{"id":169,"date":"2026-01-08T00:02:46","date_gmt":"2026-01-07T16:02:46","guid":{"rendered":"https:\/\/jiangqvweihuan.cn\/?p=169"},"modified":"2026-03-19T19:01:28","modified_gmt":"2026-03-19T11:01:28","slug":"%e5%89%8d%e7%bc%80%e5%92%8c%e5%92%8c%e5%b7%ae%e5%88%86","status":"publish","type":"post","link":"https:\/\/jiangqvweihuan.cn\/index.php\/2026\/01\/08\/%e5%89%8d%e7%bc%80%e5%92%8c%e5%92%8c%e5%b7%ae%e5%88%86\/","title":{"rendered":"\u7b97\u6cd5\u5fc3\u4f20\u00b7\u524d\u7f00\u548c\u4e0e\u5dee\u5206"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">\u4e00\u3001\u4e00\u7ef4\u524d\u7f00\u548c\u4e0e\u4e00\u7ef4\u5dee\u5206<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u4e00\u7ef4\u524d\u7f00\u548c<\/h3>\n\n\n\n<p>\u4ec0\u4e48\u662f\u524d\u7f00\u548c\u5462\uff1f<\/p>\n\n\n\n<p>\u524d\u7f00\u548c\uff08Prefix Sum) \u987e\u540d\u601d\u4e49\u5c31\u662f\u6784\u5efa\u4e00\u4e2a\u65b0\u7684\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5143\u7d20\u8868\u793a\u539f\u59cb\u6570\u7ec4\u4ece\u8d77\u59cb\u4f4d\u7f6e\u5230\u5f53\u524d\u4f4d\u7f6e\u7684<strong>\u7d2f\u52a0\u548c<\/strong>\u3002\u662f\u4e00\u79cd\u91cd\u8981\u7684\u6570\u636e\u9884\u5904\u7406\u6280\u672f,\u8fd9\u79cd\u901a\u8fc7\u7a7a\u95f4\u6362\u65f6\u95f4\u7684\u601d\u60f3\uff0c\u5c06\u533a\u95f4\u6c42\u548c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ece<strong>O(n)<\/strong>\u964d\u5230<strong>O(1)<\/strong>\uff0c\u53ef\u4ee5\u5feb\u901f\u6c42\u51fa\u6570\u7ec4<strong>\u4efb\u610f\u533a\u95f4\u7684\u548c<\/strong>,\u662f\u89e3\u51b3\u8bb8\u591a<strong>\u533a\u95f4\/\u5b50\u6570\u7ec4\u95ee\u9898<\/strong>\u7684\u5229\u5668\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u76f8\u6bd4\u8f83\u5176\u4ed6\u7b97\u6cd5\u800c\u8a00\uff0c\u524d\u7f00\u548c\u66f4\u50cf\u662f\u4e00\u79cd\u89e3\u9898\u7684\u6280\u5de7\uff0c\u4e00\u79cd\u4f18\u5316\u65b9\u5f0f\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u5f15\u5165<\/p>\n\n\n\n<p>\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a<strong>n<\/strong>\u7684\u6574\u6570\u5e8f\u5217\uff0c\u63a5\u4e0b\u6765\u518d\u8f93\u5165<strong>m<\/strong>\u4e2a\u8be2\u95ee\uff0c\u6bcf\u4e2a\u8be2\u95ee\u8f93\u5165\u4e00\u5bf9<strong>l<\/strong>,<strong>r<\/strong>\u3002\u5bf9\u4e8e\u6bcf\u4e2a\u8be2\u95ee\uff0c\u8f93\u51fa\u539f\u5e8f\u5217\u4e2d\u4ece\u7b2cl\u4e2a\u6570\u5230\u7b2cr\u4e2a\u6570\u7684\u548c\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6b21\u95ee\u9898\uff0c\u6211\u4eec\u4e0d\u96be\u60f3\u5230\u66b4\u529b\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>const int N = 1e5 + 10;  \/\/\u79d1\u5b66\u8ba1\u6570\u6cd5,\u5373100010\nint a&#91;N];\nint n,m;\ncin &gt;&gt; n &gt;&gt; m;           \/\/\u8f93\u5165n,m\nfor(int i = 1; i &lt;= n; i++)\n{\n    cin &gt;&gt; a&#91;i];\n}\nwhile(m--)\n{\n    int l, r;\n    int sum = 0;\n    cin &gt;&gt; l &gt;&gt; r;\n    for(int i = l; i &lt;= r; i++)\n    { \n        sum += a&#91;i];\n    }\n    cout &lt;&lt; sum &lt;&lt; endl;\n}<\/code><\/pre>\n\n\n\n<p>\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u6bb5\u4ee3\u7801\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a<strong>O(n * m)<\/strong>\uff0c\u5982\u679cn\u548cm\u7684\u7a0d\u5927\u4e00\u4e9b\uff0c\u4ee3\u7801\u5c31\u4f1a\u8d85\u65f6\u3002<\/p>\n\n\n\n<p>\u8fd9\u65f6\u5019\u6211\u4eec\u5982\u679c\u60f3\u4f18\u5316\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e00\u7ef4\u524d\u7f00\u548c\u3002<\/p>\n\n\n\n<p>\u9996\u5148\u505a\u4e00\u4e2a\u9884\u5904\u7406\uff0c\u5b9a\u4e49\u4e00\u4e2a<strong>pref[n + 1]<\/strong>\u6570\u7ec4\uff0c<strong>pref[i]<\/strong>\u4ee3\u8868a\u6570\u7ec4\u4e2d\u524di\u4e2a\u6570\u7684\u548c\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>const int N = 1e5 + 10;\nint pref&#91;N], a&#91;N];       \/\/pref&#91;i]=a&#91;1]+a&#91;2]+a&#91;3].....a&#91;i];\nfor(int i = 1;i &lt;= n; i++)\n{ \n    pref&#91;i] = pref&#91;i - 1] + a&#91;i];   \n}<\/code><\/pre>\n\n\n\n<p>\u5bf9\u4e8e\u67e5\u8be2\u64cd\u4f5c\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cin &gt;&gt; l &gt;&gt; r;\ncout &lt;&lt; pref&#91;r] - pref&#91;l-1] &lt;&lt; endl;<\/code><\/pre>\n\n\n\n<p>\u67e5\u8be2\u64cd\u4f5c\u4f18\u5316\u4e3acout &lt;&lt; pref[r] &#8211; pref[l-1] &lt;&lt; endl\uff0c\u53ef\u4ee5\u53c2\u8003\u4e0b\u56fe\u7406\u89e3\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/\u4e00\u7ef4\u524d\u7f00\u548c\u533a\u95f4\u6c42\u548c-2.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"777\" height=\"306\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/\u4e00\u7ef4\u524d\u7f00\u548c\u533a\u95f4\u6c42\u548c-2.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-181\"  sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/div><\/figure>\n\n\n\n<p>\u4e00\u7ef4\u524d\u7f00\u548c\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u9884\u5904\u7406O(n),\u67e5\u8be2O(1)\uff0cO(n + m)\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u6548\u7387\u6bd4\u8f83\u9ad8\u6548\uff0c\u540e\u7eed\u4e5f\u4f1a\u6709\u4e00\u4e9b\u5176\u4ed6\u7684\u89e3\u6cd5\uff0c\u6bd4\u5982\u8bf4\u7ebf\u6bb5\u6811\uff0c\u6811\u72b6\u6570\u7ec4\u7b49\uff0c\u524d\u7f00\u548c\u7684\u8fd0\u884c\u65f6\u95f4\u662f\u6700\u77ed\u7684\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.\u4e00\u7ef4\u5dee\u5206<\/h3>\n\n\n\n<p>\u5dee\u5206\uff08Difference\uff09\u53ef\u4ee5\u7406\u89e3\u4e3a\u524d\u7f00\u548c\u7684<strong>\u9006\u8fd0\u7b97<\/strong>\uff0c\u5c31\u662f\u5c06\u6570\u5217\u4e2d<strong>\u6bcf\u4e00\u9879\u5206\u522b\u4e0e\u524d\u4e00\u9879\u505a\u5dee<\/strong>\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-3.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"590\" height=\"306\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-3.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-182\"  sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/div><\/figure>\n\n\n\n<p>\u90a3\u4e48\u6211\u4eec\u5982\u4f55\u5f97\u5230\u4e00\u7ef4\u5dee\u5206\u6570\u7ec4\u5462\uff1f<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-4.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"658\" height=\"211\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-4.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-183\"  sizes=\"auto, (max-width: 658px) 100vw, 658px\" \/><\/div><\/figure>\n\n\n\n<p>\u4e00\u7ef4\u5dee\u5206\u6570\u7ec4\u7684\u6709\u4ec0\u4e48\u7528\u5904\u5462\uff1f<\/p>\n\n\n\n<p>\u540c\u6837\u7684\uff0c\u6211\u4eec\u601d\u8003\u4e00\u4e2a\u8fd9\u6837\u7684\u95ee\u9898\uff1a<\/p>\n\n\n\n<p>\u6709n\u4e2a\u6574\u6570\uff0c\u7ed9\u5b9am\u4e2a\u533a\u95f4[l, r ]\uff0c\u8ba9\u6211\u4eec\u628aa\u6570\u7ec4\u4e2d\u7684[l, r] \u533a\u95f4\u4e2d\u7684\u6bcf\u4e00\u4e2a\u6570\u90fd\u52a0\u4e0ac\uff0c\u5373 a[l] + c , a[l + 1] + c , a[l + 2] + c, &#8230;&#8230; ,a[r] + c\uff0c\u6700\u540e\u6309\u987a\u5e8f\u7ed9\u51fa\u6539\u53d8\u540e\u7684\u6570\u7ec4\u3002<\/p>\n\n\n\n<p>\u66b4\u529b\u6cd5\u548c\u4e4b\u524d\u7684\u524d\u7f00\u548c\u57fa\u672c\u601d\u8def\u76f8\u540c\uff0c\u540c\u6837\u7684\uff0c\u95ee\u9898\u662f\u65f6\u95f4\u590d\u6742\u5ea6<strong>O(n * m)<\/strong>\u592a\u9ad8\u3002<\/p>\n\n\n\n<p>\u8fd9\u65f6\u5019\u6211\u4eec\u8981\u8003\u8651\u5dee\u5206\u505a\u6cd5\u3002<\/p>\n\n\n\n<p>a\u6570\u7ec4\u662fdiff\u6570\u7ec4\u7684\u524d\u7f00\u548c\u6570\u7ec4\uff0c\u5bf9diff\u6570\u7ec4\u7684diff[i]\u7684\u4fee\u6539\uff0c\u4f1a\u5f71\u54cd\u5230a\u6570\u7ec4\u4e2d\u4ecea[i]\u53ca\u5f80\u540e\u7684\u6bcf\u4e00\u4e2a\u6570\u3002<\/p>\n\n\n\n<p>\u9996\u5148\u8ba9\u5dee\u5206diff\u6570\u7ec4\u4e2d\u7684 diff[l] + c \uff0c\u901a\u8fc7\u524d\u7f00\u548c\u8fd0\u7b97\uff0ca\u6570\u7ec4\u4ece\u4e0b\u6807<strong>l<\/strong>\u5411\u540e\u90fd\u591a\u4e86\u4e00\u4e2a\u6211\u4eec\u6dfb\u52a0\u7684<strong>c<\/strong>\uff0c<\/p>\n\n\n\n<p>\u5373 a[l] + c ,a[l + 1] + c, &#8230;&#8230; ,a[n] + c;<\/p>\n\n\n\n<p>\u7136\u540e\u6211\u4eec\u628a\u5dee\u5206diff\u6570\u7ec4\u4e2d\u7684 diff[r + 1] &#8211; c\u901a\u8fc7\u524d\u7f00\u548c\u8fd0\u7b97\uff0ca\u6570\u7ec4\u4ece\u4e0b\u6807<strong>r<\/strong>\u5411\u540e\u51cf\u53bb\u4e00\u4e2a<strong>c<\/strong>\uff0c<\/p>\n\n\n\n<p>\u5373 a[r + 1] &#8211; c,a[r + 2] &#8211; c, &#8230;&#8230; ,a[n] &#8211; c;<\/p>\n\n\n\n<p>\u6b64\u65f6\u6211\u4eec\u53d1\u73b0\uff0c\u5bf9diff\u6570\u7ec4\u505a\u524d\u7f00\u548c\u5f97\u5230\u7684\u65b0\u6570\u7ec4\u5373\u4e3a\u4ec5\u6539\u53d8\u533a\u95f4<strong>l<\/strong>\u5230<strong>r<\/strong>\u4e14\u6539\u53d8\u91cf\u4e3a<strong>c<\/strong>\u7684\u6570\u7ec4\u3002<\/p>\n\n\n\n<p>\u56e0\u6b64\u6211\u4eec\u5f97\u5230\u7ed3\u8bba\uff1a<\/p>\n\n\n\n<p>\u8981\u5bf9\u6570\u7ec4\u533a\u95f4 [ l, r ] \u7684\u6570\u636e\u90fd\u52a0\u4e0ac\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u8981\u5bf9\u5dee\u5206\u6570\u7ec4\u8fdb\u884c\u64cd\u4f5c\u5373\u53ef\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em><strong>\u5373 l\u5230\u524d\u4e00\u4e2a\u5143\u7d20\u7684\u8ddd\u79bb+c\uff0cr\u5230\u540e\u4e00\u4e2a\u5143\u7d20\u7684\u8ddd\u79bb-<\/strong><\/em><strong><em>c<\/em><\/strong><\/li>\n<\/ul>\n\n\n\n<pre class=\"wp-block-code\"><code>diff&#91;l] += c;\ndiff&#91;r+1] -= c;<\/code><\/pre>\n\n\n\n<p>\u8fd9\u91cc\u7ed9\u51fa\u6838\u5fc3\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\/\/ \u6838\u5fc31\uff1a\u6784\u5efa\u5dee\u5206\u6570\u7ec4\n\/\/ diff&#91;i] = a&#91;i] - a&#91;i-1]\uff0c\u5176\u4e2ddiff&#91;1] = a&#91;1]\uff08\u56e0\u4e3aa&#91;0]=0\uff09\nfor (int i = 1; i &lt;= n; ++i) {\n    diff&#91;i] = a&#91;i] - a&#91;i - 1];\n}\n\/\/ \u6838\u5fc32\uff1a\u6267\u884cm\u6b21\u533a\u95f4\u52a0\u64cd\u4f5c\nfor (int i = 0; i &lt; m; ++i) {\n    int l, r, c;\n    cin &gt;&gt; l &gt;&gt; r &gt;&gt; c; \n    \/\/ \u5dee\u5206\u6570\u7ec4\u7684\u6838\u5fc3\u64cd\u4f5c\n    diff&#91;l] += c;          \/\/ l\u4f4d\u7f6e\u52a0c\uff0c\u5f71\u54cd\u540e\u9762\u6240\u6709\u5143\u7d20\n    diff&#91;r + 1] -= c;      \/\/ r+1\u4f4d\u7f6e\u51cfc\uff0c\u62b5\u6d88r\u4e4b\u540e\u7684\u5f71\u54cd\n}\n\/\/ \u6838\u5fc33\uff1a\u901a\u8fc7\u524d\u7f00\u548c\u8fd8\u539f\u6570\u7ec4\n\/\/ \u5bf9\u5dee\u5206\u6570\u7ec4\u6c42\u524d\u7f00\u548c\uff0c\u5f97\u5230\u4fee\u6539\u540e\u7684\u539f\u6570\u7ec4\nfor (int i = 1; i &lt;= n; ++i) {\n    diff&#91;i] += diff&#91;i - 1];  \/\/ \u524d\u7f00\u548c\uff1a\u5f53\u524d\u503c = \u524d\u4e00\u4e2a\u503c + \u5f53\u524d\u5dee\u503c\n    cout &lt;&lt; diff&#91;i] &lt;&lt; \" \";\n}\n<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e8c\u3001\u4e8c\u7ef4\u524d\u7f00\u548c\u548c\u4e8c\u7ef4\u5dee\u5206<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u4e8c\u7ef4\u524d\u7f00\u548c<\/h3>\n\n\n\n<p>\u4e8c\u7ef4\u524d\u7f00\u548c\uff08\u4e5f\u79f0\u4e3a\u79ef\u5206\u56fe\uff09\u662f\u5bf9\u4e8c\u7ef4\u6570\u7ec4\uff08\u77e9\u9635\uff09\u7684\u4e00\u79cd\u9884\u5904\u7406\u65b9\u6cd5\uff0c\u53ef\u4ee5<strong>\u5feb\u901f\u8ba1\u7b97\u4efb\u610f\u5b50\u77e9\u9635\u7684\u5143\u7d20\u548c<\/strong>\u3002<\/p>\n\n\n\n<p>\u5f15\u5165<\/p>\n\n\n\n<p>\u8f93\u5165\u4e00\u4e2an\u884cm\u5217\u7684\u6574\u6570\u77e9\u9635\uff0c\u518d\u8f93\u5165q\u4e2a\u8be2\u95ee\uff0c\u6bcf\u4e2a\u8be2\u95ee\u5305\u542b\u56db\u4e2a\u6574\u6570x1, y1, x2, y2\uff0c\u8868\u793a\u4e00\u4e2a\u5b50\u77e9\u9635\u7684\u5de6\u4e0a\u89d2\u5750\u6807\u548c\u53f3\u4e0b\u89d2\u5750\u6807\u3002\u5bf9\u4e8e\u6bcf\u4e2a\u8be2\u95ee\u8f93\u51fa\u5b50\u77e9\u9635\u4e2d\u6240\u6709\u6570\u7684\u548c\u3002<\/p>\n\n\n\n<p>\u540c\u4e00\u7ef4\u524d\u7f00\u548c\u4e00\u6837\uff0c\u6211\u4eec\u5148\u6765\u5b9a\u4e49\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4diff[n + 1][m + 1]\u3002<\/p>\n\n\n\n<p>diff[i][j] \u8868\u793a\u4e8c\u7ef4\u6570\u7ec4\u4e2d\uff0c\u5de6\u4e0a\u89d2(1, 1)\u5230\u53f3\u4e0b\u89d2(i, j)\u6240\u5305\u56f4\u7684\u77e9\u9635\u5143\u7d20\u7684\u548c\u3002<\/p>\n\n\n\n<p>\u90a3\u4e48\u4e8c\u7ef4\u524d\u7f00\u548c\u7684\u516c\u5f0f\u5c31\u662f\uff1a<em><strong>pref[i][j] = pref[i &#8211; 1][j] + pref[i][j &#8211; 1 ] + <em><strong>a[i] [j]<\/strong><\/em> &#8211; pref[i &#8211; 1][j &#8211; 1]<\/strong><\/em><\/p>\n\n\n\n<p>\u53ef\u4ee5\u501f\u52a9\u4e0b\u56fe\u7406\u89e3\u8bb0\u5fc6\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-6.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"317\" height=\"280\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-6.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-186\"  sizes=\"auto, (max-width: 317px) 100vw, 317px\" \/><\/div><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-7.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"797\" height=\"185\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-7.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-187\"  sizes=\"auto, (max-width: 797px) 100vw, 797px\" \/><\/div><\/figure>\n\n\n\n<p>\u6574\u4e2a\u5916\u56f4\u84dd\u8272\u77e9\u5f62\u9762\u79ef<strong>pref[i][j]<\/strong><\/p>\n\n\n\n<p>=&nbsp;<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\">\u7eff\u8272\u9762\u79efpref[i &#8211; 1][j]<\/mark>&nbsp;+&nbsp;<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-amber-color\">\u9ec4\u8272\u9762\u79efpref[i][j &#8211; 1]<\/mark>&nbsp;&nbsp;+&nbsp;<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>\u5c0f\u65b9\u5757\u7684\u9762\u79ef<\/strong>a[i][j]<\/mark> &#8211; <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">\u91cd\u590d\u52a0\u7684\u7ea2\u8272\u7684\u9762\u79efpref[i &#8211; 1][j &#8211; 1]<\/mark>;<\/p>\n\n\n\n<p>\u901a\u8fc7\u56fe\u4f8b\u6211\u4eec\u518d\u6b21\u5370\u8bc1\u4e86\u4e0a\u8ff0\u4e8c\u7ef4\u524d\u7f00\u548c\u516c\u5f0f\u7684\u5408\u7406\u6027\u4e0e\u6b63\u786e\u6027\u3002<\/p>\n\n\n\n<p>\u6709\u4e86\u4ee5\u4e0a\u57fa\u7840\uff0c\u73b0\u5728\u82e5\u8981\u6c42\u67d0\u4e00\u533a\u57df\u7684\u5b50\u77e9\u9635\u548c\u600e\u4e48\u529e\uff1f<\/p>\n\n\n\n<p>\u6211\u4eec\u4ece\u56fe\u4f8b\u4e2d\u7406\u89e3\u5982\u4f55\u6c42\u4ee5(x1, y1)\u4e3a\u5de6\u4e0a\u89d2\uff0c(x2, y2)\u4e3a\u53f3\u4e0b\u89d2\u7684\u5b50\u77e9\u9635\u7684\u548c\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-9.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"925\" height=\"184\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-9.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-190\"  sizes=\"auto, (max-width: 925px) 100vw, 925px\" \/><\/div><\/figure>\n\n\n\n<p>\u5206\u6790\u5f97\u51fa\u4ee5(x1, y1)\u4e3a\u5de6\u4e0a\u89d2\uff0c(x2, y2)\u4e3a\u53f3\u4e0b\u89d2\u7684\u5b50\u77e9\u9635\u7684\u548c\u4e3a\uff1a<br><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">pref[x2, y2]<\/mark> &#8211; <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">pref[x1 &#8211; 1, y2]<\/mark> &#8211; <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-amber-color\">pref[x2, y1 &#8211; 1]<\/mark> + <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">pref[x1 &#8211; 1, y1 &#8211; 1]<\/mark><\/p>\n\n\n\n<p>\u4e0b\u9762\u7ed9\u51fa\u4ee5\u4e0a\u5b50\u77e9\u9635\u95ee\u9898\u6838\u5fc3\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\/\/ \u8ba1\u7b97\u524d\u7f00\u548c\nfor (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n        pref&#91;i]&#91;j] = pref&#91;i - 1]&#91;j] + pref&#91;i]&#91;j - 1] - pref&#91;i - 1]&#91;j - 1] + a&#91;i]&#91;j];\n    }\n}\nwhile (q--) {\n    int x1, y1, x2, y2;\n    cin &gt;&gt; x1 &gt;&gt; y1 &gt;&gt; x2 &gt;&gt; y2;  \n    \/\/ \u8ba1\u7b97\u5b50\u77e9\u9635\u548c\uff0c#define ll long long\n    ll sum = pref&#91;x2]&#91;y2] - pref&#91;x1 - 1]&#91;y2] - pref&#91;x2]&#91;y1 - 1] + pref&#91;x1 - 1]&#91;y1 - 1];\n    cout &lt;&lt; sum &lt;&lt; endl;\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">2.\u4e8c\u7ef4\u5dee\u5206<\/h3>\n\n\n\n<p>\u4e8c\u7ef4\u5dee\u5206\u662f\u5904\u7406<strong>\u4e8c\u7ef4\u77e9\u9635\u533a\u95f4\u64cd\u4f5c<\/strong>\u7684\u9ad8\u6548\u5de5\u5177\uff0c\u53ef\u4ee5\u770b\u4f5c<strong>\u4e8c\u7ef4\u524d\u7f00\u548c\u7684\u9006\u8fd0\u7b97<\/strong>\u3002\u7ed9\u5b9a\u4e00\u4e2a m\u00d7n \u7684\u539f\u59cb\u77e9\u9635 A\uff0c\u5176\u5bf9\u5e94\u7684\u4e8c\u7ef4\u5dee\u5206\u77e9\u9635 D \u662f\u4e00\u4e2a (m+2)\u00d7(n+2) \u7684\u77e9\u9635,\u5176\u4e2d\u8d85\u51fa\u8fb9\u754c\u7684 A[i][j] \u89c6\u4e3a 0\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u4e00\u4e2a\u7ed9\u5b9a\u7684\u4e8c\u7ef4\u6570\u7ec4 a[][]\uff0c\u5b83\u7684\u4e8c\u7ef4\u5dee\u5206\u6570\u7ec4 diff[i][j] \u53ef\u4ee5\u7528\u5982\u4e0b\u516c\u5f0f\u8ba1\u7b97\uff1a<\/p>\n\n\n\n<p>\u2460 <strong><em>diff[0][0]=a[0][0]<\/em><\/strong><\/p>\n\n\n\n<p>\u2461 <strong><em>diff[0][j]=a[0][j]-a[0][j &#8211; 1]<\/em><\/strong><\/p>\n\n\n\n<p>\u2462 <strong><em>diff[i][0]=a[i][0]-a[i &#8211; 1][0]<\/em><\/strong><\/p>\n\n\n\n<p>\u2463 \u9012\u63a8\u5f0f\uff1a<strong><em>diff[i][j]=a[i][j]-a[i &#8211; 1][j]-a[i][j &#8211; 1]+a[i &#8211; 1][j &#8211; 1]<\/em><\/strong><\/p>\n\n\n\n<p>\u4e0a\u8ff0\u516c\u5f0f\u662f\u901a\u8fc7\u4e8c\u7ef4\u6570\u7ec4 a[][]\u662f\u4e8c\u7ef4\u5dee\u5206\u6570\u7ec4\u7684\u524d\u7f00\u548c\u8fd9\u4e2a\u6761\u4ef6\u63a8\u5bfc\u51fa\u6765\u7684\u3002<\/p>\n\n\n\n<p>\u501f\u52a9\u56fe\u4f8b\u7406\u89e3\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-10.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"487\" height=\"331\" data-original=\"https:\/\/jiangqvweihuan.cn\/wp-content\/uploads\/2026\/01\/image-10.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-194\" style=\"width:487px;height:auto\"  sizes=\"auto, (max-width: 487px) 100vw, 487px\" \/><\/div><\/figure>\n\n\n\n<pre class=\"wp-block-code\"><code><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">diff&#91;x\u2081]&#91;y\u2081]<\/mark>     += c; \/\/ \u5bf9\u5dee\u5206\u6570\u7ec4\u7684\u66f4\u65b0\uff0c\u5f71\u54cd\u4ece(x1,y1)\u5230\u53f3\u4e0b\u89d2\u7684\u6574\u4e2a\u533a\u57df\n<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\">diff&#91;x\u2081]&#91;y\u2082+1]<\/mark>   -= c; \/\/ \u5bf9\u5dee\u5206\u6570\u7ec4\u7684\u66f4\u65b0\uff0c\u4fee\u6b63\u4ece(x1,y2+1)\u5230\u53f3\u4e0b\u89d2\u7684\u533a\u57df\n<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">diff&#91;x\u2082+1]&#91;y\u2081]<\/mark>   -= c; \/\/ \u5bf9\u5dee\u5206\u6570\u7ec4\u7684\u66f4\u65b0\uff0c\u4fee\u6b63\u4ece(x2+1,y1)\u5230\u53f3\u4e0b\u89d2\u7684\u533a\u57df\n<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">diff&#91;x\u2082+1]&#91;y\u2082+1]<\/mark> += c; \/\/ \u5bf9\u5dee\u5206\u6570\u7ec4\u7684\u66f4\u65b0\uff0c\u4fee\u6b63\u4ece(x2+1,y2+1)\u5230\u53f3\u4e0b\u89d2\u7684\u533a\u57df<\/code><\/pre>\n\n\n\n<p>\u76f8\u5173\u4e60\u9898\u63a8\u8350\u6211\u4f1a\u62bd\u65f6\u95f4\u5177\u4f53\u7814\u7a76\u7136\u540e\u5217\u51fa\u6765\u94fe\u63a5\uff0c\u6211\u4eec\u53ea\u505a\u7cbe\u54c1\uff0c\u795d\u5929\u5929\u5f00\u5fc3\uff01<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e09\u3001\u7ecf\u5178\u4f8b\u9898<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u4e00\u7ef4\u524d\u7f00\u548c<\/h3>\n\n\n\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1114\">P1114 \u201c\u975e\u5e38\u7537\u5973\u201d\u8ba1\u5212 &#8211; \u6d1b\u8c37<\/a><br><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1182\">P1182 \u6570\u5217\u5206\u6bb5 Section II &#8211; \u6d1b\u8c37<\/a><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.\u4e00\u7ef4\u5dee\u5206<\/h3>\n\n\n\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/P1083\">P1083 [NOIP 2012 \u63d0\u9ad8\u7ec4] \u501f\u6559\u5ba4 &#8211; \u6d1b\u8c37<\/a><br><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.\u4e8c\u7ef4\u524d\u7f00\u548c<\/h3>\n\n\n\n<h3 class=\"wp-block-heading\">4.\u4e8c\u7ef4\u5dee\u5206<\/h3>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u3001\u4e00\u7ef4\u524d\u7f00\u548c\u4e0e\u4e00\u7ef4\u5dee\u5206 1.\u4e00\u7ef4\u524d\u7f00\u548c \u4ec0\u4e48\u662f\u524d\u7f00\u548c\u5462\uff1f \u524d\u7f00\u548c\uff08Prefix Sum) \u987e\u540d\u601d\u4e49\u5c31\u662f\u6784\u5efa\u4e00 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":156,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,15,12],"tags":[],"class_list":["post-169","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-article","category-algorithm-template","category-programming-algorithm-road"],"_links":{"self":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/169","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/comments?post=169"}],"version-history":[{"count":18,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/169\/revisions"}],"predecessor-version":[{"id":351,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/169\/revisions\/351"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/media\/156"}],"wp:attachment":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/media?parent=169"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/categories?post=169"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/tags?post=169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}