{"id":76269,"date":"2026-02-22T03:49:34","date_gmt":"2026-02-21T19:49:34","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/76269.html"},"modified":"2026-02-22T03:49:34","modified_gmt":"2026-02-21T19:49:34","slug":"%e3%80%90%e4%b8%87%e5%ad%97%e6%80%bb%e7%bb%93%e3%80%91c%e7%ae%97%e6%b3%95%e7%ab%9e%e8%b5%9b%e6%a0%b8%e5%bf%83%e5%9f%ba%e7%a1%80%e7%ae%97%e6%b3%95%e5%85%a8%e5%ae%b6%e6%a1%b6%ef%bc%9a%e4%bb%8e","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/76269.html","title":{"rendered":"\u3010\u4e07\u5b57\u603b\u7ed3\u3011C++\u7b97\u6cd5\u7ade\u8d5b\u6838\u5fc3\u57fa\u7840\u7b97\u6cd5\u5168\u5bb6\u6876\uff1a\u4ece\u4e8c\u5206\u5230\u62df\u9635\uff0c\u6536\u85cf\u8fd9\u4e00\u7bc7\u5c31\u591f\u4e86\uff01"},"content":{"rendered":"<p>\u6458\u8981&#xff1a;\u8fd8\u5728\u4e3a\u7b97\u6cd5\u57fa\u7840\u4e0d\u7262\u800c\u82e6\u607c&#xff1f;\u9762\u5bf9\u9898\u76ee\u4e0d\u77e5\u9053\u7528\u4ec0\u4e48\u7b97\u6cd5&#xff1f;\u672c\u6587\u5c06 algorithm \u7ade\u8d5b\u4e2d\u6700\u6838\u5fc3\u7684\u57fa\u7840\u7b97\u6cd5\u4e00\u7f51\u6253\u5c3d&#xff01;\u4ece\u6700\u5e38\u7528\u7684\u4e8c\u5206\u3001\u5c3a\u53d6&#xff0c;\u5230\u6570\u636e\u5904\u7406\u795e\u5668\u524d\u7f00\u548c\u3001\u5dee\u5206\u3001\u79bb\u6563\u5316&#xff0c;\u518d\u5230\u8fdb\u9636\u7684\u500d\u589e\u3001ST\u8868&#xff0c;\u6700\u540e\u6df1\u5165\u63a2\u8ba8\u8d2a\u5fc3\u7b97\u6cd5\u80cc\u540e\u7684\u6570\u5b66\u539f\u7406\u2014\u2014\u62df\u9635\u3002\u65e0\u8bba\u4f60\u662f\u84dd\u6865\u676f\u3001\u8003\u7814\u673a\u8bd5\u8fd8\u662f ACMer&#xff0c;\u8fd9\u7bc7\u6587\u7ae0\u90fd\u662f\u4f60\u7684\u5175\u5668\u5e93\u3002\u5efa\u8bae\u6536\u85cf\u53cd\u590d\u7814\u8bfb&#xff01;<\/p>\n<hr \/>\n<h3>\u76ee\u5f55<\/h3>\n<li>\u641c\u7d20\u4e0e\u904d\u5386\u7684\u827a\u672f\n<ul>\n<li>\u5c3a\u53d6\u6cd5&#xff08;\u53cc\u6307\u9488&#xff09;&#xff1a;\u50cf\u866b\u5b50\u4e00\u6837\u8815\u52a8<\/li>\n<li>\u4e8c\u5206\u6cd5&#xff1a;\u4e0d\u4ec5\u662f\u67e5\u627e&#xff0c;\u66f4\u662f\u7b54\u6848\u7684\u5224\u5b9a<\/li>\n<li>\u4e09\u5206\u6cd5&#xff1a;\u51f8\u51fd\u6570\u7684\u6781\u503c\u514b\u661f<\/li>\n<\/ul>\n<\/li>\n<li>\u6570\u636e\u7684\u9b54\u6cd5\u5904\u7406\n<ul>\n<li>\u524d\u7f00\u548c\u4e0e\u5dee\u5206&#xff1a;\u533a\u95f4\u64cd\u4f5c\u7684 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          O(1)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u79d8\u5bc6<\/li>\n<li>\u79bb\u6563\u5316&#xff1a;\u8ba9 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>            10<\/p>\n<p>            9<\/p>\n<p>          10^9<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u6570\u636e\u4e56\u4e56\u6392\u961f<\/li>\n<li>\u6392\u5217\u4e0e\u6392\u5e8f&#xff1a;STL \u7684\u9a9a\u64cd\u4f5c<\/li>\n<\/ul>\n<\/li>\n<li>\u5206\u6cbb\u4e0e\u500d\u589e\n<ul>\n<li>\u5206\u6cbb\u6cd5&#xff1a;\u5927\u4e8b\u5316\u5c0f&#xff0c;\u5404\u4e2a\u51fb\u7834<\/li>\n<li>\u500d\u589e\u6cd5\u4e0e ST \u8868&#xff1a;\u9884\u5904\u7406\u7684 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>            2<\/p>\n<p>            k<\/p>\n<p>          2^k<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u667a\u6167<\/li>\n<\/ul>\n<\/li>\n<li>\u8d2a\u5fc3\u7684\u6570\u5b66\u4e4b\u7f8e\n<ul>\n<li>\u8d2a\u5fc3\u6cd5&#xff1a;\u5c40\u90e8\u6700\u4f18\u80fd\u5426\u63a8\u51fa\u5168\u5c40\u6700\u4f18&#xff1f;<\/li>\n<li>\u62df\u9635&#xff08;Matroid&#xff09;&#xff1a;\u8d2a\u5fc3\u7b97\u6cd5\u7684\u7406\u8bba\u57fa\u77f3&#xff08;\u786c\u6838&#xff01;&#xff09;<\/li>\n<\/ul>\n<\/li>\n<li>\u603b\u7ed3\u4e0e\u5237\u9898\u8def\u7ebf<\/li>\n<hr \/>\n<h3>\u7b2c\u4e00\u90e8\u5206&#xff1a;\u641c\u7d22\u4e0e\u904d\u5386\u7684\u827a\u672f<\/h3>\n<h4>1. \u5c3a\u53d6\u6cd5&#xff08;\u53cc\u6307\u9488 \/ Two Pointers&#xff09;<\/h4>\n<p>\u6838\u5fc3\u601d\u60f3&#xff1a;\u5229\u7528\u533a\u95f4\u7684\u5355\u8c03\u6027&#xff0c;\u901a\u8fc7\u4e24\u4e2a\u6307\u9488&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        L<\/p>\n<p>        ,<\/p>\n<p>        R<\/p>\n<p>       L, R<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0077em\">R<\/span><\/span><\/span><\/span><\/span>&#xff09;\u50cf\u6bdb\u6bdb\u866b\u4e00\u6837\u5728\u6570\u7ec4\u4e0a\u8815\u52a8&#xff0c;\u907f\u514d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>         N<\/p>\n<p>         2<\/p>\n<p>        )<\/p>\n<p>       O(N^2)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7684\u66b4\u529b\u679a\u4e3e\u3002<\/p>\n<p>\u6a21\u7248\u4ee3\u7801&#xff1a;<\/p>\n<p><span class=\"token comment\">\/\/ \u5bfb\u627e\u6ee1\u8db3\u6761\u4ef6\u7684\u6700\u77ed\u533a\u95f4\u957f\u5ea6<\/span><br \/>\n<span class=\"token keyword\">void<\/span> <span class=\"token function\">solve<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> S<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> a<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> l <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> r <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> sum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> res <span class=\"token operator\">&#061;<\/span> n <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token boolean\">true<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>r <span class=\"token operator\">&lt;<\/span> n <span class=\"token operator\">&amp;&amp;<\/span> sum <span class=\"token operator\">&lt;<\/span> S<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/ 1. \u53f3\u6307\u9488\u4e0d\u65ad\u8fdb\u98df<\/span><br \/>\n            sum <span class=\"token operator\">&#043;&#061;<\/span> a<span class=\"token punctuation\">[<\/span>r<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>sum <span class=\"token operator\">&lt;<\/span> S<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">break<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u65e0\u6cd5\u6ee1\u8db3&#xff0c;\u7ed3\u675f<\/span><br \/>\n        res <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">min<\/span><span class=\"token punctuation\">(<\/span>res<span class=\"token punctuation\">,<\/span> r <span class=\"token operator\">&#8211;<\/span> l<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u66f4\u65b0\u7b54\u6848<\/span><br \/>\n        sum <span class=\"token operator\">-&#061;<\/span> a<span class=\"token punctuation\">[<\/span>l<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ 2. \u5de6\u6307\u9488\u6536\u7f29<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u76f8\u5e94\u4f8b\u9898&#xff1a;P1638 \u901b\u753b\u5c55<\/p>\n<h4>2. \u4e8c\u5206\u6cd5&#xff08;Binary Search&#xff09;<\/h4>\n<p>\u75db\u70b9&#xff1a;\u5f88\u591a\u4eba\u5199\u4e8c\u5206\u6b7b\u5faa\u73af\u3002 \u79d8\u7c4d&#xff1a;\u8ba4\u51c6\u4e00\u4e2a\u6a21\u7248&#xff08;while(l &lt; r)&#xff09;&#xff0c;\u4e0d\u8981\u6df7\u7528\u3002<\/p>\n<p>\u6574\u6570\u4e8c\u5206\u6a21\u7248&#xff08;\u67e5\u627e\u5de6\u8fb9\u754c&#xff09;&#xff1a;<\/p>\n<p><span class=\"token comment\">\/\/ \u68c0\u67e5 x \u662f\u5426\u6ee1\u8db3\u67d0\u79cd\u6027\u8d28<\/span><br \/>\n<span class=\"token keyword\">bool<\/span> <span class=\"token function\">check<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> x<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/* &#8230; *\/<\/span> <span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">bsearch_1<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> l<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> r<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>l <span class=\"token operator\">&lt;<\/span> r<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> mid <span class=\"token operator\">&#061;<\/span> l <span class=\"token operator\">&#043;<\/span> r <span class=\"token operator\">&gt;&gt;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token function\">check<\/span><span class=\"token punctuation\">(<\/span>mid<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> r <span class=\"token operator\">&#061;<\/span> mid<span class=\"token punctuation\">;<\/span>    <span class=\"token comment\">\/\/ \u7b54\u6848\u5728\u5de6\u8fb9&#xff0c;\u5305\u542b mid<\/span><br \/>\n        <span class=\"token keyword\">else<\/span> l <span class=\"token operator\">&#061;<\/span> mid <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span>           <span class=\"token comment\">\/\/ \u7b54\u6848\u5728\u53f3\u8fb9<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> l<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u8fdb\u9636&#xff1a;\u4e8c\u5206\u4e0d\u4ec5\u67e5\u4e0b\u6807&#xff0c;\u66f4\u591a\u65f6\u5019\u7528\u4e8e\u4e8c\u5206\u7b54\u6848&#xff08;\u4f8b\u5982&#xff1a;\u6700\u5927\u5316\u6700\u5c0f\u503c&#xff09;\u3002<\/p>\n<h4>3. \u4e09\u5206\u6cd5&#xff08;Ternary Search&#xff09;<\/h4>\n<p>\u573a\u666f&#xff1a;\u4e8c\u5206\u89e3\u51b3\u5355\u8c03\u51fd\u6570&#xff0c;\u4e09\u5206\u89e3\u51b3\u51f8\u51fd\u6570\/\u51f9\u51fd\u6570&#xff08;\u6c42\u6781\u503c&#xff09;\u3002 \u539f\u7406&#xff1a;\u53d6\u4e24\u4e2a\u70b9 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        m<\/p>\n<p>        1<\/p>\n<p>        ,<\/p>\n<p>        m<\/p>\n<p>        2<\/p>\n<p>       m1, m2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span> \u5c06\u533a\u95f4\u4e09\u7b49\u5206&#xff0c;\u6bd4\u8f83\u51fd\u6570\u503c\u820d\u53bb\u4e00\u6bb5\u3002<\/p>\n<p><span class=\"token keyword\">double<\/span> l <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> r <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1000<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>r <span class=\"token operator\">&#8211;<\/span> l <span class=\"token operator\">&gt;<\/span> <span class=\"token number\">1e-7<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">double<\/span> m1 <span class=\"token operator\">&#061;<\/span> l <span class=\"token operator\">&#043;<\/span> <span class=\"token punctuation\">(<\/span>r <span class=\"token operator\">&#8211;<\/span> l<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">double<\/span> m2 <span class=\"token operator\">&#061;<\/span> r <span class=\"token operator\">&#8211;<\/span> <span class=\"token punctuation\">(<\/span>r <span class=\"token operator\">&#8211;<\/span> l<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token function\">f<\/span><span class=\"token punctuation\">(<\/span>m1<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;<\/span> <span class=\"token function\">f<\/span><span class=\"token punctuation\">(<\/span>m2<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> l <span class=\"token operator\">&#061;<\/span> m1<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u6c42\u6700\u5927\u503c\u65f6&#xff0c;m1 \u80af\u5b9a\u4e0d\u662f\u5cf0\u503c<\/span><br \/>\n    <span class=\"token keyword\">else<\/span> r <span class=\"token operator\">&#061;<\/span> m2<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<hr \/>\n<h3>\u7b2c\u4e8c\u90e8\u5206&#xff1a;\u6570\u636e\u7684\u9b54\u6cd5\u5904\u7406<\/h3>\n<h4>1. \u524d\u7f00\u548c\u4e0e\u5dee\u5206<\/h4>\n<p>\u8fd9\u662f\u4e00\u5bf9\u4e92\u9006\u8fd0\u7b97&#xff0c;\u4e13\u95e8\u5904\u7406\u533a\u95f4\u95ee\u9898\u3002<\/p>\n<ul>\n<li>\u524d\u7f00\u548c&#xff1a;\u5feb\u901f\u6c42\u533a\u95f4\u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>        O(1)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002 <\/p>\n<ul>\n<li>\u516c\u5f0f&#xff1a;sum[i] &#061; sum[i-1] &#043; a[i]<\/li>\n<li>\u67e5\u8be2 [L, R]&#xff1a;sum[R] &#8211; sum[L-1]<\/li>\n<\/ul>\n<\/li>\n<li>\u5dee\u5206&#xff1a;\u5feb\u901f\u8fdb\u884c\u533a\u95f4\u4fee\u6539 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>        O(1)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002 <\/p>\n<ul>\n<li>\u64cd\u4f5c&#xff1a;\u7ed9\u533a\u95f4 [L, R] \u5168\u90e8\u52a0\u4e0a k\u3002<\/li>\n<li>\u6838\u5fc3&#xff1a;diff[L] &#043;&#061; k; diff[R&#043;1] -&#061; k;<\/li>\n<li>\u6700\u540e\u505a\u4e00\u6b21\u524d\u7f00\u548c\u8fd8\u539f\u539f\u6570\u7ec4\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h4>2. \u79bb\u6563\u5316&#xff08;Discretization&#xff09;<\/h4>\n<p>\u573a\u666f&#xff1a;\u6570\u503c\u5f88\u5927&#xff08;\u5982 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>         9<\/p>\n<p>       10^9<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u4f46\u6570\u636e\u4e2a\u6570\u5f88\u5c11&#xff08;\u5982 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>         5<\/p>\n<p>       10^5<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u4e14\u6211\u4eec\u53ea\u5173\u5fc3\u4ed6\u4eec\u7684\u5927\u5c0f\u5173\u7cfb&#xff0c;\u4e0d\u5173\u5fc3\u5177\u4f53\u6570\u503c\u3002<\/p>\n<p>STL \u4e09\u6b65\u8d70\u6a21\u7248&#xff1a;<\/p>\n<p>vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> a<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u5b58\u539f\u59cb\u6570\u636e<\/span><br \/>\nvector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> b <span class=\"token operator\">&#061;<\/span> a<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u5907\u4efd<\/span><br \/>\n<span class=\"token function\">sort<\/span><span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">.<\/span><span class=\"token function\">begin<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ 1. \u6392\u5e8f<\/span><br \/>\nb<span class=\"token punctuation\">.<\/span><span class=\"token function\">erase<\/span><span class=\"token punctuation\">(<\/span><span class=\"token function\">unique<\/span><span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">.<\/span><span class=\"token function\">begin<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ 2. \u53bb\u91cd<\/span><\/p>\n<p><span class=\"token comment\">\/\/ 3. \u67e5\u627e x \u79bb\u6563\u5316\u540e\u7684\u503c&#xff08;\u6392\u540d&#xff09;<\/span><br \/>\n<span class=\"token keyword\">int<\/span> <span class=\"token function\">get_id<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> x<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> <span class=\"token function\">lower_bound<\/span><span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">.<\/span><span class=\"token function\">begin<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> x<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#8211;<\/span> b<span class=\"token punctuation\">.<\/span><span class=\"token function\">begin<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>3. \u6392\u5e8f\u4e0e\u6392\u5217<\/h4>\n<p>\u9664\u4e86 sort&#xff0c;C&#043;&#043; STL \u8fd8\u6709\u4e00\u4e2a\u795e\u5668&#xff1a;next_permutation&#xff08;\u751f\u6210\u4e0b\u4e00\u4e2a\u5b57\u5178\u5e8f\u6392\u5217&#xff09;\u3002 \u5168\u6392\u5217\u6a21\u7248&#xff1a;<\/p>\n<p>vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> p <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">do<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/\/ \u5904\u7406\u5f53\u524d\u6392\u5217<\/span><br \/>\n<span class=\"token punctuation\">}<\/span> <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token function\">next_permutation<\/span><span class=\"token punctuation\">(<\/span>p<span class=\"token punctuation\">.<\/span><span class=\"token function\">begin<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> p<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>\u6709\u611f\u5174\u8da3\u7684\u53ef\u4ee5\u53c2\u8003\u6211\u4e4b\u524d\u7684\u4e00\u7bc7\u6587\u7ae0\u7b97\u6cd5\u968f\u7b14&#xff1a;\u6d1b\u8c37P1036 \u9009\u6570<\/p>\n<hr \/>\n<h3>\u7b2c\u4e09\u90e8\u5206&#xff1a;\u5206\u6cbb\u4e0e\u500d\u589e<\/h3>\n<h4>1. \u5206\u6cbb\u6cd5&#xff08;Divide and Conquer&#xff09;<\/h4>\n<p>\u53e3\u8bc0&#xff1a;\u5206\u800c\u6cbb\u4e4b&#xff0c;\u5408\u4e8c\u4e3a\u4e00\u3002 \u7ecf\u5178\u6848\u4f8b&#xff1a;\u5f52\u5e76\u6392\u5e8f\u6c42\u9006\u5e8f\u5bf9\u3001\u5feb\u901f\u5e42\u3002<\/p>\n<ul>\n<li>\u5feb\u901f\u5e42&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>          a<\/p>\n<p>          b<\/p>\n<p>        a^b<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u53ef\u4ee5\u5728 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         b<\/p>\n<p>         )<\/p>\n<p>        O(\\\\log b)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5185\u7b97\u51fa\u3002<span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> <span class=\"token function\">qpow<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> a<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> b<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> res <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>b <span class=\"token operator\">&amp;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span> res <span class=\"token operator\">&#061;<\/span> res <span class=\"token operator\">*<\/span> a<span class=\"token punctuation\">;<\/span><br \/>\n        a <span class=\"token operator\">&#061;<\/span> a <span class=\"token operator\">*<\/span> a<span class=\"token punctuation\">;<\/span><br \/>\n        b <span class=\"token operator\">&gt;&gt;&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> res<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span>\n <\/li>\n<\/ul>\n<h4>2. \u500d\u589e\u6cd5\u4e0e ST \u8868<\/h4>\n<p>\u500d\u589e\u601d\u60f3&#xff1a;\u4efb\u4f55\u6570\u5b57\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        2<\/p>\n<p>       2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span> \u7684\u5e42\u6b21\u4e4b\u548c\u3002 ST \u8868&#xff08;Sparse Table&#xff09;&#xff1a;\u89e3\u51b3 RMQ&#xff08;\u533a\u95f4\u6700\u503c\u67e5\u8be2&#xff09; \u95ee\u9898&#xff0c;\u4e0d\u652f\u6301\u4fee\u6539&#xff0c;\u4f46\u67e5\u8be2\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>       O(1)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7684&#xff01;<\/p>\n<p>\u6838\u5fc3\u8f6c\u79fb\u65b9\u7a0b&#xff1a; <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         [<\/p>\n<p>         i<\/p>\n<p>         ]<\/p>\n<p>         [<\/p>\n<p>         j<\/p>\n<p>         ]<\/p>\n<p>         &#061;<\/p>\n<p>         max<\/p>\n<p>         \u2061<\/p>\n<p>         (<\/p>\n<p>         f<\/p>\n<p>         [<\/p>\n<p>         i<\/p>\n<p>         ]<\/p>\n<p>         [<\/p>\n<p>         j<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>         ]<\/p>\n<p>         ,<\/p>\n<p>         f<\/p>\n<p>         [<\/p>\n<p>         i<\/p>\n<p>         &#043;<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         &lt;<\/p>\n<p>         &lt;<\/p>\n<p>         (<\/p>\n<p>         j<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>         ]<\/p>\n<p>         [<\/p>\n<p>         j<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>         ]<\/p>\n<p>         )<\/p>\n<p>        f[i][j] &#061; \\\\max(f[i][j-1], f[i &#043; (1 &lt;&lt; (j-1))][j-1])<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mclose\">]<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose\">]<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">max<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mclose\">]<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">]<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">))]<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">])<\/span><\/span><\/span><\/span><\/span><\/span> \u610f\u601d\u662f&#xff1a;\u4ee5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         i<\/p>\n<p>        i<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6595em\"><\/span><span class=\"mord mathnormal\">i<\/span><\/span><\/span><\/span><\/span> \u5f00\u59cb\u957f\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          2<\/p>\n<p>          j<\/p>\n<p>        2^j<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8247em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8247em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u6700\u5927\u503c&#xff0c;\u7b49\u4e8e\u5de6\u534a\u8fb9\u7684\u6700\u5927\u503c\u548c\u53f3\u534a\u8fb9\u7684\u6700\u5927\u503c\u53d6 max\u3002 \u76f8\u5e94\u4f8b\u9898&#xff1a;P3865 \u3010\u6a21\u677f\u3011ST \u8868 &amp; RMQ \u95ee\u9898<\/p>\n<hr \/>\n<h3>\u7b2c\u56db\u90e8\u5206&#xff1a;\u8d2a\u5fc3\u7684\u6570\u5b66\u4e4b\u7f8e<\/h3>\n<h4>1. \u8d2a\u5fc3\u6cd5&#xff08;Greedy&#xff09;<\/h4>\n<p>\u5b9a\u4e49&#xff1a;\u987e\u540d\u601d\u4e49\u5c31\u662f\u5728\u6bcf\u4e00\u6b65\u9009\u62e9\u4e2d\u90fd\u91c7\u53d6\u5f53\u524d\u72b6\u6001\u4e0b\u6700\u597d\u6216\u6700\u4f18&#xff08;\u5373\u6700\u6709\u5229&#xff09;\u7684\u9009\u62e9&#xff0c;\u4ece\u800c\u5e0c\u671b\u5bfc\u81f4\u7ed3\u679c\u662f\u5168\u5c40\u6700\u597d\u6216\u6700\u4f18\u7684\u7b97\u6cd5\u3002 \u96be\u70b9&#xff1a;\u8bc1\u660e\u3002\u5f88\u591a\u65f6\u5019\u76f4\u89c9\u662f\u9519\u7684&#xff0c;\u9700\u8981\u901a\u8fc7\u201c\u4ea4\u6362\u8bba\u8bc1\u6cd5\u201d\u6216\u201c\u53cd\u8bc1\u6cd5\u201d\u6765\u8bc1\u660e\u3002<\/p>\n<h4>2. \u62df\u9635&#xff08;Matroid&#xff09;&#xff1a;\u8d2a\u5fc3\u7684\u7406\u8bba\u57fa\u77f3<\/h4>\n<p>\u4f60\u662f\u5426\u7591\u60d1\u8fc7&#xff1a;\u4e3a\u4ec0\u4e48\u6700\u5c0f\u751f\u6210\u6811&#xff08;Kruskal&#xff09;\u53ef\u4ee5\u7528\u8d2a\u5fc3&#xff1f;\u4e3a\u4ec0\u4e48\u786c\u5e01\u627e\u96f6\u95ee\u9898\u5728\u67d0\u4e9b\u9762\u989d\u4e0b\u8d2a\u5fc3\u4f1a\u5931\u6548&#xff1f; \u8fd9\u4e00\u5207\u7684\u80cc\u540e&#xff0c;\u90fd\u6709\u4e00\u4e2a\u6570\u5b66\u7ed3\u6784\u5728\u652f\u6491\u2014\u2014\u62df\u9635\u3002<\/p>\n<p>\u4ec0\u4e48\u662f\u62df\u9635&#xff1f; \u7b80\u5355\u6765\u8bf4&#xff0c;\u62df\u9635 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        M<\/p>\n<p>        &#061;<\/p>\n<p>        (<\/p>\n<p>        S<\/p>\n<p>        ,<\/p>\n<p>        I<\/p>\n<p>        )<\/p>\n<p>       M &#061; (S, I)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u4e8c\u5143\u7ec4&#xff0c;\u5176\u4e2d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        S<\/p>\n<p>       S<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><\/span> \u662f\u6709\u9650\u96c6\u5408&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        I<\/p>\n<p>       I<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><\/span><\/span><\/span><\/span> \u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        S<\/p>\n<p>       S<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><\/span> \u7684\u5b50\u96c6\u65cf&#xff08;\u79f0\u4e3a\u72ec\u7acb\u96c6&#xff09;&#xff0c;\u6ee1\u8db3\u4ee5\u4e0b\u4e09\u4e2a\u516c\u7406&#xff1a;<\/p>\n<li>\u975e\u7a7a\u6027&#xff1a;\u7a7a\u96c6\u5c5e\u4e8e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         I<\/p>\n<p>        I<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<li>\u9057\u4f20\u6027&#xff1a;\u5982\u679c\u4f60\u662f\u72ec\u7acb\u7684&#xff0c;\u90a3\u4f60\u7684\u5b50\u96c6\u4e5f\u662f\u72ec\u7acb\u7684\u3002<\/li>\n<li>\u4ea4\u6362\u6027&#xff08;\u6269\u5145\u6027&#xff09;&#xff1a;\u5982\u679c\u4e24\u4e2a\u72ec\u7acb\u96c6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         A<\/p>\n<p>        A<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span><\/span> \u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         B<\/p>\n<p>        B<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0502em\">B<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4e14 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2223<\/p>\n<p>         A<\/p>\n<p>         \u2223<\/p>\n<p>         &lt;<\/p>\n<p>         \u2223<\/p>\n<p>         B<\/p>\n<p>         \u2223<\/p>\n<p>        |A| &lt; |B|<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0502em\">B<\/span><span class=\"mord\">\u2223<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u90a3\u4e48\u4e00\u5b9a\u5b58\u5728\u4e00\u4e2a\u5143\u7d20 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>         \u2208<\/p>\n<p>         B<\/p>\n<p>         \u2212<\/p>\n<p>         A<\/p>\n<p>        x \\\\in B-A<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5782em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0502em\">B<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4f7f\u5f97 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         \u222a<\/p>\n<p>         {<\/p>\n<p>         x<\/p>\n<p>         }<\/p>\n<p>        A \\\\cup \\\\{x\\\\}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u222a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span> \u4e5f\u662f\u72ec\u7acb\u7684\u3002<\/li>\n<p>\u7ed3\u8bba&#xff1a;<\/p>\n<p>\u5982\u679c\u4e00\u4e2a\u6700\u4f18\u5316\u95ee\u9898\u53ef\u4ee5\u88ab\u5efa\u6a21\u4e3a\u201c\u5728\u62df\u9635\u4e2d\u5bfb\u627e\u6743\u503c\u6700\u5927\u7684\u72ec\u7acb\u96c6\u201d&#xff0c;\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u4e00\u5b9a\u80fd\u5f97\u5230\u5168\u5c40\u6700\u4f18\u89e3\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;<\/p>\n<ul>\n<li>\u56fe\u7684\u751f\u6210\u6811\u5c31\u662f\u4e00\u4e2a\u62df\u9635&#xff08;\u56fe\u5f62\u62df\u9635&#xff09;\u3002\u6240\u4ee5 Kruskal \u7b97\u6cd5&#xff08;\u8d2a\u5fc3&#xff09;\u662f\u6b63\u786e\u7684\u3002<\/li>\n<li>\u7ebf\u6027\u57fa\u4e5f\u662f\u4e00\u4e2a\u62df\u9635&#xff08;\u5411\u91cf\u62df\u9635&#xff09;\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u7b2c\u4e94\u90e8\u5206&#xff1a;\u603b\u7ed3<\/h3>\n<p>\u57fa\u7840\u7b97\u6cd5\u662f\u5730\u57fa&#xff0c;\u5730\u57fa\u4e0d\u7262&#xff0c;\u5730\u52a8\u5c71\u6447\u3002<\/p>\n<ul>\n<li>\u9047\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>          10<\/p>\n<p>          9<\/p>\n<p>        10^9<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6570\u636e&#xff1f;\u5148\u60f3\u79bb\u6563\u5316\u3002<\/li>\n<li>\u6c42\u533a\u95f4\u6700\u503c&#xff1f;\u9759\u6001\u7528 ST\u8868&#xff0c;\u52a8\u6001\u7528\u7ebf\u6bb5\u6811&#xff08;\u8fdb\u9636&#xff09;\u3002<\/li>\n<li>\u5224\u5b9a\u7b54\u6848\u5bb9\u6613&#xff0c;\u6c42\u7b54\u6848\u96be&#xff1f;\u4e8c\u5206\u7b54\u6848\u3002<\/li>\n<li>\u8d2a\u5fc3\u62ff\u4e0d\u51c6&#xff1f;\u8bd5\u7740\u5f80\u62df\u9635\u7684\u4ea4\u6362\u6027\u4e0a\u60f3\u4e00\u60f3\u3002<\/li>\n<\/ul>\n<p>\u5e0c\u671b\u8fd9\u7bc7\u603b\u7ed3\u80fd\u6210\u4e3a\u4f60\u5237\u9898\u8def\u4e0a\u7684\u706f\u5854&#xff01;<\/p>\n<p>\u8bc4\u8bba\u533a\u6253\u5361&#xff1a;\u4f60\u89c9\u5f97\u6700\u96be\u7406\u89e3\u7684\u57fa\u7840\u7b97\u6cd5\u662f\u54ea\u4e2a&#xff1f;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6458\u8981&#xff1a;\u8fd8\u5728\u4e3a\u7b97\u6cd5\u57fa\u7840\u4e0d\u7262\u800c\u82e6\u607c&#xff1f;\u9762\u5bf9\u9898\u76ee\u4e0d\u77e5\u9053\u7528\u4ec0\u4e48\u7b97\u6cd5&#xff1f;\u672c\u6587\u5c06 algorithm 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