97\u53e3\u7c7bDP\uff0c\u5982<code>dp[i] = min(dp[j]) + cost<\/code>\uff0c<code>j<\/code>\u5728\u6ed1\u52a8\u533a\u95f4\u5185\u3002<\/li>\n\n\n\n<li><strong>\u659c\u7387\u4f18\u5316<\/strong>\uff1a\u5c06\u8f6c\u79fb\u65b9\u7a0b\u8f6c\u5316\u4e3a\u76f4\u7ebf\u5f62\u5f0f\uff0c\u7ef4\u62a4\u51f8\u5305\uff0c\u7528\u4e8e<code>dp[i] = min(dp[j] + (x[i]-x[j])^2)<\/code>\u7c7b\u95ee\u9898\u3002<\/li>\n\n\n\n<li><strong>\u56db\u8fb9\u5f62\u4e0d\u7b49\u5f0f\u4f18\u5316<\/strong>\uff1a\u7528\u4e8e\u533a\u95f4DP\uff0c\u5f53\u51b3\u7b56\u70b9\u5177\u6709\u5355\u8c03\u6027\u65f6\uff0c\u53ef\u5c06\u590d\u6742\u5ea6\u4eceO(n^3)\u964d\u5230O(n^2)\u3002<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u5341\u3001\u603b\u7ed3\u4e0e\u5fc3\u6cd5<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">10.1 \u5982\u4f55\u8bc6\u522bDP\u95ee\u9898\uff1f<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u95ee\u9898\u6c42\u6700\u503c\u3001\u65b9\u6848\u6570<\/li>\n\n\n\n<li>\u5b58\u5728\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d<\/li>\n\n\n\n<li>\u72b6\u6001\u53ef\u7531\u4e4b\u524d\u72b6\u6001\u63a8\u5bfc<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">10.2 DP\u89e3\u9898\u56db\u6b65\u6cd5<\/h3>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li><strong>\u5b9a\u4e49\u72b6\u6001<\/strong>\uff1a\u60f3\u6e05\u695a\u4e00\u7ef4\u8fd8\u662f\u591a\u7ef4\uff0c\u6bcf\u4e2a\u72b6\u6001\u4ee3\u8868\u4ec0\u4e48\u3002<\/li>\n\n\n\n<li><strong>\u5199\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1a\u601d\u8003\u201c\u6700\u540e\u4e00\u6b65\u201d\u6216\u201c\u5f53\u524d\u51b3\u7b56\u201d\uff0c\u7528\u4e4b\u524d\u72b6\u6001\u8868\u793a\u5f53\u524d\u72b6\u6001\u3002<\/li>\n\n\n\n<li><strong>\u521d\u59cb\u5316<\/strong>\uff1a\u8fb9\u754c\u6761\u4ef6\u5f80\u5f80\u5c31\u662f\u9012\u63a8\u7684\u8d77\u70b9\u3002<\/li>\n\n\n\n<li><strong>\u786e\u5b9a\u987a\u5e8f<\/strong>\uff1a\u6309\u72b6\u6001\u4f9d\u8d56\u5173\u7cfb\u51b3\u5b9a\u5faa\u73af\u65b9\u5411\u3002<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">10.3 \u5e38\u89c1DP\u6a21\u578b<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u4e00\u7ef4\u7ebf\u6027DP\uff1a\u6590\u6ce2\u90a3\u5951\u3001\u722c\u697c\u68af\u3001\u6700\u5927\u5b50\u5e8f\u548c<\/li>\n\n\n\n<li>\u4e8c\u7ef4\u7f51\u683cDP\uff1a\u8def\u5f84\u6570\u3001\u6700\u5c0f\u8def\u5f84\u548c<\/li>\n\n\n\n<li>\u80cc\u5305DP\uff1a0-1\u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305<\/li>\n\n\n\n<li>\u533a\u95f4DP\uff1a\u77f3\u5b50\u5408\u5e76\u3001\u56de\u6587\u5206\u5272<\/li>\n\n\n\n<li>\u6811\u5f62DP\uff1a\u6811\u4e0a\u7684\u6700\u5927\u72ec\u7acb\u96c6\u3001\u6811\u5f62\u80cc\u5305<\/li>\n\n\n\n<li>\u72b6\u6001\u538b\u7f29DP\uff1aTSP\u3001\u96c6\u5408\u8986\u76d6<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">10.4 \u6700\u540e\u7684\u8bdd<\/h3>\n\n\n\n<p>\u52a8\u6001\u89c4\u5212\u4e0d\u662f\u201c\u80cc\u6a21\u677f\u201d\u7684\u7b97\u6cd5\uff0c\u800c\u662f\u201c\u627e\u89c4\u5f8b\u201d\u7684\u827a\u672f\u3002\u5f53\u4f60\u9762\u5bf9\u4e00\u9053\u65b0\u9898\u65f6\uff0c\u4e0d\u8981\u6025\u7740\u5199\u4ee3\u7801\uff0c\u5148\u7528\u624b\u5de5\u63a8\u5bfc\u5c0f\u89c4\u6a21\u60c5\u51b5\uff0c\u89c2\u5bdf\u72b6\u6001\u4e4b\u95f4\u7684\u9012\u63a8\u5173\u7cfb\u3002\u591a\u505a\u7ecf\u5178\u9898\uff0c\u6162\u6162\u4f60\u5c31\u4f1a\u53d1\u73b0\uff0c\u5f88\u591aDP\u9898\u7684\u672c\u8d28\u90fd\u662f\u76f8\u901a\u7684\u3002<\/p>\n\n\n\n<p><strong>\u7eb8\u4e0a\u5f97\u6765\u7ec8\u89c9\u6d45\uff0c\u7edd\u77e5\u6b64\u4e8b\u8981\u8eac\u884c\u3002<\/strong><\/p>\n\n\n\n<p>\u4e0b\u9762\u7ed9\u51fa\u76f8\u5173\u7ecf\u5178\u4f8b\u9898\u4f9b\u5927\u5bb6\u8bad\u7ec3\uff1a<br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1115\">P1115 \u6700\u5927\u5b50\u6bb5\u548c &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/AT_dp_a\">AT_dp_a Frog 1 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1002\">P1002 [NOIP 2002 \u666e\u53ca\u7ec4] \u8fc7\u6cb3\u5352 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1006\">P1006 [NOIP 2008 \u63d0\u9ad8\u7ec4] \u4f20\u7eb8\u6761 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/AT_dp_y\">AT_dp_y Grid 2 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1048\">P1048 [NOIP 2005 \u666e\u53ca\u7ec4] \u91c7\u836f &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1616\">P1616 \u75af\u72c2\u7684\u91c7\u836f &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1880\">P1880 [NOI1995] \u77f3\u5b50\u5408\u5e76 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1063\">P1063 [NOIP 2006 \u63d0\u9ad8\u7ec4] \u80fd\u91cf\u9879\u94fe &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1352\">P1352 \u6ca1\u6709\u4e0a\u53f8\u7684\u821e\u4f1a &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P2014\">P2014 [CTSC1997] \u9009\u8bfe &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1896\">P1896 [SCOI2005] \u4e92\u4e0d\u4fb5\u72af &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P2704\">P2704 [NOI2001] \u70ae\u5175\u9635\u5730 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/AT_dp_s\">AT_dp_s Digit Sum &#8211; \u6d1b\u8c37<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u52a8\u6001\u89c4\u5212\uff0c\u6c5f\u6e56\u4eba\u79f0\u201cDP\u201d\uff0c\u662f\u7b97\u6cd5\u9762\u8bd5\u4e2d\u7684\u201c\u62e6\u8def\u864e\u201d\uff0c\u4e5f\u662f\u7ade\u8d5b\u9009\u624b\u7684\u201c\u770b\u5bb6\u672c\u9886\u201d\u3002\u5f88\u591a\u4eba\u89c9\u5f97DP\u96be\uff0c\u662f\u56e0\u4e3a\u5b83\u4e0d [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":156,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,15,12],"tags":[],"class_list":["post-404","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-article","category-algorithm-template","category-programming-algorithm-road"],"_links":{"self":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/404","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/comments?post=404"}],"version-history":[{"count":6,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/404\/revisions"}],"predecessor-version":[{"id":420,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/404\/revisions\/420"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/media\/156"}],"wp:attachment":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/media?parent=404"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/categories?post=404"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/tags?post=404"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}