{"id":97454,"date":"2026-08-29T14:48:54","date_gmt":"2026-08-29T06:48:54","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/97454.html"},"modified":"2026-08-29T14:48:54","modified_gmt":"2026-08-29T06:48:54","slug":"%e3%80%90%e5%89%8d%e7%bc%80%e5%92%8c%e6%a8%a1%e6%9d%bf%e3%80%91%ef%bc%9a%e5%ad%90%e7%9f%a9%e9%98%b5%e7%9a%84%e5%92%8c%ef%bc%88%e5%89%8d%e7%bc%80%e5%92%8c%ef%bc%89-%e9%a2%98%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/97454.html","title":{"rendered":"\u3010\u524d\u7f00\u548c\u6a21\u677f\u3011\uff1a\u5b50\u77e9\u9635\u7684\u548c\uff08\u524d\u7f00\u548c\uff09\u2014\u2014\u9898\u89e3"},"content":{"rendered":"<h2><span style=\"color:#494949\">\u00a0\u00a0&#x1f44b; \u6b22\u8fce\u9605\u8bfb<\/span><\/h2>\n<p><span style=\"color:#494949\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"1080\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260829064849-6a9280d19dc52.jpg\" width=\"1920\" \/><\/span><\/p>\n<p><span style=\"color:#494949\">&#x1f3af; 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\u5de6 &#043; \u4e0a &#8211; \u5de6\u4e0a\u89d2 &#043; \u81ea\u5df1&#xff09;&#xff1a;<\/span><\/p>\n<table>\n<tr><span style=\"color:#494949\">i\\\\j<\/span><span style=\"color:#494949\">0<\/span><span style=\"color:#494949\">1<\/span><span style=\"color:#494949\">2<\/span><span style=\"color:#494949\">3<\/span><\/tr>\n<tbody>\n<tr>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color:#494949\">1<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">1<\/span><\/td>\n<td><span style=\"color:#494949\">3<\/span><\/td>\n<td><span style=\"color:#494949\">6<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color:#494949\">2<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">5<\/span><\/td>\n<td><span style=\"color:#494949\">12<\/span><\/td>\n<td><span style=\"color:#494949\">21<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color:#494949\">3<\/span><\/td>\n<td><span style=\"color:#494949\">0<\/span><\/td>\n<td><span style=\"color:#494949\">12<\/span><\/td>\n<td><span style=\"color:#494949\">27<\/span><\/td>\n<td><span style=\"color:#494949\">45<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color:#494949\">\u67e5\u8be2\u00a0[1,1]-[2,2]&#xff1a;dp[2][2] &#8211; dp[0][2] &#8211; dp[2][0] &#043; dp[0][0] &#061; 12 &#8211; 0 &#8211; 0 &#043; 0 &#061; 12&#xff08;&#061;\u00a01&#043;2&#043;4&#043;5&#xff09;<\/span><\/p>\n<h3><span style=\"color:#1c7892\">\u4e09\u3001\u6b63\u786e\u6027\u8bf4\u660e&#xff08;\u7b80\u5355\u7248\u672c&#xff09;<\/span><\/h3>\n<ul>\n<li><span style=\"color:#494949\">\u6784\u5efa\u516c\u5f0f\u81ea\u6d3d&#xff1a;dp[i][j]\u00a0\u8868\u793a\u4ece (1,1) \u5230 (i,j) \u7684\u77e9\u5f62\u548c\u3002dp[i][j-1] &#043; dp[i-1][j]\u00a0\u8986\u76d6\u4e86\u5de6\u5757\u4e0e\u4e0a\u5757&#xff0c;\u4f46\u5de6\u4e0a\u89d2\u5757\u88ab\u52a0\u4e86\u4e24\u6b21&#xff0c;\u51cf\u53bb\u00a0dp[i-1][j-1]\u00a0\u8865\u56de&#xff0c;\u518d\u52a0\u81ea\u8eab\u00a0v[i][j]\u2014\u2014\u5bb9\u65a5\u4fdd\u8bc1\u4e0d\u91cd\u4e0d\u6f0f\u3002<\/span><\/li>\n<li><span style=\"color:#494949\">\u67e5\u8be2\u516c\u5f0f\u6b63\u786e&#xff1a;(x1,y1)-(x2,y2)\u00a0&#061; \u5927\u5757\u00a0dp[x2][y2]\u00a0\u51cf\u53bb\u5de6\u4fa7\u591a\u51fa\u7684\u7ad6\u6761\u00a0dp[x1-1][y2]\u00a0\u4e0e\u4e0a\u65b9\u591a\u51fa\u7684\u6a2a\u6761\u00a0dp[x2][y1-1]&#xff0c;\u4f46\u5de6\u4e0a\u89d2\u5757\u88ab\u51cf\u4e86\u4e24\u6b21&#xff0c;\u9700\u52a0\u56de\u00a0dp[x1-1][y1-1]\u2014\u2014\u56db\u9879\u7ec4\u5408\u6070\u597d\u7559\u4e0b\u76ee\u6807\u5b50\u77e9\u9635\u3002<\/span><\/li>\n<li><span style=\"color:#494949\">\u4e0b\u6807\u4ece 1 \u8d77\u6b65\u5b89\u5168&#xff1a;x1-1\u3001y1-1\u00a0\u6700\u5c0f\u4e3a 0&#xff0c;dp[0][*]\u3001dp[*][0]\u00a0\u6052\u4e3a 0&#xff0c;\u65e0\u9700\u7279\u5224&#xff0c;\u6240\u6709\u5408\u6cd5\u67e5\u8be2\u7edf\u4e00\u9002\u7528&#xff0c;\u4e0d\u6f0f\u4e0d\u9519\u3002<\/span><\/li>\n<\/ul>\n<p><span style=\"color:#494949\">&#x1f4cc; \u4f8b\u5b50&#xff1a;\u5bb9\u65a5\u4e3a\u4ec0\u4e48\u6210\u7acb&#xff08;\u4ee5\u6784\u5efa dp[2][2] \u4e3a\u4f8b&#xff09;<\/span><\/p>\n<p><span style=\"color:#494949\">dp[2][2] &#061; dp[2][1] &#043; dp[1][2] &#8211; dp[1][1] &#043; v[2][2]\u00a0&#061; (1&#043;4) &#043; (1&#043;2) &#8211; (1) &#043; 5 &#061; 12\u2014\u2014\u5de6\u5757\u00a0{1,4}\u00a0\u52a0\u4e0a\u5757\u00a0{1,2}\u00a0\u65f6&#xff0c;\u89d2\u4e0a\u00a01\u00a0\u88ab\u52a0\u4e86\u4e24\u6b21&#xff0c;\u51cf\u53bb\u4e00\u6b21\u8865\u56de&#xff0c;\u518d\u52a0\u81ea\u5df1\u00a05&#xff0c;\u6070\u597d\u662f\u00a0{1,2,4,5}\u00a0\u7684\u548c&#xff0c;\u4e0d\u91cd\u4e0d\u6f0f\u3002<\/span><\/p>\n<h3><span style=\"color:#1c7892\">\u56db\u3001\u5b9e\u73b0\u7ec6\u8282&#xff08;\u8fb9\u754c\u9632\u62a4&#xff09;<\/span><\/h3>\n<ul>\n<li><span style=\"color:#494949\">\u521d\u59cb\u5316&#xff1a;v\u3001dp\u00a0\u5747\u5f00\u00a0(n&#043;1)\u00d7(m&#043;1)&#xff0c;\u7b2c 0 \u884c\/\u5217\u9ed8\u8ba4 0\u3002<\/span><\/li>\n<li><span style=\"color:#494949\">\u8fb9\u754c\u9632\u62a4&#xff1a;\u4e0b\u6807\u4ece 1 \u5f00\u59cb\u907f\u514d\u00a0x1-1\u3001y1-1\u00a0\u8d8a\u754c&#xff08;dp[0][*] &#061; 0\u00a0\u515c\u5e95&#xff09;&#xff1b;\u7d2f\u52a0\u548c\u7528\u00a0long long&#xff08;1e3\u00d71e3\u00a0\u4e2a\u00a01e9\u00a0\u76f8\u52a0\u8fbe\u00a01e15&#xff0c;\u8fdc\u8d85 int \u4e0a\u9650\u00a02.1e9&#xff09;&#xff1b;\u6ce8\u610f Windows \u4e0b\u00a0long\u00a0\u662f 32 \u4f4d&#xff0c;\u8de8\u5e73\u53f0\u5e94\u7edf\u4e00\u7528\u00a0long long&#xff1b;\u5927\u91cf\u67e5\u8be2\u65f6\u5173\u95ed\u6d41\u540c\u6b65\u63d0\u5347 IO \u901f\u5ea6\u3002<\/span><\/li>\n<li><span style=\"color:#494949\">\u590d\u6742\u5ea6&#xff1a;\u65f6\u95f4 O(n\u00d7m &#043; q)&#xff08;\u9884\u5904\u7406 O(n\u00d7m)&#xff0c;\u6bcf\u6b21\u67e5\u8be2 O(1)&#xff09;&#xff0c;\u7a7a\u95f4 O(n\u00d7m)&#xff08;\u4e24\u4e2a\u77e9\u9635&#xff09;\u3002<\/span><\/li>\n<li><span style=\"color:#494949\">\u5173\u952e\u64cd\u4f5c&#xff1a;dp[i][j] &#061; dp[i][j-1] &#043; dp[i-1][j] &#8211; dp[i-1][j-1] &#043; v[i][j]&#xff08;\u6784\u5efa&#xff09;\u3001dp[x2][y2] &#8211; dp[x1-1][y2] &#8211; dp[x2][y1-1] &#043; dp[x1-1][y1-1]&#xff08;\u67e5\u8be2&#xff09;\u3002<\/span><\/li>\n<\/ul>\n<h3><span style=\"color:#1c7892\">\u4e94\u3001\u8fd4\u56de\u503c&#xff08;\u76ee\u6807\u6620\u5c04&#xff09;<\/span><\/h3>\n<ul>\n<li><span style=\"color:#494949\">\u6bcf\u6b21\u67e5\u8be2\u8f93\u51fa\u56db\u9879\u5bb9\u65a5\u516c\u5f0f\u7684\u7ed3\u679c&#xff1a;\u5b50\u77e9\u9635 (x1,y1)-(x2,y2) \u7684\u5143\u7d20\u4e4b\u548c&#xff0c;\u5bf9\u5e94\u9898\u76ee&#034;\u8f93\u51fa\u6bcf\u6b21\u8be2\u95ee\u7684\u5b50\u77e9\u9635\u548c&#034;\u3002<\/span><\/li>\n<\/ul>\n<h2><span style=\"color:#494949\">\u4e09.\u4ee3\u7801<\/span><\/h2>\n<p>#include &lt;iostream&gt;<br \/>\n#include &lt;vector&gt;<br \/>\nusing namespace std;<\/p>\n<p>int main()<br \/>\n{<br \/>\n    ios::sync_with_stdio(false);    \/\/ \u5173\u95ed\u6d41\u540c\u6b65&#xff0c;\u52a0\u901f IO<br \/>\n    cin.tie(0);<\/p>\n<p>    int n, m, q;                    \/\/ n \u884c m \u5217\u77e9\u9635&#xff0c;q \u6b21\u8be2\u95ee<br \/>\n    cin &gt;&gt; n &gt;&gt; m &gt;&gt; q;<\/p>\n<p>    \/\/ \u539f\u77e9\u9635&#xff1a;\u4e0b\u6807\u4ece 1 \u5f00\u59cb&#xff0c;\u7b2c 0 \u884c\/\u5217\u95f2\u7f6e&#xff08;\u4fdd\u6301\u516c\u5f0f\u7edf\u4e00&#xff09;<br \/>\n    vector&lt;vector&lt;long long&gt;&gt; v(n &#043; 1, vector&lt;long long&gt;(m &#043; 1));<br \/>\n    for (int i &#061; 1; i &lt;&#061; n; i&#043;&#043;)<br \/>\n    {<br \/>\n        for (int j &#061; 1; j &lt;&#061; m; j&#043;&#043;)<br \/>\n        {<br \/>\n            cin &gt;&gt; v[i][j];<br \/>\n        }<br \/>\n    }<\/p>\n<p>    \/\/ 1. \u9884\u5904\u7406\u4e8c\u7ef4\u524d\u7f00\u548c&#xff1a;dp[i][j] &#061; \u4ece (1,1) \u5230 (i,j) \u7684\u77e9\u5f62\u548c<br \/>\n    vector&lt;vector&lt;long long&gt;&gt; dp(n &#043; 1, vector&lt;long long&gt;(m &#043; 1));<br \/>\n    for (int i &#061; 1; i &lt;&#061; n; i&#043;&#043;)<br \/>\n    {<br \/>\n        for (int j &#061; 1; j &lt;&#061; m; j&#043;&#043;)<br \/>\n        {<br \/>\n            \/\/ \u5bb9\u65a5&#xff1a;\u5de6\u5757 &#043; \u4e0a\u5757 &#8211; \u91cd\u53e0\u89d2 &#043; \u81ea\u8eab<br \/>\n            dp[i][j] &#061; dp[i][j &#8211; 1] &#043; dp[i &#8211; 1][j] &#8211; dp[i &#8211; 1][j &#8211; 1] &#043; v[i][j];<br \/>\n        }<br \/>\n    }<\/p>\n<p>    \/\/ 2. \u5b50\u77e9\u9635\u67e5\u8be2&#xff1a;O(1) \u56db\u6b21\u76f8\u52a0\u51cf<br \/>\n    int x1, y1, x2, y2;             \/\/ \u5b50\u77e9\u9635\u4e24\u4e2a\u89d2\u5750\u6807<br \/>\n    for (int i &#061; 0; i &lt; q; i&#043;&#043;)<br \/>\n    {<br \/>\n        cin &gt;&gt; x1 &gt;&gt; y1 &gt;&gt; x2 &gt;&gt; y2;<br \/>\n        \/\/ \u5927\u5757 &#8211; \u5de6\u4fa7\u7ad6\u6761 &#8211; \u4e0a\u65b9\u6a2a\u6761 &#043; \u5de6\u4e0a\u89d2\u5757&#xff08;\u88ab\u51cf\u4e24\u6b21\u9700\u52a0\u56de&#xff09;<br \/>\n        cout &lt;&lt; dp[x2][y2] &#8211; dp[x1 &#8211; 1][y2] &#8211; dp[x2][y1 &#8211; 1] &#043; dp[x1 &#8211; 1][y1 &#8211; 1] &lt;&lt; &#039;\\\\n&#039;;<br \/>\n    }<br \/>\n    return 0;<br \/>\n} <\/p>\n<h2><span style=\"color:#494949\">\u56db\u3001\u6613\u9519\u70b9\u5206\u6790<\/span><\/h2>\n<h4><span style=\"color:#1a439c\">\u96be\u70b91&#xff1a;\u6784\u5efa\u516c\u5f0f\u7684\u56db\u9879\u6765\u6e90&#xff08;\u5bb9\u65a5&#xff09;<\/span><\/h4>\n<p>dp[i][j] &#061; dp[i][j &#8211; 1] &#043; dp[i &#8211; 1][j] &#8211; dp[i &#8211; 1][j &#8211; 1] &#043; v[i][j]; <\/p>\n<p><span style=\"color:#494949\">dp[i][j-1]\u00a0\u8986\u76d6\u5de6\u8fb9\u5757&#xff0c;dp[i-1][j]\u00a0\u8986\u76d6\u4e0a\u8fb9\u5757&#xff0c;\u4f46\u5de6\u4e0a\u89d2\u7684\u00a0dp[i-1][j-1]\u00a0\u88ab\u4e24\u5757\u540c\u65f6\u5305\u542b\u3001\u52a0\u4e86\u4e24\u6b21&#xff0c;\u5fc5\u987b\u51cf\u53bb\u4e00\u6b21&#xff1b;\u518d\u52a0\u4e0a\u81ea\u8eab\u00a0v[i][j]\u3002\u6f0f\u6389\u00a0&#8211; dp[i-1][j-1]\u00a0\u6216\u00a0&#043; v[i][j]\u00a0\u90fd\u4f1a\u8ba9\u524d\u7f00\u548c\u6574\u4f53\u9519\u4f4d&#xff0c;\u4e14\u9519\u8bef\u4f1a\u9010\u683c\u7d2f\u79ef&#xff0c;\u6700\u7ec8\u67e5\u8be2\u5168\u90e8\u51fa\u9519\u3002<\/span><\/p>\n<h4><span style=\"color:#1a439c\">\u96be\u70b92&#xff1a;\u67e5\u8be2\u516c\u5f0f\u4e0e\u6784\u5efa\u516c\u5f0f\u7684\u5bb9\u65a5\u65b9\u5411\u76f8\u53cd<\/span><\/h4>\n<p>dp[x2][y2] &#8211; dp[x1 &#8211; 1][y2] &#8211; dp[x2][y1 &#8211; 1] &#043; dp[x1 &#8211; 1][y1 &#8211; 1] <\/p>\n<p><span style=\"color:#494949\">\u6784\u5efa\u662f&#034;\u5de6 &#043; \u4e0a &#8211; \u91cd\u53e0 &#043; \u81ea\u8eab&#034;&#xff08;\u5408\u5e76&#xff09;&#xff0c;\u67e5\u8be2\u662f&#034;\u5927\u5757 &#8211; \u5de6\u6761 &#8211; \u4e0a\u6761 &#043; \u91cd\u53e0\u89d2&#034;&#xff08;\u5265\u79bb&#xff09;\u3002\u56db\u4e2a\u4e0b\u6807\u6781\u6613\u5199\u6df7&#xff1a;dp[x2][y2]\u00a0\u662f\u5927\u5757\u53f3\u4e0b\u89d2&#xff0c;\u4e24\u4e2a\u51cf\u9879\u5206\u522b\u7528\u00a0x1-1\u3001y1-1\u00a0\u4e0e\u00a0x2\u00a0\u4ea4\u53c9&#xff0c;\u52a0\u9879\u662f\u00a0(x1-1, y1-1)\u3002\u5efa\u8bae\u6309&#034;\u53f3\u4e0b &#8211; \u5de6\u754c &#8211; \u4e0a\u754c &#043; \u5de6\u4e0a\u89d2&#034;\u8bb0\u5fc6&#xff0c;\u5e76\u9010\u9879\u6838\u5bf9\u4e0b\u6807\u3002<\/span><\/p>\n<h4><span style=\"color:#1a439c\">\u96be\u70b93&#xff1a;\u4e3a\u4ec0\u4e48\u51cf\u4e24\u6b21\u7684\u89d2\u8981\u52a0\u56de\u6765<\/span><\/h4>\n<p>&#043; dp[x1 &#8211; 1][y1 &#8211; 1]   \/\/ \u8fd9\u4e2a&#034;&#043;&#034;\u7684\u6765\u6e90 <\/p>\n<p><span style=\"color:#494949\">\u5927\u5757\u51cf\u53bb\u5de6\u7ad6\u6761\u00a0dp[x1-1][y2]\u00a0\u65f6&#xff0c;\u628a\u5de6\u4e0a\u89d2\u5757\u00a0dp[x1-1][y1-1]\u00a0\u4e5f\u51cf\u4e86\u4e00\u6b21&#xff1b;\u518d\u51cf\u4e0a\u6a2a\u6761\u00a0dp[x2][y1-1]\u00a0\u65f6&#xff0c;\u540c\u4e00\u4e2a\u89d2\u5757\u88ab\u51cf\u4e86\u7b2c\u4e8c\u6b21\u3002\u5b83\u672c\u4e0d\u5c5e\u4e8e\u76ee\u6807\u5b50\u77e9\u9635&#xff0c;\u5e94\u53ea\u88ab\u51cf\u4e00\u6b21&#xff0c;\u6240\u4ee5\u5fc5\u987b\u52a0\u56de\u4e00\u6b21&#xff08;\u5bb9\u65a5\u7684&#034;\u52a0\u56de\u91cd\u53e0&#034;&#xff09;\u3002\u6f0f\u6389\u8fd9\u4e2a\u00a0&#043;\u00a0\u4f1a\u5c11\u52a0\u4e00\u4e2a\u89d2\u5757&#xff0c;\u4ec5\u5f53\u00a0x1&gt;1 &amp;&amp; y1&gt;1\u00a0\u65f6\u9519\u8bef\u624d\u66b4\u9732&#xff0c;\u6781\u96be\u6392\u67e5\u3002<\/span><\/p>\n<h4><span style=\"color:#1a439c\">\u96be\u70b94&#xff1a;\u4e0b\u6807\u4ece 1 \u5f00\u59cb\u4e0e\u7b2c 0 \u884c\/\u5217\u7684\u515c\u5e95<\/span><\/h4>\n<p>vector&lt;vector&lt;long long&gt;&gt; dp(n &#043; 1, vector&lt;long long&gt;(m &#043; 1));<br \/>\n\/\/ \u67e5\u8be2\u65f6 dp[x1 &#8211; 1][y1 &#8211; 1]&#xff0c;x1&#061;1 \u65f6\u8bbf\u95ee dp[0][0] <\/p>\n<p><span style=\"color:#494949\">\u82e5\u4ece 0 \u5f00\u59cb\u5b58&#xff0c;\u67e5\u8be2\u00a0x1&#061;1\u00a0\u65f6\u00a0dp[x1-1]\u00a0\u5373\u00a0dp[-1]&#xff0c;\u8d8a\u754c\u8bbf\u95ee\u3002\u4ece 1 \u5f00\u59cb\u540e\u7b2c 0 \u884c\/\u5217\u5929\u7136\u4e3a 0&#xff0c;x1-1&#061;0\u00a0\u65f6\u516c\u5f0f\u81ea\u52a8\u9000\u5316&#xff08;dp[0][y2] &#061; 0&#xff09;&#xff0c;\u65e0\u9700\u4efb\u4f55\u7279\u5224\u2014\u2014\u8fd9\u662f&#034;\u591a\u5f00\u4e00\u5708&#034;\u7684\u7ecf\u5178\u624b\u6cd5\u3002<\/span><\/p>\n<h2><span style=\"color:#494949\">\u4e94\u3001\u6d41\u7a0b\u56fe<\/span><\/h2>\n<p><span style=\"color:#494949\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"1343\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260829064850-6a9280d2aff56.png\" width=\"430\" \/><\/span><\/p>\n<h3 style=\"background-color:transparent\"><span style=\"color:#494949\">\u00a0&#x1f3af; 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