{"id":103941,"date":"2020-02-20T18:13:40","date_gmt":"2020-02-20T10:13:40","guid":{"rendered":"http:\/\/4563.org\/?p=103941"},"modified":"2020-02-20T18:13:40","modified_gmt":"2020-02-20T10:13:40","slug":"%e5%8a%a8%e6%80%81%e8%a7%84%e5%88%92%e7%ae%97%e6%b3%95%e8%af%a6%e8%a7%a3","status":"publish","type":"post","link":"http:\/\/4563.org\/?p=103941","title":{"rendered":"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3"},"content":{"rendered":"
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\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3 <\/h1>\n

<\/p>\n

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\u8cc7\u6df1\u5927\u4f6c : fudonglai <\/span> <\/i> 54<\/span> <\/div>\n
<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<\/p><\/div>\n
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\u8fd9\u7bc7\u6587\u7ae0\u662f\u6211\u4eec\u53f7\u534a\u5e74\u524d\u4e00\u7bc7 200 \u591a\u8d5e\u8d4f\u7684\u6210\u540d\u4e4b\u4f5c\u300c\u52a8\u6001\u89c4\u5212\u8be6\u89e3\u300d\u7684\u8fdb\u9636\u7248\u3002\u7531\u4e8e\u8d26\u53f7\u8fc1\u79fb\u7684\u539f\u56e0\uff0c\u65e7\u6587\u65e0\u6cd5\u88ab\u641c\u7d22\u5230\uff0c\u6240\u4ee5\u6211\u6da6\u8272\u4e86\u672c\u6587\uff0c\u5e76\u6dfb\u52a0\u4e86\u66f4\u591a\u5e72\u8d27\u5185\u5bb9\uff0c\u5e0c\u671b\u672c\u6587\u6210\u4e3a\u89e3\u51b3\u52a8\u6001\u89c4\u5212\u7684\u4e00\u90e8\u300c\u6307\u5bfc\u65b9\u9488\u300d\u3002<\/p>\n

\u518d\u8bf4\u53e5\u9898\u5916\u8bdd\uff0c\u6211\u4eec\u7684\u516c\u4f17\u53f7\u5f00\u53f7\u81f3\u4eca\u5199\u4e86\u8d77\u7801\u5341\u51e0\u7bc7\u6587\u7ae0\u62c6\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u6211\u90fd\u6574\u7406\u5230\u4e86\u516c\u4f17\u53f7\u83dc\u5355\u7684\u300c\u6587\u7ae0\u76ee\u5f55\u300d\u4e2d\uff0c\u5b83\u4eec\u90fd\u63d0\u5230\u4e86\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6846\u67b6\u601d\u7ef4\uff0c\u672c\u6587\u5c31\u7cfb\u7edf\u603b\u7ed3\u4e00\u4e0b<\/strong>\u3002\u8fd9\u6bb5\u65f6\u95f4\u672c\u4eba\u4e5f\u4ece\u975e\u79d1\u73ed\u5c0f\u767d\u6210\u957f\u5230\u5237\u901a\u534a\u4e2a LeetCode\uff0c\u6240\u4ee5\u6211\u603b\u7ed3\u7684\u5957\u8def\u53ef\u80fd\u4e0d\u9002\u5408\u5404\u8def\u5927\u795e\uff0c\u4f46\u662f\u5e94\u8be5\u9002\u5408\u5927\u4f17\uff0c\u6bd5\u7adf\u6211\u81ea\u5df1\u4e5f\u662f\u4e00\u8def\u6478\u722c\u6eda\u6253\u8fc7\u6765\u7684\u3002<\/p>\n

\u7b97\u6cd5\u6280\u5de7\u5c31\u90a3\u51e0\u4e2a\u5957\u8def\uff0c\u5982\u679c\u4f60\u5fc3\u91cc\u6709\u6570\uff0c\u5c31\u4f1a\u8f7b\u677e\u5f88\u591a\uff0c\u672c\u6587\u5c31\u6765\u6252\u4e00\u6252\u52a8\u6001\u89c4\u5212\u7684\u88e4\u5b50\uff0c\u5f62\u6210\u4e00\u5957\u89e3\u51b3\u8fd9\u7c7b\u95ee\u9898\u7684\u601d\u7ef4\u6846\u67b6\u3002\u5e9f\u8bdd\u4e0d\u591a\u8bf4\u4e86\uff0c\u4e0a\u5e72\u8d27\u3002<\/p>\n

\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e00\u822c\u5f62\u5f0f\u5c31\u662f\u6c42\u6700\u503c<\/strong>\u3002\u52a8\u6001\u89c4\u5212\u5176\u5b9e\u662f\u8fd0\u7b79\u5b66\u7684\u4e00\u79cd\u6700\u4f18\u5316\u65b9\u6cd5\uff0c\u53ea\u4e0d\u8fc7\u5728\u8ba1\u7b97\u673a\u95ee\u9898\u4e0a\u5e94\u7528\u6bd4\u8f83\u591a\uff0c\u6bd4\u5982\u8bf4\u8ba9\u4f60\u6c42\u6700\u957f<\/strong>\u9012\u589e\u5b50\u5e8f\u5217\u5440\uff0c\u6700\u5c0f<\/strong>\u7f16\u8f91\u8ddd\u79bb\u5440\u7b49\u7b49\u3002<\/p>\n

\u65e2\u7136\u662f\u8981\u6c42\u6700\u503c\uff0c\u6838\u5fc3\u95ee\u9898\u662f\u4ec0\u4e48\u5462\uff1f\u6c42\u89e3\u52a8\u6001\u89c4\u5212\u7684\u6838\u5fc3\u95ee\u9898\u662f\u7a77\u4e3e<\/strong>\u3002\u56e0\u4e3a\u8981\u6c42\u6700\u503c\uff0c\u80af\u5b9a\u8981\u628a\u6240\u6709\u53ef\u884c\u7684\u7b54\u6848\u7a77\u4e3e\u51fa\u6765\uff0c\u7136\u540e\u5728\u5176\u4e2d\u627e\u6700\u503c\u5457\u3002<\/p>\n

\u52a8\u6001\u89c4\u5212\u5c31\u8fd9\u4e48\u7b80\u5355\uff0c\u5c31\u662f\u7a77\u4e3e\u5c31\u5b8c\u4e8b\u4e86\uff1f\u6211\u770b\u5230\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u5f88\u96be\u554a\uff01<\/p>\n

\u9996\u5148\uff0c\u52a8\u6001\u89c4\u5212\u7684\u7a77\u4e3e\u6709\u70b9\u7279\u522b\uff0c\u56e0\u4e3a\u8fd9\u7c7b\u95ee\u9898\u5b58\u5728\u300c\u91cd\u53e0\u5b50\u95ee\u9898\u300d<\/strong>\uff0c\u5982\u679c\u66b4\u529b\u7a77\u4e3e\u7684\u8bdd\u6548\u7387\u4f1a\u6781\u5176\u4f4e\u4e0b\uff0c\u6240\u4ee5\u9700\u8981\u300c\u5907\u5fd8\u5f55\u300d\u6216\u8005\u300c DP table \u300d\u6765\u4f18\u5316\u7a77\u4e3e\u8fc7\u7a0b\uff0c\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u8ba1\u7b97\u3002<\/p>\n

\u800c\u4e14\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e00\u5b9a\u4f1a\u5177\u5907\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d<\/strong>\uff0c\u624d\u80fd\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u503c\u5f97\u5230\u539f\u95ee\u9898\u7684\u6700\u503c\u3002<\/p>\n

\u53e6\u5916\uff0c\u867d\u7136\u52a8\u6001\u89c4\u5212\u7684\u6838\u5fc3\u601d\u60f3\u5c31\u662f\u7a77\u4e3e\u6c42\u6700\u503c\uff0c\u4f46\u662f\u95ee\u9898\u53ef\u4ee5\u5343\u53d8\u4e07\u5316\uff0c\u7a77\u4e3e\u6240\u6709\u53ef\u884c\u89e3\u5176\u5b9e\u5e76\u4e0d\u662f\u4e00\u4ef6\u5bb9\u6613\u7684\u4e8b\uff0c\u53ea\u6709\u5217\u51fa\u6b63\u786e\u7684\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/strong>\u300d\u624d\u80fd\u6b63\u786e\u5730\u7a77\u4e3e\u3002<\/p>\n

\u4ee5\u4e0a\u63d0\u5230\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5c31\u662f\u52a8\u6001\u89c4\u5212\u4e09\u8981\u7d20\u3002\u5177\u4f53\u4ec0\u4e48\u610f\u601d\u7b49\u4f1a\u4f1a\u4e3e\u4f8b\u8be6\u89e3\uff0c\u4f46\u662f\u5728\u5b9e\u9645\u7684\u7b97\u6cd5\u95ee\u9898\u4e2d\uff0c\u5199\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u662f\u6700\u56f0\u96be\u7684<\/strong>\uff0c\u8fd9\u4e5f\u5c31\u662f\u4e3a\u4ec0\u4e48\u5f88\u591a\u670b\u53cb\u89c9\u5f97\u52a8\u6001\u89c4\u5212\u95ee\u9898\u56f0\u96be\u7684\u539f\u56e0\uff0c\u6211\u6765\u63d0\u4f9b\u6211\u7814\u7a76\u51fa\u6765\u7684\u4e00\u4e2a\u601d\u7ef4\u6846\u67b6\uff0c\u8f85\u52a9\u4f60\u601d\u8003\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a<\/p>\n

\u660e\u786e\u300c\u72b6\u6001\u300d -> \u5b9a\u4e49 dp \u6570\u7ec4 \/\u51fd\u6570\u7684\u542b\u4e49 -> \u660e\u786e\u300c\u9009\u62e9\u300d-> \u660e\u786e base case\u3002<\/p>\n

\u4e0b\u9762\u901a\u8fc7\u6590\u6ce2\u90a3\u5951\u6570\u5217\u95ee\u9898\u548c\u51d1\u96f6\u94b1\u95ee\u9898\u6765\u8be6\u89e3\u52a8\u6001\u89c4\u5212\u7684\u57fa\u672c\u539f\u7406\u3002\u524d\u8005\u4e3b\u8981\u662f\u8ba9\u4f60\u660e\u767d\u4ec0\u4e48\u662f\u91cd\u53e0\u5b50\u95ee\u9898\uff08\u6590\u6ce2\u90a3\u5951\u6570\u5217\u4e25\u683c\u6765\u8bf4\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff09\uff0c\u540e\u8005\u4e3b\u8981\u4e3e\u96c6\u4e2d\u4e8e\u5982\u4f55\u5217\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002<\/p>\n

\u8bf7\u8bfb\u8005\u4e0d\u8981\u5acc\u5f03\u8fd9\u4e2a\u4f8b\u5b50\u7b80\u5355\uff0c\u53ea\u6709\u7b80\u5355\u7684\u4f8b\u5b50\u624d\u80fd\u8ba9\u4f60\u628a\u7cbe\u529b\u5145\u5206\u96c6\u4e2d\u5728\u7b97\u6cd5\u80cc\u540e\u7684\u901a\u7528\u601d\u60f3\u548c\u6280\u5de7\u4e0a\uff0c\u800c\u4e0d\u4f1a\u88ab\u90a3\u4e9b\u9690\u6666\u7684\u7ec6\u8282\u95ee\u9898\u641e\u7684\u83ab\u540d\u5176\u5999<\/strong>\u3002\u60f3\u8981\u56f0\u96be\u7684\u4f8b\u5b50\uff0c\u5386\u53f2\u6587\u7ae0\u91cc\u6709\u7684\u662f\u3002<\/p>\n

\u4e00\u3001\u6590\u6ce2\u90a3\u5951\u6570\u5217<\/h3>\n

1\u3001\u66b4\u529b\u9012\u5f52<\/strong><\/p>\n

\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u6570\u5b66\u5f62\u5f0f\u5c31\u662f\u9012\u5f52\u7684\uff0c\u5199\u6210\u4ee3\u7801\u5c31\u662f\u8fd9\u6837\uff1a<\/p>\n

int fib(int N) {     if (N == 1 || N == 2) return 1;     return fib(N - 1) + fib(N - 2); } <\/code><\/pre>\n
\u8fd9\u4e2a\u4e0d\u7528\u591a\u8bf4\u4e86\uff0c\u5b66\u6821\u8001\u5e08\u8bb2\u9012\u5f52\u7684\u65f6\u5019\u4f3c\u4e4e\u90fd\u662f\u62ff\u8fd9\u4e2a\u4e3e\u4f8b\u3002\u6211\u4eec\u4e5f\u77e5\u9053\u8fd9\u6837\u5199\u4ee3\u7801\u867d\u7136\u7b80\u6d01\u6613\u61c2\uff0c\u4f46\u662f\u5341\u5206\u4f4e\u6548\uff0c\u4f4e\u6548\u5728\u54ea\u91cc\uff1f\u5047\u8bbe n = 20\uff0c\u8bf7\u753b\u51fa\u9012\u5f52\u6811\u3002<\/p>\n
PS\uff1a\u4f46\u51e1\u9047\u5230\u9700\u8981\u9012\u5f52\u7684\u95ee\u9898\uff0c\u6700\u597d\u90fd\u753b\u51fa\u9012\u5f52\u6811\uff0c\u8fd9\u5bf9\u4f60\u5206\u6790\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\uff0c\u5bfb\u627e\u7b97\u6cd5\u4f4e\u6548\u7684\u539f\u56e0\u90fd\u6709\u5de8\u5927\u5e2e\u52a9\u3002<\/p>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n
\u8fd9\u4e2a\u9012\u5f52\u6811\u600e\u4e48\u7406\u89e3\uff1f\u5c31\u662f\u8bf4\u60f3\u8981\u8ba1\u7b97\u539f\u95ee\u9898 f(20)<\/code>\uff0c\u6211\u5c31\u5f97\u5148\u8ba1\u7b97\u51fa\u5b50\u95ee\u9898 f(19)<\/code> \u548c f(18)<\/code>\uff0c\u7136\u540e\u8981\u8ba1\u7b97 f(19)<\/code>\uff0c\u6211\u5c31\u8981\u5148\u7b97\u51fa\u5b50\u95ee\u9898 f(18)<\/code> \u548c f(17)<\/code>\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002\u6700\u540e\u9047\u5230 f(1)<\/code> \u6216\u8005 f(2)<\/code> \u7684\u65f6\u5019\uff0c\u7ed3\u679c\u5df2\u77e5\uff0c\u5c31\u80fd\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\uff0c\u9012\u5f52\u6811\u4e0d\u518d\u5411\u4e0b\u751f\u957f\u4e86\u3002<\/p>\n
\u9012\u5f52\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u600e\u4e48\u8ba1\u7b97\uff1f\u5b50\u95ee\u9898\u4e2a\u6570\u4e58\u4ee5\u89e3\u51b3\u4e00\u4e2a\u5b50\u95ee\u9898\u9700\u8981\u7684\u65f6\u95f4\u3002<\/strong><\/p>\n
\u5b50\u95ee\u9898\u4e2a\u6570\uff0c\u5373\u9012\u5f52\u6811\u4e2d\u8282\u70b9\u7684\u603b\u6570\u3002\u663e\u7136\u4e8c\u53c9\u6811\u8282\u70b9\u603b\u6570\u4e3a\u6307\u6570\u7ea7\u522b\uff0c\u6240\u4ee5\u5b50\u95ee\u9898\u4e2a\u6570\u4e3a O(2^n)\u3002<\/p>\n
\u89e3\u51b3\u4e00\u4e2a\u5b50\u95ee\u9898\u7684\u65f6\u95f4\uff0c\u5728\u672c\u7b97\u6cd5\u4e2d\uff0c\u6ca1\u6709\u5faa\u73af\uff0c\u53ea\u6709 f(n – 1) + f(n – 2) \u4e00\u4e2a\u52a0\u6cd5\u64cd\u4f5c\uff0c\u65f6\u95f4\u4e3a O(1)\u3002<\/p>\n
\u6240\u4ee5\uff0c\u8fd9\u4e2a\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(2^n)\uff0c\u6307\u6570\u7ea7\u522b\uff0c\u7206\u70b8\u3002<\/p>\n
\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5f88\u660e\u663e\u53d1\u73b0\u4e86\u7b97\u6cd5\u4f4e\u6548\u7684\u539f\u56e0\uff1a\u5b58\u5728\u5927\u91cf\u91cd\u590d\u8ba1\u7b97\uff0c\u6bd4\u5982 f(18)<\/code> \u88ab\u8ba1\u7b97\u4e86\u4e24\u6b21\uff0c\u800c\u4e14\u4f60\u53ef\u4ee5\u770b\u5230\uff0c\u4ee5 f(18)<\/code> \u4e3a\u6839\u7684\u8fd9\u4e2a\u9012\u5f52\u6811\u4f53\u91cf\u5de8\u5927\uff0c\u591a\u7b97\u4e00\u904d\uff0c\u4f1a\u8017\u8d39\u5de8\u5927\u7684\u65f6\u95f4\u3002\u66f4\u4f55\u51b5\uff0c\u8fd8\u4e0d\u6b62 f(18)<\/code> \u8fd9\u4e00\u4e2a\u8282\u70b9\u88ab\u91cd\u590d\u8ba1\u7b97\uff0c\u6240\u4ee5\u8fd9\u4e2a\u7b97\u6cd5\u53ca\u5176\u4f4e\u6548\u3002<\/p>\n
\u8fd9\u5c31\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u7b2c\u4e00\u4e2a\u6027\u8d28\uff1a\u91cd\u53e0\u5b50\u95ee\u9898<\/strong>\u3002\u4e0b\u9762\uff0c\u6211\u4eec\u60f3\u529e\u6cd5\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002<\/p>\n
2\u3001\u5e26\u5907\u5fd8\u5f55\u7684\u9012\u5f52\u89e3\u6cd5<\/strong><\/p>\n
\u660e\u786e\u4e86\u95ee\u9898\uff0c\u5176\u5b9e\u5c31\u5df2\u7ecf\u628a\u95ee\u9898\u89e3\u51b3\u4e86\u4e00\u534a\u3002\u5373\u7136\u8017\u65f6\u7684\u539f\u56e0\u662f\u91cd\u590d\u8ba1\u7b97\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u9020\u4e00\u4e2a\u300c\u5907\u5fd8\u5f55\u300d\uff0c\u6bcf\u6b21\u7b97\u51fa\u67d0\u4e2a\u5b50\u95ee\u9898\u7684\u7b54\u6848\u540e\u522b\u6025\u7740\u8fd4\u56de\uff0c\u5148\u8bb0\u5230\u300c\u5907\u5fd8\u5f55\u300d\u91cc\u518d\u8fd4\u56de\uff1b\u6bcf\u6b21\u9047\u5230\u4e00\u4e2a\u5b50\u95ee\u9898\u5148\u53bb\u300c\u5907\u5fd8\u5f55\u300d\u91cc\u67e5\u4e00\u67e5\uff0c\u5982\u679c\u53d1\u73b0\u4e4b\u524d\u5df2\u7ecf\u89e3\u51b3\u8fc7\u8fd9\u4e2a\u95ee\u9898\u4e86\uff0c\u76f4\u63a5\u628a\u7b54\u6848\u62ff\u51fa\u6765\u7528\uff0c\u4e0d\u8981\u518d\u8017\u65f6\u53bb\u8ba1\u7b97\u4e86\u3002<\/p>\n
\u4e00\u822c\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4\u5145\u5f53\u8fd9\u4e2a\u300c\u5907\u5fd8\u5f55\u300d\uff0c\u5f53\u7136\u4f60\u4e5f\u53ef\u4ee5\u4f7f\u7528\u54c8\u5e0c\u8868\uff08\u5b57\u5178\uff09\uff0c\u601d\u60f3\u90fd\u662f\u4e00\u6837\u7684\u3002<\/p>\n
int fib(int N) {     if (N < 1) return 0;     \/\/ \u5907\u5fd8\u5f55\u5168\u521d\u59cb\u5316\u4e3a 0     vector<int> memo(N + 1, 0);     \/\/ \u521d\u59cb\u5316\u6700\u7b80\u60c5\u51b5     return helper(memo, N); }   int helper(vector<int>& memo, int n) {     \/\/ base case      if (n == 1 || n == 2) return 1;     \/\/ \u5df2\u7ecf\u8ba1\u7b97\u8fc7     if (memo[n] != 0) return memo[n];     memo[n] = helper(memo, n - 1) +                  helper(memo, n - 2);     return memo[n]; } <\/code><\/pre>\n
\u73b0\u5728\uff0c\u753b\u51fa\u9012\u5f52\u6811\uff0c\u4f60\u5c31\u77e5\u9053\u300c\u5907\u5fd8\u5f55\u300d\u5230\u5e95\u505a\u4e86\u4ec0\u4e48\u3002<\/p>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n
\u5b9e\u9645\u4e0a\uff0c\u5e26\u300c\u5907\u5fd8\u5f55\u300d\u7684\u9012\u5f52\u7b97\u6cd5\uff0c\u628a\u4e00\u68f5\u5b58\u5728\u5de8\u91cf\u5197\u4f59\u7684\u9012\u5f52\u6811\u901a\u8fc7\u300c\u526a\u679d\u300d\uff0c\u6539\u9020\u6210\u4e86\u4e00\u5e45\u4e0d\u5b58\u5728\u5197\u4f59\u7684\u9012\u5f52\u56fe\uff0c\u6781\u5927\u51cf\u5c11\u4e86\u5b50\u95ee\u9898\uff08\u5373\u9012\u5f52\u56fe\u4e2d\u8282\u70b9\uff09\u7684\u4e2a\u6570\u3002<\/p>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n
\u9012\u5f52\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u600e\u4e48\u7b97\uff1f\u5b50\u95ee\u9898\u4e2a\u6570\u4e58\u4ee5\u89e3\u51b3\u4e00\u4e2a\u5b50\u95ee\u9898\u9700\u8981\u7684\u65f6\u95f4\u3002<\/p>\n
\u5b50\u95ee\u9898\u4e2a\u6570\uff0c\u5373\u56fe\u4e2d\u8282\u70b9\u7684\u603b\u6570\uff0c\u7531\u4e8e\u672c\u7b97\u6cd5\u4e0d\u5b58\u5728\u5197\u4f59\u8ba1\u7b97\uff0c\u5b50\u95ee\u9898\u5c31\u662f f(1)<\/code>, f(2)<\/code>, f(3)<\/code> … f(20)<\/code>\uff0c\u6570\u91cf\u548c\u8f93\u5165\u89c4\u6a21 n = 20 \u6210\u6b63\u6bd4\uff0c\u6240\u4ee5\u5b50\u95ee\u9898\u4e2a\u6570\u4e3a O(n)\u3002<\/p>\n
\u89e3\u51b3\u4e00\u4e2a\u5b50\u95ee\u9898\u7684\u65f6\u95f4\uff0c\u540c\u4e0a\uff0c\u6ca1\u6709\u4ec0\u4e48\u5faa\u73af\uff0c\u65f6\u95f4\u4e3a O(1)\u3002<\/p>\n
\u6240\u4ee5\uff0c\u672c\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f O(n)\u3002\u6bd4\u8d77\u66b4\u529b\u7b97\u6cd5\uff0c\u662f\u964d\u7ef4\u6253\u51fb\u3002<\/p>\n
\u81f3\u6b64\uff0c\u5e26\u5907\u5fd8\u5f55\u7684\u9012\u5f52\u89e3\u6cd5\u7684\u6548\u7387\u5df2\u7ecf\u548c\u8fed\u4ee3\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u4e00\u6837\u4e86\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u89e3\u6cd5\u548c\u8fed\u4ee3\u7684\u52a8\u6001\u89c4\u5212\u5df2\u7ecf\u5dee\u4e0d\u591a\u4e86\uff0c\u53ea\u4e0d\u8fc7\u8fd9\u79cd\u65b9\u6cd5\u53eb\u505a\u300c\u81ea\u9876\u5411\u4e0b\u300d\uff0c\u52a8\u6001\u89c4\u5212\u53eb\u505a\u300c\u81ea\u5e95\u5411\u4e0a\u300d\u3002<\/p>\n
\u5565\u53eb\u300c\u81ea\u9876\u5411\u4e0b\u300d\uff1f\u6ce8\u610f\u6211\u4eec\u521a\u624d\u753b\u7684\u9012\u5f52\u6811\uff08\u6216\u8005\u8bf4\u56fe\uff09\uff0c\u662f\u4ece\u4e0a\u5411\u4e0b\u5ef6\u4f38\uff0c\u90fd\u662f\u4ece\u4e00\u4e2a\u89c4\u6a21\u8f83\u5927\u7684\u539f\u95ee\u9898\u6bd4\u5982\u8bf4 f(20)<\/code>\uff0c\u5411\u4e0b\u9010\u6e10\u5206\u89e3\u89c4\u6a21\uff0c\u76f4\u5230 f(1)<\/code> \u548c f(2)<\/code> \u89e6\u5e95\uff0c\u7136\u540e\u9010\u5c42\u8fd4\u56de\u7b54\u6848\uff0c\u8fd9\u5c31\u53eb\u300c\u81ea\u9876\u5411\u4e0b\u300d\u3002<\/p>\n
\u5565\u53eb\u300c\u81ea\u5e95\u5411\u4e0a\u300d\uff1f\u53cd\u8fc7\u6765\uff0c\u6211\u4eec\u76f4\u63a5\u4ece\u6700\u5e95\u4e0b\uff0c\u6700\u7b80\u5355\uff0c\u95ee\u9898\u89c4\u6a21\u6700\u5c0f\u7684 f(1)<\/code> \u548c f(2)<\/code> \u5f00\u59cb\u5f80\u4e0a\u63a8\uff0c\u76f4\u5230\u63a8\u5230\u6211\u4eec\u60f3\u8981\u7684\u7b54\u6848 f(20)<\/code>\uff0c\u8fd9\u5c31\u662f\u52a8\u6001\u89c4\u5212\u7684\u601d\u8def\uff0c\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u52a8\u6001\u89c4\u5212\u4e00\u822c\u90fd\u8131\u79bb\u4e86\u9012\u5f52\uff0c\u800c\u662f\u7531\u5faa\u73af\u8fed\u4ee3\u5b8c\u6210\u8ba1\u7b97\u3002<\/p>\n
3\u3001dp \u6570\u7ec4\u7684\u8fed\u4ee3\u89e3\u6cd5<\/strong><\/p>\n
\u6709\u4e86\u4e0a\u4e00\u6b65\u300c\u5907\u5fd8\u5f55\u300d\u7684\u542f\u53d1\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u8fd9\u4e2a\u300c\u5907\u5fd8\u5f55\u300d\u72ec\u7acb\u51fa\u6765\u6210\u4e3a\u4e00\u5f20\u8868\uff0c\u5c31\u53eb\u505a DP table \u5427\uff0c\u5728\u8fd9\u5f20\u8868\u4e0a\u5b8c\u6210\u300c\u81ea\u5e95\u5411\u4e0a\u300d\u7684\u63a8\u7b97\u5c82\u4e0d\u7f8e\u54c9\uff01<\/p>\n
int fib(int N) {     vector<int> dp(N + 1, 0);     \/\/ base case     dp[1] = dp[2] = 1;     for (int i = 3; i <= N; i++)         dp[i] = dp[i - 1] + dp[i - 2];     return dp[N]; } <\/code><\/pre>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n
\u753b\u4e2a\u56fe\u5c31\u5f88\u597d\u7406\u89e3\u4e86\uff0c\u800c\u4e14\u4f60\u53d1\u73b0\u8fd9\u4e2a DP table \u7279\u522b\u50cf\u4e4b\u524d\u90a3\u4e2a\u300c\u526a\u679d\u300d\u540e\u7684\u7ed3\u679c\uff0c\u53ea\u662f\u53cd\u8fc7\u6765\u7b97\u800c\u5df2\u3002\u5b9e\u9645\u4e0a\uff0c\u5e26\u5907\u5fd8\u5f55\u7684\u9012\u5f52\u89e3\u6cd5\u4e2d\u7684\u300c\u5907\u5fd8\u5f55\u300d\uff0c\u6700\u7ec8\u5b8c\u6210\u540e\u5c31\u662f\u8fd9\u4e2a DP table\uff0c\u6240\u4ee5\u8bf4\u8fd9\u4e24\u79cd\u89e3\u6cd5\u5176\u5b9e\u662f\u5dee\u4e0d\u591a\u7684\uff0c\u5927\u90e8\u5206\u60c5\u51b5\u4e0b\uff0c\u6548\u7387\u4e5f\u57fa\u672c\u76f8\u540c\u3002<\/p>\n
\u8fd9\u91cc\uff0c\u5f15\u51fa\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u8fd9\u4e2a\u540d\u8bcd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u63cf\u8ff0\u95ee\u9898\u7ed3\u6784\u7684\u6570\u5b66\u5f62\u5f0f\uff1a<\/p>\n
$$f(n) = begin{cases} \u200b 1, n = 1, 2  \u200b f(n – 1) + f(n – 2), n > 2 end{cases}$$<\/p>\n
\u4e3a\u5565\u53eb\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\uff1f\u4e3a\u4e86\u542c\u8d77\u6765\u9ad8\u7aef\u3002\u4f60\u628a f(n) \u60f3\u505a\u4e00\u4e2a\u72b6\u6001 n\uff0c\u8fd9\u4e2a\u72b6\u6001 n \u662f\u7531\u72b6\u6001 n – 1 \u548c\u72b6\u6001 n – 2 \u76f8\u52a0\u8f6c\u79fb\u800c\u6765\uff0c\u8fd9\u5c31\u53eb\u72b6\u6001\u8f6c\u79fb\uff0c\u4ec5\u6b64\u800c\u5df2\u3002<\/p>\n
\u4f60\u4f1a\u53d1\u73b0\uff0c\u4e0a\u9762\u7684\u51e0\u79cd\u89e3\u6cd5\u4e2d\u7684\u6240\u6709\u64cd\u4f5c\uff0c\u4f8b\u5982 return f(n – 1) + f(n – 2)\uff0cdp[i] = dp[i – 1] + dp[i – 2]\uff0c\u4ee5\u53ca\u5bf9\u5907\u5fd8\u5f55\u6216 DP table \u7684\u521d\u59cb\u5316\u64cd\u4f5c\uff0c\u90fd\u662f\u56f4\u7ed5\u8fd9\u4e2a\u65b9\u7a0b\u5f0f\u7684\u4e0d\u540c\u8868\u73b0\u5f62\u5f0f\u3002\u53ef\u89c1\u5217\u51fa\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u7684\u91cd\u8981\u6027\uff0c\u5b83\u662f\u89e3\u51b3\u95ee\u9898\u7684\u6838\u5fc3\u3002\u5f88\u5bb9\u6613\u53d1\u73b0\uff0c\u5176\u5b9e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f4\u63a5\u4ee3\u8868\u7740\u66b4\u529b\u89e3\u6cd5\u3002<\/p>\n
\u5343\u4e07\u4e0d\u8981\u770b\u4e0d\u8d77\u66b4\u529b\u89e3\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u6700\u56f0\u96be\u7684\u5c31\u662f\u5199\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff0c\u5373\u8fd9\u4e2a\u66b4\u529b\u89e3\u3002\u4f18\u5316\u65b9\u6cd5\u65e0\u975e\u662f\u7528\u5907\u5fd8\u5f55\u6216\u8005 DP table\uff0c\u518d\u65e0\u5965\u5999\u53ef\u8a00\u3002<\/p>\n
\u8fd9\u4e2a\u4f8b\u5b50\u7684\u6700\u540e\uff0c\u8bb2\u4e00\u4e2a\u7ec6\u8282\u4f18\u5316\u3002\u7ec6\u5fc3\u7684\u8bfb\u8005\u4f1a\u53d1\u73b0\uff0c\u6839\u636e\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u5f53\u524d\u72b6\u6001\u53ea\u548c\u4e4b\u524d\u7684\u4e24\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u5176\u5b9e\u5e76\u4e0d\u9700\u8981\u90a3\u4e48\u957f\u7684\u4e00\u4e2a DP table \u6765\u5b58\u50a8\u6240\u6709\u7684\u72b6\u6001\uff0c\u53ea\u8981\u60f3\u529e\u6cd5\u5b58\u50a8\u4e4b\u524d\u7684\u4e24\u4e2a\u72b6\u6001\u5c31\u884c\u4e86\u3002\u6240\u4ee5\uff0c\u53ef\u4ee5\u8fdb\u4e00\u6b65\u4f18\u5316\uff0c\u628a\u7a7a\u95f4\u590d\u6742\u5ea6\u964d\u4e3a O(1)\uff1a<\/p>\n
int fib(int n) {     if (n == 2 || n == 1)          return 1;     int prev = 1, curr = 1;     for (int i = 3; i <= n; i++) {         int sum = prev + curr;         prev = curr;         curr = sum;     }     return curr; } <\/code><\/pre>\n
\u6709\u4eba\u4f1a\u95ee\uff0c\u52a8\u6001\u89c4\u5212\u7684\u53e6\u4e00\u4e2a\u91cd\u8981\u7279\u6027\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d\uff0c\u600e\u4e48\u6ca1\u6709\u6d89\u53ca\uff1f\u4e0b\u9762\u4f1a\u6d89\u53ca\u3002\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u4f8b\u5b50\u4e25\u683c\u6765\u8bf4\u4e0d\u7b97\u52a8\u6001\u89c4\u5212\uff0c\u56e0\u4e3a\u6ca1\u6709\u6d89\u53ca\u6c42\u6700\u503c\uff0c\u4ee5\u4e0a\u65e8\u5728\u6f14\u793a\u7b97\u6cd5\u8bbe\u8ba1\u87ba\u65cb\u4e0a\u5347\u7684\u8fc7\u7a0b\u3002\u4e0b\u9762\uff0c\u770b\u7b2c\u4e8c\u4e2a\u4f8b\u5b50\uff0c\u51d1\u96f6\u94b1\u95ee\u9898\u3002<\/p>\n
\u4e8c\u3001\u51d1\u96f6\u94b1\u95ee\u9898<\/h3>\n
\u5148\u770b\u4e0b\u9898\u76ee\uff1a\u7ed9\u4f60 k<\/code> \u79cd\u9762\u503c\u7684\u786c\u5e01\uff0c\u9762\u503c\u5206\u522b\u4e3a c1, c2 ... ck<\/code>\uff0c\u6bcf\u79cd\u786c\u5e01\u7684\u6570\u91cf\u65e0\u9650\uff0c\u518d\u7ed9\u4e00\u4e2a\u603b\u91d1\u989d amount<\/code>\uff0c\u95ee\u4f60\u6700\u5c11<\/strong>\u9700\u8981\u51e0\u679a\u786c\u5e01\u51d1\u51fa\u8fd9\u4e2a\u91d1\u989d\uff0c\u5982\u679c\u4e0d\u53ef\u80fd\u51d1\u51fa\uff0c\u7b97\u6cd5\u8fd4\u56de -1\u3002\u7b97\u6cd5\u7684\u51fd\u6570\u7b7e\u540d\u5982\u4e0b\uff1a<\/p>\n
\/\/ coins \u4e2d\u662f\u53ef\u9009\u786c\u5e01\u9762\u503c\uff0camount \u662f\u76ee\u6807\u91d1\u989d int coinChange(int[] coins, int amount); <\/code><\/pre>\n
\u6bd4\u5982\u8bf4 k = 3<\/code>\uff0c\u9762\u503c\u5206\u522b\u4e3a 1\uff0c2\uff0c5\uff0c\u603b\u91d1\u989d amount = 11<\/code>\u3002\u90a3\u4e48\u6700\u5c11\u9700\u8981 3 \u679a\u786c\u5e01\u51d1\u51fa\uff0c\u5373 11 = 5 + 5 + 1\u3002<\/p>\n
\u4f60\u8ba4\u4e3a\u8ba1\u7b97\u673a\u5e94\u8be5\u5982\u4f55\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff1f\u663e\u7136\uff0c\u5c31\u662f\u628a\u6240\u6709\u80af\u80fd\u7684\u51d1\u786c\u5e01\u65b9\u6cd5\u90fd\u7a77\u4e3e\u51fa\u6765\uff0c\u7136\u540e\u627e\u627e\u770b\u6700\u5c11\u9700\u8981\u591a\u5c11\u679a\u786c\u5e01\u3002<\/p>\n
1\u3001\u66b4\u529b\u9012\u5f52<\/strong><\/p>\n
\u9996\u5148\uff0c\u8fd9\u4e2a\u95ee\u9898\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u5177\u6709\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d\u7684\u3002\u8981\u7b26\u5408\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d\uff0c\u5b50\u95ee\u9898\u95f4\u5fc5\u987b\u4e92\u76f8\u72ec\u7acb<\/strong>\u3002\u5565\u53eb\u76f8\u4e92\u72ec\u7acb\uff1f\u4f60\u80af\u5b9a\u4e0d\u60f3\u770b\u6570\u5b66\u8bc1\u660e\uff0c\u6211\u7528\u4e00\u4e2a\u76f4\u89c2\u7684\u4f8b\u5b50\u6765\u8bb2\u89e3\u3002<\/p>\n
\u6bd4\u5982\u8bf4\uff0c\u4f60\u7684\u539f\u95ee\u9898\u662f\u8003\u51fa\u6700\u9ad8\u7684\u603b\u6210\u7ee9\uff0c\u90a3\u4e48\u4f60\u7684\u5b50\u95ee\u9898\u5c31\u662f\u8981\u628a\u8bed\u6587\u8003\u5230\u6700\u9ad8\uff0c\u6570\u5b66\u8003\u5230\u6700\u9ad8\u2026\u2026 \u4e3a\u4e86\u6bcf\u95e8\u8bfe\u8003\u5230\u6700\u9ad8\uff0c\u4f60\u8981\u628a\u6bcf\u95e8\u8bfe\u76f8\u5e94\u7684\u9009\u62e9\u9898\u5206\u6570\u62ff\u5230\u6700\u9ad8\uff0c\u586b\u7a7a\u9898\u5206\u6570\u62ff\u5230\u6700\u9ad8\u2026\u2026 \u5f53\u7136\uff0c\u6700\u7ec8\u5c31\u662f\u4f60\u6bcf\u95e8\u8bfe\u90fd\u662f\u6ee1\u5206\uff0c\u8fd9\u5c31\u662f\u6700\u9ad8\u7684\u603b\u6210\u7ee9\u3002<\/p>\n
\u5f97\u5230\u4e86\u6b63\u786e\u7684\u7ed3\u679c\uff1a\u6700\u9ad8\u7684\u603b\u6210\u7ee9\u5c31\u662f\u603b\u5206\u3002\u56e0\u4e3a\u8fd9\u4e2a\u8fc7\u7a0b\u7b26\u5408\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u201c\u6bcf\u95e8\u79d1\u76ee\u8003\u5230\u6700\u9ad8\u201d\u8fd9\u4e9b\u5b50\u95ee\u9898\u662f\u4e92\u76f8\u72ec\u7acb\uff0c\u4e92\u4e0d\u5e72\u6270\u7684\u3002<\/p>\n
\u4f46\u662f\uff0c\u5982\u679c\u52a0\u4e00\u4e2a\u6761\u4ef6\uff1a\u4f60\u7684\u8bed\u6587\u6210\u7ee9\u548c\u6570\u5b66\u6210\u7ee9\u4f1a\u4e92\u76f8\u5236\u7ea6\uff0c\u6b64\u6d88\u5f7c\u957f\u3002\u8fd9\u6837\u7684\u8bdd\uff0c\u663e\u7136\u4f60\u80fd\u8003\u5230\u7684\u6700\u9ad8\u603b\u6210\u7ee9\u5c31\u8fbe\u4e0d\u5230\u603b\u5206\u4e86\uff0c\u6309\u521a\u624d\u90a3\u4e2a\u601d\u8def\u5c31\u4f1a\u5f97\u5230\u9519\u8bef\u7684\u7ed3\u679c\u3002\u56e0\u4e3a\u5b50\u95ee\u9898\u5e76\u4e0d\u72ec\u7acb\uff0c\u8bed\u6587\u6570\u5b66\u6210\u7ee9\u65e0\u6cd5\u540c\u65f6\u6700\u4f18\uff0c\u6240\u4ee5\u6700\u4f18\u5b50\u7ed3\u6784\u88ab\u7834\u574f\u3002<\/p>\n
\u56de\u5230\u51d1\u96f6\u94b1\u95ee\u9898\uff0c\u4e3a\u4ec0\u4e48\u8bf4\u5b83\u7b26\u5408\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u6bd4\u5982\u4f60\u60f3\u6c42 amount = 11<\/code> \u65f6\u7684\u6700\u5c11\u786c\u5e01\u6570\uff08\u539f\u95ee\u9898\uff09\uff0c\u5982\u679c\u4f60\u77e5\u9053\u51d1\u51fa amount = 10<\/code> \u7684\u6700\u5c11\u786c\u5e01\u6570\uff08\u5b50\u95ee\u9898\uff09\uff0c\u4f60\u53ea\u9700\u8981\u628a\u5b50\u95ee\u9898\u7684\u7b54\u6848\u52a0\u4e00\uff08\u518d\u9009\u4e00\u679a\u9762\u503c\u4e3a 1 \u7684\u786c\u5e01\uff09\u5c31\u662f\u539f\u95ee\u9898\u7684\u7b54\u6848\uff0c\u56e0\u4e3a\u786c\u5e01\u7684\u6570\u91cf\u662f\u6ca1\u6709\u9650\u5236\u7684\uff0c\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u76f8\u4e92\u5236\uff0c\u662f\u4e92\u76f8\u72ec\u7acb\u7684\u3002<\/p>\n
\u90a3\u4e48\uff0c\u65e2\u7136\u77e5\u9053\u4e86\u8fd9\u662f\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5c31\u8981\u601d\u8003\u5982\u4f55\u5217\u51fa\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/strong>\uff1f<\/p>\n
\u5148\u786e\u5b9a\u300c\u72b6\u6001\u300d<\/strong>\uff0c\u4e5f\u5c31\u662f\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u4e2d\u53d8\u5316\u7684\u53d8\u91cf\u3002\u7531\u4e8e\u786c\u5e01\u6570\u91cf\u65e0\u9650\uff0c\u6240\u4ee5\u552f\u4e00\u7684\u72b6\u6001\u5c31\u662f\u76ee\u6807\u91d1\u989d amount<\/code>\u3002<\/p>\n
\u7136\u540e\u786e\u5b9a dp<\/code> \u51fd\u6570\u7684\u5b9a\u4e49<\/strong>\uff1a\u5f53\u524d\u7684\u76ee\u6807\u91d1\u989d\u662f n<\/code>\uff0c\u81f3\u5c11\u9700\u8981 dp(n)<\/code> \u4e2a\u786c\u5e01\u51d1\u51fa\u8be5\u91d1\u989d\u3002<\/p>\n
\u7136\u540e\u786e\u5b9a\u300c\u9009\u62e9\u300d\u5e76\u62e9\u4f18<\/strong>\uff0c\u4e5f\u5c31\u662f\u5bf9\u4e8e\u6bcf\u4e2a\u72b6\u6001\uff0c\u53ef\u4ee5\u505a\u51fa\u4ec0\u4e48\u9009\u62e9\u6539\u53d8\u5f53\u524d\u72b6\u6001\u3002\u5177\u4f53\u5230\u8fd9\u4e2a\u95ee\u9898\uff0c\u65e0\u8bba\u5f53\u7684\u76ee\u6807\u91d1\u989d\u662f\u591a\u5c11\uff0c\u9009\u62e9\u5c31\u662f\u4ece\u9762\u989d\u5217\u8868 coins<\/code> \u4e2d\u9009\u62e9\u4e00\u4e2a\u786c\u5e01\uff0c\u7136\u540e\u76ee\u6807\u91d1\u989d\u5c31\u4f1a\u51cf\u5c11\uff1a<\/p>\n
# \u4f2a\u7801\u6846\u67b6 def coinChange(coins: List[int], amount: int):     # \u5b9a\u4e49\uff1a\u8981\u51d1\u51fa\u91d1\u989d n\uff0c\u81f3\u5c11\u8981 dp(n) \u4e2a\u786c\u5e01     def dp(n):         # \u505a\u9009\u62e9\uff0c\u9009\u62e9\u9700\u8981\u786c\u5e01\u6700\u5c11\u7684\u90a3\u4e2a\u7ed3\u679c         for coin in coins:             res = min(res, 1 + dp(n - coin))         return res     # \u6211\u4eec\u8981\u6c42\u7684\u95ee\u9898\u662f dp(amount)     return dp(amount) <\/code><\/pre>\n
\u6700\u540e\u660e\u786e base case<\/strong>\uff0c\u663e\u7136\u76ee\u6807\u91d1\u989d\u4e3a 0 \u65f6\uff0c\u6240\u9700\u786c\u5e01\u6570\u91cf\u4e3a 0 \uff1b\u5f53\u76ee\u6807\u91d1\u989d\u5c0f\u4e8e 0 \u65f6\uff0c\u65e0\u89e3\uff0c\u8fd4\u56de -1\uff1a<\/p>\n
def coinChange(coins: List[int], amount: int):      def dp(n):         # base case         if n == 0: return 0         if n < 0: return -1         # \u6c42\u6700\u5c0f\u503c\uff0c\u6240\u4ee5\u521d\u59cb\u5316\u4e3a\u6b63\u65e0\u7a77         res = float('INF')         for coin in coins:             subproblem = dp(n - coin)             # \u5b50\u95ee\u9898\u65e0\u89e3\uff0c\u8df3\u8fc7             if subproblem == -1: continue             res = min(res, 1 + subproblem)          return res if res != float('INF') else -1          return dp(amount) <\/code><\/pre>\n
\u81f3\u6b64\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5176\u5b9e\u5df2\u7ecf\u5b8c\u6210\u4e86\uff0c\u4ee5\u4e0a\u7b97\u6cd5\u5df2\u7ecf\u662f\u66b4\u529b\u89e3\u6cd5\u4e86\uff0c\u4ee5\u4e0a\u4ee3\u7801\u7684\u6570\u5b66\u5f62\u5f0f\u5c31\u662f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a<\/p>\n
$$ dp(n) = begin{cases} \u200b 0, n = 0  \u200b -1, n < 0 \u200b min{dp(n – coin) + 1 | coin in coins}, n > 0 end{cases}$$<\/p>\n
\u81f3\u6b64\uff0c\u8fd9\u4e2a\u95ee\u9898\u5176\u5b9e\u5c31\u89e3\u51b3\u4e86\uff0c\u53ea\u4e0d\u8fc7\u9700\u8981\u6d88\u9664\u4e00\u4e0b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6bd4\u5982 amount = 11, coins = {1,2,5}<\/code> \u65f6\u753b\u51fa\u9012\u5f52\u6811\u770b\u770b\uff1a<\/p>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n
\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\uff1a\u5b50\u95ee\u9898\u603b\u6570 x \u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u65f6\u95f4<\/strong>\u3002<\/p>\n
\u5b50\u95ee\u9898\u603b\u6570\u4e3a\u9012\u5f52\u6811\u8282\u70b9\u4e2a\u6570\uff0c\u8fd9\u4e2a\u6bd4\u8f83\u96be\u770b\u51fa\u6765\uff0c\u662f O(n^k)\uff0c\u603b\u4e4b\u662f\u6307\u6570\u7ea7\u522b\u7684\u3002\u6bcf\u4e2a\u5b50\u95ee\u9898\u4e2d\u542b\u6709\u4e00\u4e2a for \u5faa\u73af\uff0c\u590d\u6742\u5ea6\u4e3a O(k)\u3002\u6240\u4ee5\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(k * n^k)\uff0c\u6307\u6570\u7ea7\u522b\u3002<\/p>\n
2\u3001\u5e26\u5907\u5fd8\u5f55\u7684\u9012\u5f52<\/strong><\/p>\n
\u53ea\u9700\u8981\u7a0d\u52a0\u4fee\u6539\uff0c\u5c31\u53ef\u4ee5\u901a\u8fc7\u5907\u5fd8\u5f55\u6d88\u9664\u5b50\u95ee\u9898\uff1a<\/p>\n
def coinChange(coins: List[int], amount: int):     # \u5907\u5fd8\u5f55     memo = dict()     def dp(n):         # \u67e5\u5907\u5fd8\u5f55\uff0c\u907f\u514d\u91cd\u590d\u8ba1\u7b97         if n in memo: return memo[n]          if n == 0: return 0         if n < 0: return -1         res = float('INF')         for coin in coins:             subproblem = dp(n - coin)             if subproblem == -1: continue             res = min(res, 1 + subproblem)                  # \u8bb0\u5165\u5907\u5fd8\u5f55         memo[n] = res if res != float('INF') else -1         return memo[n]          return dp(amount) <\/code><\/pre>\n
\u4e0d\u753b\u56fe\u4e86\uff0c\u5f88\u663e\u7136\u300c\u5907\u5fd8\u5f55\u300d\u5927\u5927\u51cf\u5c0f\u4e86\u5b50\u95ee\u9898\u6570\u76ee\uff0c\u5b8c\u5168\u6d88\u9664\u4e86\u5b50\u95ee\u9898\u7684\u5197\u4f59\uff0c\u6240\u4ee5\u5b50\u95ee\u9898\u603b\u6570\u4e0d\u4f1a\u8d85\u8fc7\u91d1\u989d\u6570 n\uff0c\u5373\u5b50\u95ee\u9898\u6570\u76ee\u4e3a O(n)\u3002\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\u7684\u65f6\u95f4\u4e0d\u53d8\uff0c\u4ecd\u662f O(k)\uff0c\u6240\u4ee5\u603b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f O(kn)\u3002<\/p>\n
3\u3001dp \u6570\u7ec4\u7684\u8fed\u4ee3\u89e3\u6cd5<\/strong><\/p>\n
\u5f53\u7136\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u81ea\u5e95\u5411\u4e0a\u4f7f\u7528 dp table \u6765\u6d88\u9664\u91cd\u53e0\u5b50\u95ee\u9898\uff0cdp<\/code> \u6570\u7ec4\u7684\u5b9a\u4e49\u548c\u521a\u624d dp<\/code> \u51fd\u6570\u7c7b\u4f3c\uff0c\u5b9a\u4e49\u4e5f\u662f\u4e00\u6837\u7684\uff1a<\/p>\n
dp[i] = x<\/code> \u8868\u793a\uff0c\u5f53\u76ee\u6807\u91d1\u989d\u4e3a i<\/code> \u65f6\uff0c\u81f3\u5c11\u9700\u8981 x<\/code> \u679a\u786c\u5e01<\/strong>\u3002<\/p>\n
int coinChange(vector<int>& coins, int amount) {     \/\/ \u6570\u7ec4\u5927\u5c0f\u4e3a amount + 1\uff0c\u521d\u59cb\u503c\u4e5f\u4e3a amount + 1     vector<int> dp(amount + 1, amount + 1);     \/\/ base case     dp[0] = 0;     for (int i = 0; i < dp.size(); i++) {         \/\/ \u5185\u5c42 for \u5728\u6c42\u6240\u6709\u5b50\u95ee\u9898 + 1 \u7684\u6700\u5c0f\u503c         for (int coin : coins) {             \/\/ \u5b50\u95ee\u9898\u65e0\u89e3\uff0c\u8df3\u8fc7             if (i - coin < 0) continue;             dp[i] = min(dp[i], 1 + dp[i - coin]);         }     }     return (dp[amount] == amount + 1) ? -1 : dp[amount]; } <\/code><\/pre>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n
PS\uff1a\u4e3a\u5565 dp<\/code> \u6570\u7ec4\u521d\u59cb\u5316\u4e3a amount + 1<\/code> \u5462\uff0c\u56e0\u4e3a\u51d1\u6210 amount<\/code> \u91d1\u989d\u7684\u786c\u5e01\u6570\u6700\u591a\u53ea\u53ef\u80fd\u7b49\u4e8e amount<\/code>\uff08\u5168\u7528 1 \u5143\u9762\u503c\u7684\u786c\u5e01\uff09\uff0c\u6240\u4ee5\u521d\u59cb\u5316\u4e3a amount + 1<\/code> \u5c31\u76f8\u5f53\u4e8e\u521d\u59cb\u5316\u4e3a\u6b63\u65e0\u7a77\uff0c\u4fbf\u4e8e\u540e\u7eed\u53d6\u6700\u5c0f\u503c\u3002<\/p>\n
\u4e09\u3001\u6700\u540e\u603b\u7ed3<\/h3>\n
\u7b2c\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u95ee\u9898\uff0c\u89e3\u91ca\u4e86\u5982\u4f55\u901a\u8fc7\u300c\u5907\u5fd8\u5f55\u300d\u6216\u8005\u300c dp table \u300d\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u9012\u5f52\u6811\uff0c\u5e76\u4e14\u660e\u786e\u4e86\u8fd9\u4e24\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u6837\u7684\uff0c\u53ea\u662f\u81ea\u9876\u5411\u4e0b\u548c\u81ea\u5e95\u5411\u4e0a\u7684\u4e0d\u540c\u800c\u5df2\u3002<\/p>\n
\u7b2c\u4e8c\u4e2a\u51d1\u96f6\u94b1\u7684\u95ee\u9898\uff0c\u5c55\u793a\u4e86\u5982\u4f55\u6d41\u7a0b\u5316\u786e\u5b9a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\uff0c\u53ea\u8981\u901a\u8fc7\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5199\u51fa\u66b4\u529b\u9012\u5f52\u89e3\uff0c\u5269\u4e0b\u7684\u4e5f\u5c31\u662f\u4f18\u5316\u9012\u5f52\u6811\uff0c\u6d88\u9664\u91cd\u53e0\u5b50\u95ee\u9898\u800c\u5df2\u3002<\/p>\n
\u5982\u679c\u4f60\u4e0d\u592a\u4e86\u89e3\u52a8\u6001\u89c4\u5212\uff0c\u8fd8\u80fd\u770b\u5230\u8fd9\u91cc\uff0c\u771f\u5f97\u7ed9\u4f60\u9f13\u638c\uff0c\u76f8\u4fe1\u4f60\u5df2\u7ecf\u638c\u63e1\u4e86\u8fd9\u4e2a\u7b97\u6cd5\u7684\u8bbe\u8ba1\u6280\u5de7\u3002<\/p>\n
\u8ba1\u7b97\u673a\u89e3\u51b3\u95ee\u9898\u5176\u5b9e\u6ca1\u6709\u4efb\u4f55\u5947\u6280\u6deb\u5de7\uff0c\u5b83\u552f\u4e00\u7684\u89e3\u51b3\u529e\u6cd5\u5c31\u662f\u7a77\u4e3e<\/strong>\uff0c\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u7b97\u6cd5\u8bbe\u8ba1\u65e0\u975e\u5c31\u662f\u5148\u601d\u8003\u201c\u5982\u4f55\u7a77\u4e3e\u201d\uff0c\u7136\u540e\u518d\u8ffd\u6c42\u201c\u5982\u4f55\u806a\u660e\u5730\u7a77\u4e3e\u201d\u3002<\/p>\n
\u5217\u51fa\u52a8\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u5c31\u662f\u5728\u89e3\u51b3\u201c\u5982\u4f55\u7a77\u4e3e\u201d\u7684\u95ee\u9898\u3002\u4e4b\u6240\u4ee5\u8bf4\u5b83\u96be\uff0c\u4e00\u662f\u56e0\u4e3a\u5f88\u591a\u7a77\u4e3e\u9700\u8981\u9012\u5f52\u5b9e\u73b0\uff0c\u4e8c\u662f\u56e0\u4e3a\u6709\u7684\u95ee\u9898\u672c\u8eab\u7684\u89e3\u7a7a\u95f4\u590d\u6742\uff0c\u4e0d\u90a3\u4e48\u5bb9\u6613\u7a77\u4e3e\u5b8c\u6574\u3002<\/p>\n
\u5907\u5fd8\u5f55\u3001DP table \u5c31\u662f\u5728\u8ffd\u6c42\u201c\u5982\u4f55\u806a\u660e\u5730\u7a77\u4e3e\u201d\u3002\u7528\u7a7a\u95f4\u6362\u65f6\u95f4\u7684\u601d\u8def\uff0c\u662f\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u4e0d\u4e8c\u6cd5\u95e8\uff0c\u9664\u6b64\u4e4b\u5916\uff0c\u8bd5\u95ee\uff0c\u8fd8\u80fd\u73a9\u51fa\u5565\u82b1\u6d3b\uff1f<\/p>\n
\u5982\u679c\u672c\u6587\u5bf9\u4f60\u6709\u5e2e\u52a9\uff0c\u6b22\u8fce\u5173\u6ce8\u6211\u7684\u516c\u4f17\u53f7 labuladong\uff0c\u81f4\u529b\u4e8e\u628a\u7b97\u6cd5\u95ee\u9898\u8bb2\u6e05\u695a\uff5e<\/p>\n
\"\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3\"<\/p>\n<\/p><\/div>\n
 \u5927\u4f6c\u6709\u8a71\u8aaa<\/b> (0<\/span>)        <\/div>\n
 <\/div>\n<\/p><\/div>\n<\/p><\/div>\n
    \n
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    \u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8be6\u89e3 \u8cc7\u6df1\u5927\u4f6c : f…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[],"tags":[],"_links":{"self":[{"href":"http:\/\/4563.org\/index.php?rest_route=\/wp\/v2\/posts\/103941"}],"collection":[{"href":"http:\/\/4563.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/4563.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/4563.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/4563.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=103941"}],"version-history":[{"count":0,"href":"http:\/\/4563.org\/index.php?rest_route=\/wp\/v2\/posts\/103941\/revisions"}],"wp:attachment":[{"href":"http:\/\/4563.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=103941"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/4563.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=103941"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/4563.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=103941"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}