{"id":97,"date":"2016-11-24T03:33:55","date_gmt":"2016-11-24T03:33:55","guid":{"rendered":"http:\/\/www.haoyuan.info\/?p=97"},"modified":"2017-01-30T16:39:31","modified_gmt":"2017-01-30T16:39:31","slug":"%ef%bc%88%e8%bd%ac%e8%bd%bd%ef%bc%89%e5%8a%a8%e6%80%81%e8%a7%84%e5%88%92%ef%bc%9a%e4%bb%8e%e6%96%b0%e6%89%8b%e5%88%b0%e4%b8%93%e5%ae%b6","status":"publish","type":"post","link":"https:\/\/haoyuan.info\/?p=97","title":{"rendered":"\uff08\u8f6c\u8f7d\uff09\u52a8\u6001\u89c4\u5212\uff1a\u4ece\u65b0\u624b\u5230\u4e13\u5bb6"},"content":{"rendered":"<p>\u4f5c\u8005\uff1aHawstein <a href=\"http:\/\/www.hawstein.com\/posts\/dp-novice-to-advanced.html\" target=\"_blank\">\u51fa\u5904\uff1ahttp:\/\/hawstein.com\/posts\/dp-novice-to-advanced.html <\/a><\/p>\n<p>\u58f0\u660e\uff1a\u672c\u6587\u91c7\u7528\u4ee5\u4e0b\u534f\u8bae\u8fdb\u884c\u6388\u6743\uff1a<a href=\"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/deed.zh\" target=\"_blank\"> \u81ea\u7531\u8f6c\u8f7d-\u975e\u5546\u7528-\u975e\u884d\u751f-\u4fdd\u6301\u7f72\u540d|Creative Commons BY-NC-ND 3.0 \uff0c<\/a><\/p>\n<p>\u8f6c\u8f7d\u8bf7\u6ce8\u660e\u4f5c\u8005\u53ca\u51fa\u5904\u3002<\/p>\n<h2 id=\"section\">\u524d\u8a00<\/h2>\n<p>\u672c\u6587\u7ffb\u8bd1\u81eaTopCoder\u4e0a\u7684\u4e00\u7bc7\u6587\u7ae0\uff1a <a href=\"http:\/\/community.topcoder.com\/tc?module=Static&amp;d1=tutorials&amp;d2=dynProg\">Dynamic Programming: From novice to advanced<\/a> \uff0c\u5e76\u975e\u4e25\u683c\u9010\u5b57\u9010\u53e5\u7ffb\u8bd1\uff0c\u5176\u4e2d\u52a0\u5165\u4e86\u81ea\u5df1\u7684\u4e00\u4e9b\u7406\u89e3\u3002\u6c34\u5e73\u6709\u9650\uff0c\u8fd8\u671b\u6307\u6458\u3002<\/p>\n<h2 id=\"section-1\">\u524d\u8a00_<\/h2>\n<p>\u6211\u4eec\u9047\u5230\u7684\u95ee\u9898\u4e2d\uff0c\u6709\u5f88\u5927\u4e00\u90e8\u5206\u53ef\u4ee5\u7528\u52a8\u6001\u89c4\u5212(\u7b80\u79f0DP)\u6765\u89e3\u3002 \u89e3\u51b3\u8fd9\u7c7b\u95ee\u9898\u53ef\u4ee5\u5f88\u5927\u5730\u63d0\u5347\u4f60\u7684\u80fd\u529b\u4e0e\u6280\u5de7\uff0c\u6211\u4f1a\u8bd5\u7740\u5e2e\u52a9\u4f60\u7406\u89e3\u5982\u4f55\u4f7f\u7528DP\u6765\u89e3\u9898\u3002 \u8fd9\u7bc7\u6587\u7ae0\u662f\u57fa\u4e8e\u5b9e\u4f8b\u5c55\u5f00\u6765\u8bb2\u7684\uff0c\u56e0\u4e3a\u5e72\u5df4\u5df4\u7684\u7406\u8bba\u5b9e\u5728\u4e0d\u597d\u7406\u89e3\u3002<\/p>\n<p>\u6ce8\u610f\uff1a\u5982\u679c\u4f60\u5bf9\u4e8e\u5176\u4e2d\u67d0\u4e00\u8282\u5df2\u7ecf\u4e86\u89e3\u5e76\u4e14\u4e0d\u60f3\u9605\u8bfb\u5b83\uff0c\u6ca1\u5173\u7cfb\uff0c\u76f4\u63a5\u8df3\u8fc7\u5b83\u5373\u53ef\u3002<\/p>\n<h2 id=\"section-2\">\u7b80\u4ecb(\u5165\u95e8)<\/h2>\n<p><strong>\u4ec0\u4e48\u662f\u52a8\u6001\u89c4\u5212\uff0c\u6211\u4eec\u8981\u5982\u4f55\u63cf\u8ff0\u5b83?<\/strong><\/p>\n<p>\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u4e00\u4e2a\u9012\u63a8\u516c\u5f0f\u53ca\u4e00\u4e2a\u6216\u591a\u4e2a\u521d\u59cb\u72b6\u6001\u3002 \u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u5c06\u7531\u4e0a\u4e00\u6b21\u5b50\u95ee\u9898\u7684\u89e3\u63a8\u51fa\u3002\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6765\u89e3\u9898\u53ea\u9700\u8981\u591a\u9879\u5f0f\u65f6\u95f4\u590d\u6742\u5ea6\uff0c \u56e0\u6b64\u5b83\u6bd4\u56de\u6eaf\u6cd5\u3001\u66b4\u529b\u6cd5\u7b49\u8981\u5feb\u8bb8\u591a\u3002<\/p>\n<p>\u73b0\u5728\u8ba9\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u4e86\u89e3\u4e00\u4e0bDP\u7684\u57fa\u672c\u539f\u7406\u3002<\/p>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u8981\u627e\u5230\u67d0\u4e2a\u72b6\u6001\u7684\u6700\u4f18\u89e3\uff0c\u7136\u540e\u5728\u5b83\u7684\u5e2e\u52a9\u4e0b\uff0c\u627e\u5230\u4e0b\u4e00\u4e2a\u72b6\u6001\u7684\u6700\u4f18\u89e3\u3002<\/p>\n<p><strong>\u201c\u72b6\u6001\u201d\u4ee3\u8868\u4ec0\u4e48\u53ca\u5982\u4f55\u627e\u5230\u5b83?<\/strong><\/p>\n<p>\u201c\u72b6\u6001\u201d\u7528\u6765\u63cf\u8ff0\u8be5\u95ee\u9898\u7684\u5b50\u95ee\u9898\u7684\u89e3\u3002\u539f\u6587\u4e2d\u6709\u4e24\u6bb5\u4f5c\u8005\u9610\u8ff0\u5f97\u4e0d\u592a\u6e05\u695a\uff0c\u8df3\u8fc7\u76f4\u63a5\u4e0a\u4f8b\u5b50\u3002<\/p>\n<p>\u5982\u679c\u6211\u4eec\u6709\u9762\u503c\u4e3a1\u5143\u30013\u5143\u548c5\u5143\u7684\u786c\u5e01\u82e5\u5e72\u679a\uff0c\u5982\u4f55\u7528\u6700\u5c11\u7684\u786c\u5e01\u51d1\u591f11\u5143\uff1f (\u8868\u9762\u4e0a\u8fd9\u9053\u9898\u53ef\u4ee5\u7528\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u4f46\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u53ef\u4ee5\u6c42\u51fa\u89e3\uff0c\u6bd4\u59821\u5143\u6362\u62102\u5143\u7684\u65f6\u5019)<\/p>\n<p>\u9996\u5148\u6211\u4eec\u601d\u8003\u4e00\u4e2a\u95ee\u9898\uff0c\u5982\u4f55\u7528\u6700\u5c11\u7684\u786c\u5e01\u51d1\u591fi\u5143(i&lt;11)\uff1f\u4e3a\u4ec0\u4e48\u8981\u8fd9\u4e48\u95ee\u5462\uff1f \u4e24\u4e2a\u539f\u56e0\uff1a1.\u5f53\u6211\u4eec\u9047\u5230\u4e00\u4e2a\u5927\u95ee\u9898\u65f6\uff0c\u603b\u662f\u4e60\u60ef\u628a\u95ee\u9898\u7684\u89c4\u6a21\u53d8\u5c0f\uff0c\u8fd9\u6837\u4fbf\u4e8e\u5206\u6790\u8ba8\u8bba\u3002 2.\u8fd9\u4e2a\u89c4\u6a21\u53d8\u5c0f\u540e\u7684\u95ee\u9898\u548c\u539f\u6765\u7684\u95ee\u9898\u662f\u540c\u8d28\u7684\uff0c\u9664\u4e86\u89c4\u6a21\u53d8\u5c0f\uff0c\u5176\u5b83\u7684\u90fd\u662f\u4e00\u6837\u7684\uff0c \u672c\u8d28\u4e0a\u5b83\u8fd8\u662f\u540c\u4e00\u4e2a\u95ee\u9898(\u89c4\u6a21\u53d8\u5c0f\u540e\u7684\u95ee\u9898\u5176\u5b9e\u662f\u539f\u95ee\u9898\u7684\u5b50\u95ee\u9898)\u3002<\/p>\n<p>\u597d\u4e86\uff0c\u8ba9\u6211\u4eec\u4ece\u6700\u5c0f\u7684i\u5f00\u59cb\u5427\u3002\u5f53i=0\uff0c\u5373\u6211\u4eec\u9700\u8981\u591a\u5c11\u4e2a\u786c\u5e01\u6765\u51d1\u591f0\u5143\u3002 \u7531\u4e8e1\uff0c3\uff0c5\u90fd\u5927\u4e8e0\uff0c\u5373\u6ca1\u6709\u6bd40\u5c0f\u7684\u5e01\u503c\uff0c\u56e0\u6b64\u51d1\u591f0\u5143\u6211\u4eec\u6700\u5c11\u9700\u89810\u4e2a\u786c\u5e01\u3002 (\u8fd9\u4e2a\u5206\u6790\u5f88\u50bb\u662f\u4e0d\u662f\uff1f\u522b\u7740\u6025\uff0c\u8fd9\u4e2a\u601d\u8def\u6709\u5229\u4e8e\u6211\u4eec\u7406\u6e05\u52a8\u6001\u89c4\u5212\u7a76\u7adf\u5728\u505a\u4e9b\u4ec0\u4e48\u3002) \u8fd9\u65f6\u5019\u6211\u4eec\u53d1\u73b0\u7528\u4e00\u4e2a\u6807\u8bb0\u6765\u8868\u793a\u8fd9\u53e5\u201c\u51d1\u591f0\u5143\u6211\u4eec\u6700\u5c11\u9700\u89810\u4e2a\u786c\u5e01\u3002\u201d\u4f1a\u6bd4\u8f83\u65b9\u4fbf\uff0c \u5982\u679c\u4e00\u76f4\u7528\u7eaf\u6587\u5b57\u6765\u8868\u8ff0\uff0c\u4e0d\u51fa\u4e00\u4f1a\u513f\u4f60\u5c31\u4f1a\u89c9\u5f97\u5f88\u7ed5\u4e86\u3002\u90a3\u4e48\uff0c \u6211\u4eec\u7528d(i)=j\u6765\u8868\u793a\u51d1\u591fi\u5143\u6700\u5c11\u9700\u8981j\u4e2a\u786c\u5e01\u3002\u4e8e\u662f\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u4e86d(0)=0\uff0c \u8868\u793a\u51d1\u591f0\u5143\u6700\u5c0f\u9700\u89810\u4e2a\u786c\u5e01\u3002\u5f53i=1\u65f6\uff0c\u53ea\u6709\u9762\u503c\u4e3a1\u5143\u7684\u786c\u5e01\u53ef\u7528\uff0c \u56e0\u6b64\u6211\u4eec\u62ff\u8d77\u4e00\u4e2a\u9762\u503c\u4e3a1\u7684\u786c\u5e01\uff0c\u63a5\u4e0b\u6765\u53ea\u9700\u8981\u51d1\u591f0\u5143\u5373\u53ef\uff0c\u800c\u8fd9\u4e2a\u662f\u5df2\u7ecf\u77e5\u9053\u7b54\u6848\u7684\uff0c \u5373d(0)=0\u3002\u6240\u4ee5\uff0cd(1)=d(1-1)+1=d(0)+1=0+1=1\u3002\u5f53i=2\u65f6\uff0c \u4ecd\u7136\u53ea\u6709\u9762\u503c\u4e3a1\u7684\u786c\u5e01\u53ef\u7528\uff0c\u4e8e\u662f\u6211\u62ff\u8d77\u4e00\u4e2a\u9762\u503c\u4e3a1\u7684\u786c\u5e01\uff0c \u63a5\u4e0b\u6765\u6211\u53ea\u9700\u8981\u518d\u51d1\u591f2-1=1\u5143\u5373\u53ef(\u8bb0\u5f97\u8981\u7528\u6700\u5c0f\u7684\u786c\u5e01\u6570\u91cf)\uff0c\u800c\u8fd9\u4e2a\u7b54\u6848\u4e5f\u5df2\u7ecf\u77e5\u9053\u4e86\u3002 \u6240\u4ee5d(2)=d(2-1)+1=d(1)+1=1+1=2\u3002\u4e00\u76f4\u5230\u8fd9\u91cc\uff0c\u4f60\u90fd\u53ef\u80fd\u4f1a\u89c9\u5f97\uff0c\u597d\u65e0\u804a\uff0c \u611f\u89c9\u50cf\u505a\u5c0f\u5b66\u751f\u7684\u9898\u76ee\u4f3c\u7684\u3002\u56e0\u4e3a\u6211\u4eec\u4e00\u76f4\u90fd\u53ea\u80fd\u64cd\u4f5c\u9762\u503c\u4e3a1\u7684\u786c\u5e01\uff01\u8010\u5fc3\u70b9\uff0c \u8ba9\u6211\u4eec\u770b\u770bi=3\u65f6\u7684\u60c5\u51b5\u3002\u5f53i=3\u65f6\uff0c\u6211\u4eec\u80fd\u7528\u7684\u786c\u5e01\u5c31\u6709\u4e24\u79cd\u4e86\uff1a1\u5143\u7684\u548c3\u5143\u7684( 5\u5143\u7684\u4ecd\u7136\u6ca1\u7528\uff0c\u56e0\u4e3a\u4f60\u9700\u8981\u51d1\u7684\u6570\u76ee\u662f3\u5143\uff015\u5143\u592a\u591a\u4e86\u4eb2)\u3002 \u65e2\u7136\u80fd\u7528\u7684\u786c\u5e01\u6709\u4e24\u79cd\uff0c\u6211\u5c31\u6709\u4e24\u79cd\u65b9\u6848\u3002\u5982\u679c\u6211\u62ff\u4e86\u4e00\u4e2a1\u5143\u7684\u786c\u5e01\uff0c\u6211\u7684\u76ee\u6807\u5c31\u53d8\u4e3a\u4e86\uff1a \u51d1\u591f3-1=2\u5143\u9700\u8981\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5373d(3)=d(3-1)+1=d(2)+1=2+1=3\u3002 \u8fd9\u4e2a\u65b9\u6848\u8bf4\u7684\u662f\uff0c\u6211\u62ff3\u4e2a1\u5143\u7684\u786c\u5e01\uff1b\u7b2c\u4e8c\u79cd\u65b9\u6848\u662f\u6211\u62ff\u8d77\u4e00\u4e2a3\u5143\u7684\u786c\u5e01\uff0c \u6211\u7684\u76ee\u6807\u5c31\u53d8\u6210\uff1a\u51d1\u591f3-3=0\u5143\u9700\u8981\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5373d(3)=d(3-3)+1=d(0)+1=0+1=1. \u8fd9\u4e2a\u65b9\u6848\u8bf4\u7684\u662f\uff0c\u6211\u62ff1\u4e2a3\u5143\u7684\u786c\u5e01\u3002\u597d\u4e86\uff0c\u8fd9\u4e24\u79cd\u65b9\u6848\u54ea\u79cd\u66f4\u4f18\u5462\uff1f \u8bb0\u5f97\u6211\u4eec\u53ef\u662f\u8981\u7528\u6700\u5c11\u7684\u786c\u5e01\u6570\u91cf\u6765\u51d1\u591f3\u5143\u7684\u3002\u6240\u4ee5\uff0c \u9009\u62e9d(3)=1\uff0c\u600e\u4e48\u6765\u7684\u5462\uff1f\u5177\u4f53\u662f\u8fd9\u6837\u5f97\u5230\u7684\uff1ad(3)=min{d(3-1)+1, d(3-3)+1}\u3002<\/p>\n<p>OK\uff0c\u7801\u4e86\u8fd9\u4e48\u591a\u5b57\u8bb2\u5177\u4f53\u7684\u4e1c\u897f\uff0c\u8ba9\u6211\u4eec\u6765\u70b9\u62bd\u8c61\u7684\u3002\u4ece\u4ee5\u4e0a\u7684\u6587\u5b57\u4e2d\uff0c \u6211\u4eec\u8981\u62bd\u51fa\u52a8\u6001\u89c4\u5212\u91cc\u975e\u5e38\u91cd\u8981\u7684\u4e24\u4e2a\u6982\u5ff5\uff1a\u72b6\u6001\u548c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002<\/p>\n<p>\u4e0a\u6587\u4e2dd(i)\u8868\u793a\u51d1\u591fi\u5143\u9700\u8981\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u6211\u4eec\u5c06\u5b83\u5b9a\u4e49\u4e3a\u8be5\u95ee\u9898\u7684\u201d\u72b6\u6001\u201d\uff0c \u8fd9\u4e2a\u72b6\u6001\u662f\u600e\u4e48\u627e\u51fa\u6765\u7684\u5462\uff1f\u6211\u5728\u53e6\u4e00\u7bc7\u6587\u7ae0 <a href=\"http:\/\/www.hawstein.com\/posts\/dp-knapsack.html\">\u52a8\u6001\u89c4\u5212\u4e4b\u80cc\u5305\u95ee\u9898(\u4e00)<\/a>\u4e2d\u5199\u8fc7\uff1a \u6839\u636e\u5b50\u95ee\u9898\u5b9a\u4e49\u72b6\u6001\u3002\u4f60\u627e\u5230\u5b50\u95ee\u9898\uff0c\u72b6\u6001\u4e5f\u5c31\u6d6e\u51fa\u6c34\u9762\u4e86\u3002 \u6700\u7ec8\u6211\u4eec\u8981\u6c42\u89e3\u7684\u95ee\u9898\uff0c\u53ef\u4ee5\u7528\u8fd9\u4e2a\u72b6\u6001\u6765\u8868\u793a\uff1ad(11)\uff0c\u5373\u51d1\u591f11\u5143\u6700\u5c11\u9700\u8981\u591a\u5c11\u4e2a\u786c\u5e01\u3002 \u90a3\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u662f\u4ec0\u4e48\u5462\uff1f\u65e2\u7136\u6211\u4eec\u7528d(i)\u8868\u793a\u72b6\u6001\uff0c\u90a3\u4e48\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u81ea\u7136\u5305\u542bd(i)\uff0c \u4e0a\u6587\u4e2d\u5305\u542b\u72b6\u6001d(i)\u7684\u65b9\u7a0b\u662f\uff1ad(3)=min{d(3-1)+1, d(3-3)+1}\u3002\u6ca1\u9519\uff0c \u5b83\u5c31\u662f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u63cf\u8ff0\u72b6\u6001\u4e4b\u95f4\u662f\u5982\u4f55\u8f6c\u79fb\u7684\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8981\u5bf9\u5b83\u62bd\u8c61\u4e00\u4e0b\uff0c<\/p>\n<p>d(i)=min{ d(i-v<sub>j<\/sub>)+1 }\uff0c\u5176\u4e2di-v<sub>j<\/sub> &gt;=0\uff0cv<sub>j<\/sub>\u8868\u793a\u7b2cj\u4e2a\u786c\u5e01\u7684\u9762\u503c;<\/p>\n<p>\u6709\u4e86\u72b6\u6001\u548c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u8fd9\u4e2a\u95ee\u9898\u57fa\u672c\u4e0a\u4e5f\u5c31\u89e3\u51b3\u4e86\u3002\u5f53\u7136\u4e86\uff0cTalk is cheap,show me the code!<\/p>\n<p>\u4f2a\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/pseudocode.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-150\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/pseudocode.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"639\" height=\"176\"  sizes=\"auto, (max-width: 639px) 100vw, 639px\" \/><\/div><\/p>\n<p>\u4e0b\u56fe\u662f\u5f53i\u4ece0\u523011\u65f6\u7684\u89e3\uff1a<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/sum.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-154\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/sum.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"449\" height=\"359\"  sizes=\"auto, (max-width: 449px) 100vw, 449px\" \/><\/div><\/p>\n<p>\u4ece\u4e0a\u56fe\u53ef\u4ee5\u5f97\u51fa\uff0c\u8981\u51d1\u591f11\u5143\u81f3\u5c11\u9700\u89813\u679a\u786c\u5e01\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u901a\u8fc7\u8ffd\u8e2a\u6211\u4eec\u662f\u5982\u4f55\u4ece\u524d\u4e00\u4e2a\u72b6\u6001\u503c\u5f97\u5230\u5f53\u524d\u72b6\u6001\u503c\u7684\uff0c \u53ef\u4ee5\u627e\u5230\u6bcf\u4e00\u6b21\u6211\u4eec\u7528\u7684\u662f\u4ec0\u4e48\u9762\u503c\u7684\u786c\u5e01\u3002\u6bd4\u5982\uff0c\u4ece\u4e0a\u9762\u7684\u56fe\u6211\u4eec\u53ef\u4ee5\u770b\u51fa\uff0c \u6700\u7ec8\u7ed3\u679cd(11)=d(10)+1(\u9762\u503c\u4e3a1)\uff0c\u800cd(10)=d(5)+1(\u9762\u503c\u4e3a5)\uff0c\u6700\u540ed(5)=d(0)+1 (\u9762\u503c\u4e3a5)\u3002\u6240\u4ee5\u6211\u4eec\u51d1\u591f11\u5143\u6700\u5c11\u9700\u8981\u76843\u679a\u786c\u5e01\u662f\uff1a1\u5143\u30015\u5143\u30015\u5143\u3002<\/p>\n<p>\u6ce8\u610f\uff1a\u539f\u6587\u4e2d\u8fd9\u91cc\u672c\u6765\u8fd8\u6709\u4e00\u6bb5\u7684\uff0c\u4f46\u6211\u53cd\u53cd\u590d\u590d\u8bfb\u4e86\u51e0\u904d\uff0c \u5927\u6982\u7684\u610f\u601d\u6211\u5df2\u7ecf\u5728\u4e0a\u6587\u4ecei=0\u5230i=3\u7684\u5206\u6790\u4e2d\u6709\u6240\u4f53\u73b0\u4e86\u3002\u4f5c\u8005\u672c\u6765\u60f3\u8bb2\u7684\u901a\u4fd7\u4e00\u4e9b\uff0c \u7ed3\u679c\u6ca1\u5199\u597d\uff0c\u53cd\u800c\u66f4\u4e0d\u597d\u61c2\uff0c\u6240\u4ee5\u8fd9\u6bb5\u4e0d\u7ffb\u8bd1\u4e86\u3002<\/p>\n<h2 id=\"section-3\">\u521d\u7ea7<\/h2>\n<p>\u4e0a\u9762\u8ba8\u8bba\u4e86\u4e00\u4e2a\u975e\u5e38\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u73b0\u5728\u8ba9\u6211\u4eec\u6765\u770b\u770b\u5bf9\u4e8e\u66f4\u590d\u6742\u7684\u95ee\u9898\uff0c \u5982\u4f55\u627e\u5230\u72b6\u6001\u4e4b\u95f4\u7684\u8f6c\u79fb\u65b9\u5f0f(\u5373\u627e\u5230\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b)\u3002 \u4e3a\u6b64\u6211\u4eec\u8981\u5f15\u5165\u4e00\u4e2a\u65b0\u8bcd\u53eb\u9012\u63a8\u5173\u7cfb\u6765\u5c06\u72b6\u6001\u8054\u7cfb\u8d77\u6765(\u8bf4\u7684\u8fd8\u662f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b)<\/p>\n<p>OK\uff0c\u4e0a\u4f8b\u5b50\uff0c\u770b\u770b\u5b83\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002<\/p>\n<p>\u4e00\u4e2a\u5e8f\u5217\u6709N\u4e2a\u6570\uff1aA[1],A[2],\u2026,A[N]\uff0c\u6c42\u51fa\u6700\u957f\u975e\u964d\u5b50\u5e8f\u5217\u7684\u957f\u5ea6\u3002 (\u8bb2DP\u57fa\u672c\u90fd\u4f1a\u8bb2\u5230\u7684\u4e00\u4e2a\u95ee\u9898LIS\uff1alongest increasing subsequence)<\/p>\n<p>\u6b63\u5982\u4e0a\u9762\u6211\u4eec\u8bb2\u7684\uff0c\u9762\u5bf9\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u9996\u5148\u8981\u5b9a\u4e49\u4e00\u4e2a\u201c\u72b6\u6001\u201d\u6765\u4ee3\u8868\u5b83\u7684\u5b50\u95ee\u9898\uff0c \u5e76\u4e14\u627e\u5230\u5b83\u7684\u89e3\u3002\u6ce8\u610f\uff0c\u5927\u90e8\u5206\u60c5\u51b5\u4e0b\uff0c\u67d0\u4e2a\u72b6\u6001\u53ea\u4e0e\u5b83\u524d\u9762\u51fa\u73b0\u7684\u72b6\u6001\u6709\u5173\uff0c \u800c\u72ec\u7acb\u4e8e\u540e\u9762\u7684\u72b6\u6001\u3002<\/p>\n<p>\u8ba9\u6211\u4eec\u6cbf\u7528\u201c\u5165\u95e8\u201d\u4e00\u8282\u91cc\u90a3\u9053\u7b80\u5355\u9898\u7684\u601d\u8def\u6765\u4e00\u6b65\u6b65\u627e\u5230\u201c\u72b6\u6001\u201d\u548c\u201c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u201d\u3002 \u5047\u5982\u6211\u4eec\u8003\u8651\u6c42A[1],A[2],\u2026,A[i]\u7684\u6700\u957f\u975e\u964d\u5b50\u5e8f\u5217\u7684\u957f\u5ea6\uff0c\u5176\u4e2di&lt;N\uff0c \u90a3\u4e48\u4e0a\u9762\u7684\u95ee\u9898\u53d8\u6210\u4e86\u539f\u95ee\u9898\u7684\u4e00\u4e2a\u5b50\u95ee\u9898(\u95ee\u9898\u89c4\u6a21\u53d8\u5c0f\u4e86\uff0c\u4f60\u53ef\u4ee5\u8ba9i=1,2,3\u7b49\u6765\u5206\u6790) \u7136\u540e\u6211\u4eec\u5b9a\u4e49d(i)\uff0c\u8868\u793a\u524di\u4e2a\u6570\u4e2d\u4ee5A[i]\u7ed3\u5c3e\u7684\u6700\u957f\u975e\u964d\u5b50\u5e8f\u5217\u7684\u957f\u5ea6\u3002OK\uff0c \u5bf9\u7167\u201c\u5165\u95e8\u201d\u4e2d\u7684\u7b80\u5355\u9898\uff0c\u4f60\u5e94\u8be5\u53ef\u4ee5\u4f30\u8ba1\u5230\u8fd9\u4e2ad(i)\u5c31\u662f\u6211\u4eec\u8981\u627e\u7684\u72b6\u6001\u3002 \u5982\u679c\u6211\u4eec\u628ad(1)\u5230d(N)\u90fd\u8ba1\u7b97\u51fa\u6765\uff0c\u90a3\u4e48\u6700\u7ec8\u6211\u4eec\u8981\u627e\u7684\u7b54\u6848\u5c31\u662f\u8fd9\u91cc\u9762\u6700\u5927\u7684\u90a3\u4e2a\u3002 \u72b6\u6001\u627e\u5230\u4e86\uff0c\u4e0b\u4e00\u6b65\u627e\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002<\/p>\n<p>\u4e3a\u4e86\u65b9\u4fbf\u7406\u89e3\u6211\u4eec\u662f\u5982\u4f55\u627e\u5230\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u7684\uff0c\u6211\u5148\u628a\u4e0b\u9762\u7684\u4f8b\u5b50\u63d0\u5230\u524d\u9762\u6765\u8bb2\u3002 \u5982\u679c\u6211\u4eec\u8981\u6c42\u7684\u8fd9N\u4e2a\u6570\u7684\u5e8f\u5217\u662f\uff1a<\/p>\n<div class=\"highlighter-rouge\">\n<pre class=\"highlight\"><code>5\uff0c3\uff0c4\uff0c8\uff0c6\uff0c7\r\n<\/code><\/pre>\n<\/div>\n<p>\u6839\u636e\u4e0a\u9762\u627e\u5230\u7684\u72b6\u6001\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\uff1a\uff08\u4e0b\u6587\u7684\u6700\u957f\u975e\u964d\u5b50\u5e8f\u5217\u90fd\u7528LIS\u8868\u793a\uff09<\/p>\n<ul>\n<li>\u524d1\u4e2a\u6570\u7684LIS\u957f\u5ea6d(1)=1(\u5e8f\u5217\uff1a5)<\/li>\n<li>\u524d2\u4e2a\u6570\u7684LIS\u957f\u5ea6d(2)=1(\u5e8f\u5217\uff1a3\uff1b3\u524d\u9762\u6ca1\u6709\u6bd43\u5c0f\u7684)<\/li>\n<li>\u524d3\u4e2a\u6570\u7684LIS\u957f\u5ea6d(3)=2(\u5e8f\u5217\uff1a3\uff0c4\uff1b4\u524d\u9762\u6709\u4e2a\u6bd4\u5b83\u5c0f\u76843\uff0c\u6240\u4ee5d(3)=d(2)+1)<\/li>\n<li>\u524d4\u4e2a\u6570\u7684LIS\u957f\u5ea6d(4)=3(\u5e8f\u5217\uff1a3\uff0c4\uff0c8\uff1b8\u524d\u9762\u6bd4\u5b83\u5c0f\u7684\u67093\u4e2a\u6570\uff0c\u6240\u4ee5 d(4)=max{d(1),d(2),d(3)}+1=3)<\/li>\n<\/ul>\n<p>OK\uff0c\u5206\u6790\u5230\u8fd9\uff0c\u6211\u89c9\u5f97\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5df2\u7ecf\u5f88\u660e\u663e\u4e86\uff0c\u5982\u679c\u6211\u4eec\u5df2\u7ecf\u6c42\u51fa\u4e86d(1)\u5230d(i-1)\uff0c \u90a3\u4e48d(i)\u53ef\u4ee5\u7528\u4e0b\u9762\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5f97\u5230\uff1a<\/p>\n<div class=\"highlighter-rouge\">\n<pre class=\"highlight\"><code>d(i) = max{1, d(j)+1},\u5176\u4e2dj&lt;i,A[j]&lt;=A[i]\r\n<\/code><\/pre>\n<\/div>\n<p>\u7528\u5927\u767d\u8bdd\u89e3\u91ca\u5c31\u662f\uff0c\u60f3\u8981\u6c42d(i)\uff0c\u5c31\u628ai\u524d\u9762\u7684\u5404\u4e2a\u5b50\u5e8f\u5217\u4e2d\uff0c \u6700\u540e\u4e00\u4e2a\u6570\u4e0d\u5927\u4e8eA[i]\u7684\u5e8f\u5217\u957f\u5ea6\u52a01\uff0c\u7136\u540e\u53d6\u51fa\u6700\u5927\u7684\u957f\u5ea6\u5373\u4e3ad(i)\u3002 \u5f53\u7136\u4e86\uff0c\u6709\u53ef\u80fdi\u524d\u9762\u7684\u5404\u4e2a\u5b50\u5e8f\u5217\u4e2d\u6700\u540e\u4e00\u4e2a\u6570\u90fd\u5927\u4e8eA[i]\uff0c\u90a3\u4e48d(i)=1\uff0c \u5373\u5b83\u81ea\u8eab\u6210\u4e3a\u4e00\u4e2a\u957f\u5ea6\u4e3a1\u7684\u5b50\u5e8f\u5217\u3002<\/p>\n<p>\u5206\u6790\u5b8c\u4e86\uff0c\u4e0a\u56fe\uff1a(\u7b2c\u4e8c\u5217\u8868\u793a\u524di\u4e2a\u6570\u4e2dLIS\u7684\u957f\u5ea6\uff0c \u7b2c\u4e09\u5217\u8868\u793a\uff0cLIS\u4e2d\u5230\u8fbe\u5f53\u524d\u8fd9\u4e2a\u6570\u7684\u4e0a\u4e00\u4e2a\u6570\u7684\u4e0b\u6807\uff0c\u6839\u636e\u8fd9\u4e2a\u53ef\u4ee5\u6c42\u51faLIS\u5e8f\u5217)<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/lis.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-155\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/lis.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"450\" height=\"228\"  sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/div><\/p>\n<p>Talk is cheap, show me the code:<\/p>\n<p>[code lang=&#8221;cpp&#8221;]#include &amp;lt;iostream&amp;gt;<br \/>\nusing namespace std;<\/p>\n<p>int lis(int A[], int n){<br \/>\n    int *d = new int[n];<br \/>\n    int len = 1;<br \/>\n    for(int i=0; i&amp;lt;n; ++i){<br \/>\n        d[i] = 1;<br \/>\n        for(int j=0; j&amp;lt;i; ++j)<br \/>\n            if(A[j]&amp;lt;=A[i] &amp;amp;&amp;amp; d[j]+1&amp;gt;d[i])<br \/>\n                d[i] = d[j] + 1;<br \/>\n        if(d[i]&amp;gt;len) len = d[i];<br \/>\n    }<br \/>\n    delete[] d;<br \/>\n    return len;<br \/>\n}<br \/>\nint main(){<br \/>\n    int A[] = {<br \/>\n        5, 3, 4, 8, 6, 7<br \/>\n    };<br \/>\n    cout&amp;lt;&amp;lt;lis(A, 6)&amp;lt;&amp;lt;endl;<br \/>\n    return 0;<br \/>\n}<br \/>\n[\/code]<\/p>\n<p>\u8be5\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662fO(n^2 )\uff0c\u5e76\u4e0d\u662f\u6700\u4f18\u7684\u89e3\u6cd5\u3002 \u8fd8\u6709\u4e00\u79cd\u5f88\u5de7\u5999\u7684\u7b97\u6cd5\u53ef\u4ee5\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u5230O(nlogn)\uff0c\u7f51\u4e0a\u5df2\u7ecf\u6709\u5404\u79cd\u6587\u7ae0\u4ecb\u7ecd\u5b83\uff0c \u8fd9\u91cc\u5c31\u4e0d\u518d\u8d58\u8ff0\u3002\u4f20\u9001\u95e8\uff1a <a href=\"http:\/\/www.felix021.com\/blog\/read.php?1587\">LIS\u7684O(nlogn)\u89e3\u6cd5<\/a>\u3002 \u6b64\u9898\u8fd8\u53ef\u4ee5\u7528\u201c\u6392\u5e8f+LCS\u201d\u6765\u89e3\uff0c\u611f\u5174\u8da3\u7684\u8bdd\u53ef\u81ea\u884cGoogle\u3002<\/p>\n<p><strong>\u7ec3\u4e60\u9898<\/strong><\/p>\n<p>\u65e0\u5411\u56feG\u6709N\u4e2a\u7ed3\u70b9(1&lt;N&lt;=1000)\u53ca\u4e00\u4e9b\u8fb9\uff0c\u6bcf\u4e00\u6761\u8fb9\u4e0a\u5e26\u6709\u6b63\u7684\u6743\u91cd\u503c\u3002 \u627e\u5230\u7ed3\u70b91\u5230\u7ed3\u70b9N\u7684\u6700\u77ed\u8def\u5f84\uff0c\u6216\u8005\u8f93\u51fa\u4e0d\u5b58\u5728\u8fd9\u6837\u7684\u8def\u5f84\u3002<\/p>\n<p>\u63d0\u793a\uff1a\u5728\u6bcf\u4e00\u6b65\u4e2d\uff0c\u5bf9\u4e8e\u90a3\u4e9b\u6ca1\u6709\u8ba1\u7b97\u8fc7\u7684\u7ed3\u70b9\uff0c \u53ca\u90a3\u4e9b\u5df2\u7ecf\u8ba1\u7b97\u51fa\u4ece\u7ed3\u70b91\u5230\u5b83\u7684\u6700\u77ed\u8def\u5f84\u7684\u7ed3\u70b9\uff0c\u5982\u679c\u5b83\u4eec\u95f4\u6709\u8fb9\uff0c \u5219\u8ba1\u7b97\u4ece\u7ed3\u70b91\u5230\u672a\u8ba1\u7b97\u7ed3\u70b9\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n<p>\u5c1d\u8bd5\u89e3\u51b3\u4ee5\u4e0b\u6765\u81eatopcoder\u7ade\u8d5b\u7684\u95ee\u9898\uff1a<\/p>\n<ul>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=4493&amp;pm=1259\">ZigZag<\/a> &#8211; 2003 TCCC Semifinals 3<\/li>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=5009&amp;pm=2402\">BadNeighbors<\/a> &#8211; 2004 TCCC Round 4<\/li>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=5006&amp;pm=1918\">FlowerGarden<\/a> &#8211; 2004 TCCC Round 1<\/li>\n<\/ul>\n<h2 id=\"section-4\">\u4e2d\u7ea7<\/h2>\n<p>\u63a5\u4e0b\u6765\uff0c\u8ba9\u6211\u4eec\u6765\u770b\u770b\u5982\u4f55\u89e3\u51b3\u4e8c\u7ef4\u7684DP\u95ee\u9898\u3002<\/p>\n<p>\u5e73\u9762\u4e0a\u6709N*M\u4e2a\u683c\u5b50\uff0c\u6bcf\u4e2a\u683c\u5b50\u4e2d\u653e\u7740\u4e00\u5b9a\u6570\u91cf\u7684\u82f9\u679c\u3002\u4f60\u4ece\u5de6\u4e0a\u89d2\u7684\u683c\u5b50\u5f00\u59cb\uff0c \u6bcf\u4e00\u6b65\u53ea\u80fd\u5411\u4e0b\u8d70\u6216\u662f\u5411\u53f3\u8d70\uff0c\u6bcf\u6b21\u8d70\u5230\u4e00\u4e2a\u683c\u5b50\u4e0a\u5c31\u628a\u683c\u5b50\u91cc\u7684\u82f9\u679c\u6536\u96c6\u8d77\u6765\uff0c \u8fd9\u6837\u4e0b\u53bb\uff0c\u4f60\u6700\u591a\u80fd\u6536\u96c6\u5230\u591a\u5c11\u4e2a\u82f9\u679c\u3002<\/p>\n<p>\u89e3\u8fd9\u4e2a\u95ee\u9898\u4e0e\u89e3\u5176\u5b83\u7684DP\u95ee\u9898\u51e0\u4e4e\u6ca1\u6709\u4ec0\u4e48\u4e24\u6837\u3002\u7b2c\u4e00\u6b65\u627e\u5230\u95ee\u9898\u7684\u201c\u72b6\u6001\u201d\uff0c \u7b2c\u4e8c\u6b65\u627e\u5230\u201c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u201d\uff0c\u7136\u540e\u57fa\u672c\u4e0a\u95ee\u9898\u5c31\u89e3\u51b3\u4e86\u3002<\/p>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u8981\u627e\u5230\u8fd9\u4e2a\u95ee\u9898\u4e2d\u7684\u201c\u72b6\u6001\u201d\u662f\u4ec0\u4e48\uff1f\u6211\u4eec\u5fc5\u987b\u6ce8\u610f\u5230\u7684\u4e00\u70b9\u662f\uff0c \u5230\u8fbe\u4e00\u4e2a\u683c\u5b50\u7684\u65b9\u5f0f\u6700\u591a\u53ea\u6709\u4e24\u79cd\uff1a\u4ece\u5de6\u8fb9\u6765\u7684(\u9664\u4e86\u7b2c\u4e00\u5217)\u548c\u4ece\u4e0a\u8fb9\u6765\u7684(\u9664\u4e86\u7b2c\u4e00\u884c)\u3002 \u56e0\u6b64\u4e3a\u4e86\u6c42\u51fa\u5230\u8fbe\u5f53\u524d\u683c\u5b50\u540e\u6700\u591a\u80fd\u6536\u96c6\u5230\u591a\u5c11\u4e2a\u82f9\u679c\uff0c \u6211\u4eec\u5c31\u8981\u5148\u53bb\u8003\u5bdf\u90a3\u4e9b\u80fd\u5230\u8fbe\u5f53\u524d\u8fd9\u4e2a\u683c\u5b50\u7684\u683c\u5b50\uff0c\u5230\u8fbe\u5b83\u4eec\u6700\u591a\u80fd\u6536\u96c6\u5230\u591a\u5c11\u4e2a\u82f9\u679c\u3002 (\u662f\u4e0d\u662f\u6709\u70b9\u7ed5\uff0c\u4f46\u8fd9\u53e5\u8bdd\u7684\u672c\u8d28\u5176\u5b9e\u662fDP\u7684\u5173\u952e\uff1a\u6b32\u6c42\u95ee\u9898\u7684\u89e3\uff0c\u5148\u8981\u53bb\u6c42\u5b50\u95ee\u9898\u7684\u89e3)<\/p>\n<p>\u7ecf\u8fc7\u4e0a\u9762\u7684\u5206\u6790\uff0c\u5f88\u5bb9\u6613\u53ef\u4ee5\u5f97\u51fa\u95ee\u9898\u7684\u72b6\u6001\u548c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002 \u72b6\u6001S[i][j]\u8868\u793a\u6211\u4eec\u8d70\u5230(i, j)\u8fd9\u4e2a\u683c\u5b50\u65f6\uff0c\u6700\u591a\u80fd\u6536\u96c6\u5230\u591a\u5c11\u4e2a\u82f9\u679c\u3002\u90a3\u4e48\uff0c \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5982\u4e0b\uff1a<\/p>\n<div class=\"highlighter-rouge\">\n<p>[code lang=&#8221;cpp&#8221;]S[i][j]=A[i][j] + max(S[i-1][j], if i&amp;gt;0 ; S[i][j-1], if j&amp;gt;0)[\/code]<\/p>\n<\/div>\n<p>\u5176\u4e2di\u4ee3\u8868\u884c\uff0cj\u4ee3\u8868\u5217\uff0c\u4e0b\u6807\u5747\u4ece0\u5f00\u59cb\uff1bA[i][j]\u4ee3\u8868\u683c\u5b50(i, j)\u5904\u7684\u82f9\u679c\u6570\u91cf\u3002<\/p>\n<p>S[i][j]\u6709\u4e24\u79cd\u8ba1\u7b97\u65b9\u5f0f\uff1a1.\u5bf9\u4e8e\u6bcf\u4e00\u884c\uff0c\u4ece\u5de6\u5411\u53f3\u8ba1\u7b97\uff0c\u7136\u540e\u4ece\u4e0a\u5230\u4e0b\u9010\u884c\u5904\u7406\uff1b2. \u5bf9\u4e8e\u6bcf\u4e00\u5217\uff0c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\uff0c\u7136\u540e\u4ece\u5de6\u5411\u53f3\u9010\u5217\u5904\u7406\u3002 \u8fd9\u6837\u505a\u7684\u76ee\u7684\u662f\u4e3a\u4e86\u5728\u8ba1\u7b97S[i][j]\u65f6\uff0cS[i-1][j]\u548cS[i][j-1]\u90fd\u5df2\u7ecf\u8ba1\u7b97\u51fa\u6765\u4e86\u3002<\/p>\n<p>\u4f2a\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/2d-dp.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-156\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/2d-dp.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"636\" height=\"123\"  sizes=\"auto, (max-width: 636px) 100vw, 636px\" \/><\/div><\/p>\n<p>\u4ee5\u4e0b\u4e24\u9053\u9898\u6765\u81eatopcoder\uff0c\u7ec3\u4e60\u7528\u7684\u3002<\/p>\n<ul>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=4709&amp;pm=1889\">AvoidRoads<\/a> &#8211; 2003 TCO Semifinals 4<\/li>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=4482&amp;pm=1592\">ChessMetric<\/a> &#8211; 2003 TCCC Round 4<\/li>\n<\/ul>\n<h2 id=\"section-5\">\u4e2d\u9ad8\u7ea7<\/h2>\n<p>\u8fd9\u4e00\u8282\u8981\u8ba8\u8bba\u7684\u662f\u5e26\u6709\u989d\u5916\u6761\u4ef6\u7684DP\u95ee\u9898\u3002<\/p>\n<p>\u4ee5\u4e0b\u7684\u8fd9\u4e2a\u95ee\u9898\u662f\u4e2a\u5f88\u597d\u7684\u4f8b\u5b50\u3002<\/p>\n<p>\u65e0\u5411\u56feG\u6709N\u4e2a\u7ed3\u70b9\uff0c\u5b83\u7684\u8fb9\u4e0a\u5e26\u6709\u6b63\u7684\u6743\u91cd\u503c\u3002<\/p>\n<p>\u4f60\u4ece\u7ed3\u70b91\u5f00\u59cb\u8d70\uff0c\u5e76\u4e14\u4e00\u5f00\u59cb\u7684\u65f6\u5019\u4f60\u8eab\u4e0a\u5e26\u6709M\u5143\u94b1\u3002\u5982\u679c\u4f60\u7ecf\u8fc7\u7ed3\u70b9i\uff0c \u90a3\u4e48\u4f60\u5c31\u8981\u82b1\u6389S[i]\u5143(\u53ef\u4ee5\u628a\u8fd9\u60f3\u8c61\u4e3a\u6536\u8fc7\u8def\u8d39)\u3002\u5982\u679c\u4f60\u6ca1\u6709\u8db3\u591f\u7684\u94b1\uff0c \u5c31\u4e0d\u80fd\u4ece\u90a3\u4e2a\u7ed3\u70b9\u7ecf\u8fc7\u3002\u5728\u8fd9\u6837\u7684\u9650\u5236\u6761\u4ef6\u4e0b\uff0c\u627e\u5230\u4ece\u7ed3\u70b91\u5230\u7ed3\u70b9N\u7684\u6700\u77ed\u8def\u5f84\u3002 \u6216\u8005\u8f93\u51fa\u8be5\u8def\u5f84\u4e0d\u5b58\u5728\u3002\u5982\u679c\u5b58\u5728\u591a\u6761\u6700\u77ed\u8def\u5f84\uff0c\u90a3\u4e48\u8f93\u51fa\u82b1\u94b1\u6570\u91cf\u6700\u5c11\u7684\u90a3\u6761\u3002 \u9650\u5236\uff1a1&lt;N&lt;=100 ; 0&lt;=M&lt;=100 ; \u5bf9\u4e8e\u6bcf\u4e2ai\uff0c0&lt;=S[i]&lt;=100\uff1b\u6b63\u5982\u6211\u4eec\u6240\u770b\u5230\u7684\uff0c \u5982\u679c\u6ca1\u6709\u989d\u5916\u7684\u9650\u5236\u6761\u4ef6(\u5728\u7ed3\u70b9\u5904\u8981\u6536\u8d39\uff0c\u8d39\u7528\u4e0d\u8db3\u8fd8\u4e0d\u7ed9\u8fc7)\uff0c\u90a3\u4e48\uff0c \u8fd9\u4e2a\u95ee\u9898\u5c31\u548c\u7ecf\u5178\u7684\u8fea\u6770\u65af\u7279\u62c9\u95ee\u9898\u4e00\u6837\u4e86(\u627e\u5230\u4e24\u7ed3\u70b9\u95f4\u7684\u6700\u77ed\u8def\u5f84)\u3002 \u5728\u7ecf\u5178\u7684\u8fea\u6770\u65af\u7279\u62c9\u95ee\u9898\u4e2d\uff0c \u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u4e00\u7ef4\u6570\u7ec4\u6765\u4fdd\u5b58\u4ece\u5f00\u59cb\u7ed3\u70b9\u5230\u6bcf\u4e2a\u7ed3\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u957f\u5ea6\uff0c \u5373M[i]\u8868\u793a\u4ece\u5f00\u59cb\u7ed3\u70b9\u5230\u7ed3\u70b9i\u7684\u6700\u77ed\u8def\u5f84\u7684\u957f\u5ea6\u3002\u7136\u800c\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c \u6211\u4eec\u8fd8\u8981\u4fdd\u5b58\u6211\u4eec\u8eab\u4e0a\u5269\u4f59\u591a\u5c11\u94b1\u8fd9\u4e2a\u4fe1\u606f\u3002\u56e0\u6b64\uff0c\u5f88\u81ea\u7136\u7684\uff0c \u6211\u4eec\u5c06\u4e00\u7ef4\u6570\u7ec4\u6269\u5c55\u4e3a\u4e8c\u7ef4\u6570\u7ec4\u3002M[i][j]\u8868\u793a\u4ece\u5f00\u59cb\u7ed3\u70b9\u5230\u7ed3\u70b9i\u7684\u6700\u77ed\u8def\u5f84\u957f\u5ea6\uff0c \u4e14\u5269\u4f59j\u5143\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u89c4\u7ea6\u5230\u539f\u59cb\u7684\u8def\u5f84\u5bfb\u627e\u95ee\u9898\u3002 \u5728\u6bcf\u4e00\u6b65\u4e2d\uff0c\u5bf9\u4e8e\u5df2\u7ecf\u627e\u5230\u7684\u6700\u77ed\u8def\u5f84\uff0c\u6211\u4eec\u627e\u5230\u5b83\u6240\u80fd\u5230\u8fbe\u7684\u4e0b\u4e00\u4e2a\u672a\u6807\u8bb0\u72b6\u6001(i,j)\uff0c \u5c06\u5b83\u6807\u8bb0\u4e3a\u5df2\u8bbf\u95ee(\u4e4b\u540e\u4e0d\u518d\u8bbf\u95ee\u8fd9\u4e2a\u7ed3\u70b9)\uff0c\u5e76\u4e14\u5728\u80fd\u5230\u8fbe\u8fd9\u4e2a\u7ed3\u70b9\u7684\u5404\u4e2a\u6700\u77ed\u8def\u5f84\u4e2d\uff0c \u627e\u5230\u52a0\u4e0a\u5f53\u524d\u8fb9\u6743\u91cd\u503c\u540e\u6700\u5c0f\u503c\u5bf9\u5e94\u7684\u8def\u5f84\uff0c\u5373\u4e3a\u8be5\u7ed3\u70b9\u7684\u6700\u77ed\u8def\u5f84\u3002 (\u5199\u8d77\u6765\u771f\u662f\u7ed5\uff0c\u5efa\u8bae\u753b\u4e2a\u56fe\u5c31\u4f1a\u660e\u4e86\u5f88\u591a)\u3002\u4e0d\u65ad\u91cd\u590d\u4e0a\u9762\u7684\u6b65\u9aa4\uff0c \u76f4\u5230\u6240\u6709\u7684\u7ed3\u70b9\u90fd\u8bbf\u95ee\u5230\u4e3a\u6b62(\u8fd9\u91cc\u7684\u8bbf\u95ee\u5e76\u4e0d\u662f\u8981\u6c42\u6211\u4eec\u8981\u7ecf\u8fc7\u5b83\uff0c \u6bd4\u5982\u6709\u4e2a\u7ed3\u70b9\u6536\u8d39\u5f88\u9ad8\uff0c\u4f60\u6ca1\u6709\u8db3\u591f\u7684\u94b1\u53bb\u7ecf\u8fc7\u5b83\uff0c\u4f46\u4f60\u5df2\u7ecf\u8bbf\u95ee\u8fc7\u5b83) \u6700\u540eMin[N-1][j]\u4e2d\u7684\u6700\u5c0f\u503c\u5373\u662f\u95ee\u9898\u7684\u7b54\u6848(\u5982\u679c\u6709\u591a\u4e2a\u6700\u5c0f\u503c\uff0c \u5373\u6709\u591a\u6761\u6700\u77ed\u8def\u5f84\uff0c\u90a3\u4e48\u9009\u62e9j\u6700\u5927\u7684\u90a3\u6761\u8def\u5f84\uff0c\u5373\uff0c\u4f7f\u4f60\u5269\u4f59\u94b1\u6570\u6700\u591a\u7684\u6700\u77ed\u8def\u5f84)\u3002<\/p>\n<p>\u4f2a\u4ee3\u7801\uff1a<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/shortest-path.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-157\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/shortest-path.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"636\" height=\"572\"  sizes=\"auto, (max-width: 636px) 100vw, 636px\" \/><\/div><\/p>\n<p>\u4e0b\u9762\u6709\u51e0\u9053topcoder\u4e0a\u7684\u9898\u4ee5\u4f9b\u7ec3\u4e60\uff1a<\/p>\n<ul>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=4705&amp;pm=1166\">Jewelry<\/a> &#8211; 2003 TCO Online Round 4<\/li>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=4555&amp;pm=1215\">StripePainter<\/a> &#8211; SRM 150 Div 1<\/li>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=5072&amp;pm=2829\">QuickSums<\/a> &#8211; SRM 197 Div 2<\/li>\n<li><a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=4630&amp;pm=1861\">ShortPalindromes<\/a> &#8211; SRM 165 Div 2<\/li>\n<\/ul>\n<h2 id=\"section-6\">\u9ad8\u7ea7<\/h2>\n<p>\u4ee5\u4e0b\u95ee\u9898\u9700\u8981\u4ed4\u7ec6\u7684\u63e3\u6469\u624d\u80fd\u5c06\u5176\u89c4\u7ea6\u4e3a\u53ef\u7528DP\u89e3\u7684\u95ee\u9898\u3002<\/p>\n<p>\u95ee\u9898\uff1a<a href=\"http:\/\/community.topcoder.com\/tc?module=ProblemDetail&amp;rd=5854&amp;pm=2940\">StarAdventure<\/a> &#8211; SRM 208 Div 1:<\/p>\n<p>\u7ed9\u5b9a\u4e00\u4e2aM\u884cN\u5217\u7684\u77e9\u9635(M*N\u4e2a\u683c\u5b50)\uff0c\u6bcf\u4e2a\u683c\u5b50\u4e2d\u653e\u7740\u4e00\u5b9a\u6570\u91cf\u7684\u82f9\u679c\u3002 \u4f60\u4ece\u5de6\u4e0a\u89d2\u7684\u683c\u5b50\u5f00\u59cb\uff0c\u53ea\u80fd\u5411\u4e0b\u6216\u5411\u53f3\u8d70\uff0c\u76ee\u7684\u5730\u662f\u53f3\u4e0b\u89d2\u7684\u683c\u5b50\u3002 \u4f60\u6bcf\u8d70\u8fc7\u4e00\u4e2a\u683c\u5b50\uff0c\u5c31\u628a\u683c\u5b50\u4e0a\u7684\u82f9\u679c\u90fd\u6536\u96c6\u8d77\u6765\u3002\u7136\u540e\u4f60\u4ece\u53f3\u4e0b\u89d2\u8d70\u56de\u5de6\u4e0a\u89d2\u7684\u683c\u5b50\uff0c \u6bcf\u6b21\u53ea\u80fd\u5411\u5de6\u6216\u662f\u5411\u4e0a\u8d70\uff0c\u540c\u6837\u7684\uff0c\u8d70\u8fc7\u4e00\u4e2a\u683c\u5b50\u5c31\u628a\u91cc\u9762\u7684\u82f9\u679c\u90fd\u6536\u96c6\u8d77\u6765\u3002 \u6700\u540e\uff0c\u4f60\u518d\u4e00\u6b21\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\uff0c\u6bcf\u8fc7\u4e00\u4e2a\u683c\u5b50\u540c\u6837\u8981\u6536\u96c6\u8d77\u91cc\u9762\u7684\u82f9\u679c (\u5982\u679c\u683c\u5b50\u91cc\u7684\u82f9\u679c\u6570\u4e3a0\uff0c\u5c31\u4e0d\u7528\u6536\u96c6)\u3002\u6c42\u4f60\u6700\u591a\u80fd\u6536\u96c6\u5230\u591a\u5c11\u82f9\u679c\u3002<\/p>\n<p>\u6ce8\u610f\uff1a\u5f53\u4f60\u7ecf\u8fc7\u4e00\u4e2a\u683c\u5b50\u65f6\uff0c\u4f60\u8981\u4e00\u6b21\u6027\u628a\u683c\u5b50\u91cc\u7684\u82f9\u679c\u90fd\u62ff\u8d70\u3002<\/p>\n<p>\u9650\u5236\u6761\u4ef6\uff1a1 &lt; N, M &lt;= 50\uff1b\u6bcf\u4e2a\u683c\u5b50\u91cc\u7684\u82f9\u679c\u6570\u91cf\u662f0\u52301000(\u5305\u542b0\u548c1000)\u3002<\/p>\n<p>\u5982\u679c\u6211\u4eec\u53ea\u9700\u8981\u4ece\u5de6\u4e0a\u89d2\u7684\u683c\u5b50\u8d70\u5230\u53f3\u4e0b\u89d2\u7684\u683c\u5b50\u4e00\u6b21\uff0c\u5e76\u4e14\u6536\u96c6\u6700\u5927\u6570\u91cf\u7684\u82f9\u679c\uff0c \u90a3\u4e48\u95ee\u9898\u5c31\u9000\u5316\u4e3a\u201c\u4e2d\u7ea7\u201d\u4e00\u8282\u91cc\u7684\u90a3\u4e2a\u95ee\u9898\u3002\u5c06\u8fd9\u91cc\u7684\u95ee\u9898\u89c4\u7ea6\u4e3a\u201c\u4e2d\u7ea7\u201d\u91cc\u7684\u7b80\u5355\u9898\uff0c \u8fd9\u6837\u4e00\u6765\u4f1a\u6bd4\u8f83\u597d\u89e3\u3002\u8ba9\u6211\u4eec\u6765\u5206\u6790\u4e00\u4e0b\u8fd9\u4e2a\u95ee\u9898\uff0c\u8981\u5982\u4f55\u89c4\u7ea6\u6216\u662f\u4fee\u6539\u624d\u80fd\u7528\u4e0aDP\u3002 \u9996\u5148\uff0c\u5bf9\u4e8e\u7b2c\u4e8c\u6b21\u4ece\u53f3\u4e0b\u89d2\u8d70\u5230\u5de6\u4e0a\u89d2\u5f97\u51fa\u7684\u8fd9\u6761\u8def\u5f84\uff0c \u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u89c6\u4e3a\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u5f97\u51fa\u7684\u8def\u5f84\uff0c\u6ca1\u6709\u4efb\u4f55\u7684\u5dee\u522b\u3002 (\u5373\u4eceB\u8d70\u5230A\u7684\u6700\u4f18\u8def\u5f84\u548c\u4eceA\u8d70\u5230B\u7684\u6700\u4f18\u8def\u5f84\u662f\u4e00\u6837\u7684)\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c \u6211\u4eec\u5f97\u5230\u4e86\u4e09\u6761\u4ece\u9876\u8d70\u5230\u5e95\u7684\u8def\u5f84\u3002\u5bf9\u4e8e\u8fd9\u4e00\u70b9\u7684\u7406\u89e3\u53ef\u4ee5\u7a0d\u5fae\u51cf\u5c0f\u95ee\u9898\u7684\u96be\u5ea6\u3002 \u4e8e\u662f\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8fd93\u6761\u8def\u5f84\u8bb0\u4e3a\u5de6\uff0c\u4e2d\uff0c\u53f3\u8def\u5f84\u3002\u5bf9\u4e8e\u4e24\u6761\u76f8\u4ea4\u8def\u5f84(\u5982\u4e0b\u56fe)\uff1a<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/intersect.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-159\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/intersect.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"148\" height=\"188\"  sizes=\"auto, (max-width: 148px) 100vw, 148px\" \/><\/div><\/p>\n<p>\u5728\u4e0d\u5f71\u54cd\u7ed3\u679c\u7684\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u89c6\u4e3a\u4e24\u6761\u4e0d\u76f8\u4ea4\u7684\u8def\u5f84\uff1a<\/p>\n<p><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/no-intersect.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-158\" data-original=\"https:\/\/www.haoyuan.info\/wp-content\/uploads\/2016\/11\/no-intersect.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" width=\"153\" height=\"190\"  sizes=\"auto, (max-width: 153px) 100vw, 153px\" \/><\/div><\/p>\n<p>\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c06\u5f97\u5230\u5de6\uff0c\u4e2d\uff0c\u53f33\u6761\u8def\u5f84\u3002\u6b64\u5916\uff0c\u5982\u679c\u6211\u4eec\u8981\u5f97\u5230\u6700\u4f18\u89e3\uff0c \u8def\u5f84\u4e4b\u95f4\u4e0d\u80fd\u76f8\u4ea4(\u9664\u4e86\u5de6\u4e0a\u89d2\u548c\u53f3\u4e0b\u89d2\u5fc5\u7136\u4f1a\u76f8\u4ea4\u7684\u683c\u5b50)\u3002\u56e0\u6b64\u5bf9\u4e8e\u6bcf\u4e00\u884cy( \u9664\u4e86\u7b2c\u4e00\u884c\u548c\u6700\u540e\u4e00\u884c)\uff0c\u4e09\u6761\u8def\u5f84\u5bf9\u5e94\u7684x\u5750\u6807\u8981\u6ee1\u8db3\uff1ax1[y] &lt; x2[y] &lt; x3[y]\u3002 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