{"id":1230,"date":"2024-08-30T18:33:00","date_gmt":"2024-08-30T10:33:00","guid":{"rendered":"https:\/\/diary.bid\/?p=1230"},"modified":"2025-08-30T18:33:40","modified_gmt":"2025-08-30T10:33:40","slug":"%e7%ac%ac%e4%b8%80%e7%ab%a0-%e5%9f%ba%e7%a1%80%e7%ae%97%e6%b3%95%ef%bc%88%e4%ba%8c%ef%bc%89","status":"publish","type":"post","link":"https:\/\/diary.bid\/?p=1230","title":{"rendered":"\u7b2c\u4e00\u7ae0 \u57fa\u7840\u7b97\u6cd5\uff08\u4e8c\uff09"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u672c\u8282\u5185\u5bb9\u4e3b\u8981\u4e3a\u9ad8\u7cbe\u5ea6\u8fd0\u7b97\u3001\u524d\u7f00\u548c\u4e0e\u5dee\u5206\u7684\u4e3b\u8981\u601d\u60f3\u548c\u4ee3\u7801\u3002<\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"toc-0\">\u9ad8\u7cbe\u5ea6<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-1\">\u57fa\u672c\u601d\u60f3<\/h3>\n\n\n\n<p>\u5e38\u8003\u7c7b\u578b\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$A+B$ \uff0c\u4e24\u4e2a\u7684\u4f4d\u6570\u5927\u7ea6\u5728 $10^6$ \u3002<\/li>\n\n\n\n<li>$A-B$ \uff0c\u4e24\u4e2a\u7684\u4f4d\u6570\u5927\u7ea6\u5728 $10^6$ \u3002<\/li>\n\n\n\n<li>$A\\times a$ \uff0c $A$ \u4f4d\u6570\u5927\u7ea6 $10^6$ \uff0c$a&lt;10^9$ \u3002<\/li>\n\n\n\n<li>$A \\div b$ \uff0c $A$ \u4f4d\u6570\u5927\u7ea6 $10^6$ \uff0c$b&lt;10^9$ \u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-2\">\u5982\u4f55\u5b58\u50a8<\/h4>\n\n\n\n<p>\u4f7f\u7528\u6570\u7ec4\u8fdb\u884c\u5b58\u50a8\uff0c\u6bcf\u4e00\u4e2a\u4f4d\u7f6e\u5b58\u4e00\u4e2a\u6570\uff0c\u4e2a\u4f4d\u6570\u5b58\u5230\u5f00\u5934\uff08\u6570\u7ec40\u4f4d\u7f6e\uff09\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-3\">\u5982\u4f55\u8fd0\u7b97<\/h4>\n\n\n\n<p>\u7ad6\u5f0f\u52a0\u51cf\u4e58\u9664\uff0c\u6bcf\u4e00\u4f4d\u662f $(A_i+B_i+t) \\% 10$ \uff0c\u5176\u4e2d $t$ \u4e3a\u4e0a\u4e00\u4f4d\u8fd0\u7b97\u7684\u8fdb\u4f4d\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-4\">\u9ad8\u7cbe\u5ea6\u52a0\u6cd5<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/2bpic.oss-cn-beijing.aliyuncs.com\/image-20250729213732222.png\" alt=\"image-20250729213732222\"\/><\/figure>\n\n\n\n<p>\u9ad8\u7cbe\u5ea6\u52a0\u6cd5\u4f7f\u7528 $t$ \u6765\u8bb0\u5f55\u8fdb\u4f4d\uff0c\u5f53\u4e0d\u8d85\u8fc7 $A.size()$ \u548c $B.size()$ \u7684\u65f6\u5019\u7ee7\u7eed\u8ba1\u7b97<br>$$<br>C_i = (t+A_i+B_i)\\%10\uff0ct = \\frac{t+A_i+B_i}{10}<br>$$<br>\u9700\u8981\u6ce8\u610f\u7684\u662f\u5faa\u73af\u5b8c\u4e86\u53ef\u80fd\u8fd8\u4f1a\u8fdb\u4e00\u4f4d $t$ \uff0c\u6240\u4ee5\u8981\u5728\u7ed3\u5c3e\u518d\u5224\u65ad\u4e00\u6b21\u3002<\/p>\n\n\n\n<p><strong>\u6838\u5fc3\u51fd\u6570\uff1a<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >vector&lt;int&gt; add(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B){\n    vector&lt;int&gt; C;\n    int t = 0;\n    for(int i = 0; i &lt; A.size() || i &lt; B.size();i++){\n        if(i &lt; A.size()) t += A&#091;i];\n        if(i &lt; B.size()) t += B&#091;i];\n        C.push_back(t % 10);\n        t \/= 10;\n    }\n    if(t) C.push_back(t);\n    return C;\n} <\/code><\/pre>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u6b64\u5904\u4f7f\u7528\u5f15\u7528\u7c7b\u578b\uff0c\u907f\u514d\u590d\u5236\u6570\u7ec4\u5bfc\u81f4\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\uff0c\u4e0b\u540c<\/p>\n<\/blockquote>\n\n\n\n<p><strong>\u5b8c\u6574\u4f7f\u7528\uff1a<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >#include &lt;iostream&gt;\n#include &lt;vector&gt;\n\nusing namespace std;\n\nconst int N = 1e6 + 10;\n\nvector&lt;int&gt; add(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B){\n    vector&lt;int&gt; C;\n    int t = 0;\n    for(int i = 0; i &lt; A.size() || i &lt; B.size();i++){\n        if(i &lt; A.size()) t += A&#091;i];\n        if(i &lt; B.size()) t += B&#091;i];\n        C.push_back(t % 10);\n        t \/= 10;\n    }\n    if(t) C.push_back(t);\n    return C;\n} \n\nint main(){\n    string a,b;\n    vector&lt;int&gt; A,B;\n\n    cin &gt;&gt; a &gt;&gt; b;\n    for(int i = a.size() - 1; i&gt;=0; i--){\n        A.push_back(a&#091;i] - &#039;0&#039;);\n    }\n    for(int i = b.size() - 1; i&gt;=0; i--){\n        B.push_back(b&#091;i] - &#039;0&#039;);\n    }\n    auto C = add(A,B);\n    for(int i = C.size() - 1;i&gt;=0;i--){\n        printf(&quot;%d&quot;,C&#091;i]);\n    }\n    return 0;\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-5\">\u9ad8\u7cbe\u5ea6\u51cf\u6cd5<\/h3>\n\n\n\n<p><strong>\u6b65\u9aa4\uff1a<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5224\u65ad $A$ \u548c $B$ \u7684\u5927\u5c0f\uff1a<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c $A&lt;B$ \uff0c\u5219 $return -jian(B,A)$ \uff1b<\/li>\n\n\n\n<li>\u5982\u679c $A\\ge B$ \uff0c\u5219\u6b63\u5e38\u8ba1\u7b97\u3002<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4e00\u4f4d\u4e00\u4f4d\u8fdb\u884c\u51cf\u6cd5\uff0c\u8bbe\u7f6e\u4e00\u4e2a\u5927\u5c0f\u4e3a $t$ \u7684\u53d8\u91cf\u8bb0\u5f55\u501f\u4f4d\uff0c\u8ba1\u7b97\u65b9\u6cd5\u5982\u4e0b\u56fe\uff1a<img decoding=\"async\" src=\"https:\/\/2bpic.oss-cn-beijing.aliyuncs.com\/image-20250730160413266.png\" alt=\"image-20250730160413266\"><\/li>\n\n\n\n<li>\u5224\u65ad\u5f97\u5230\u7684\u7b54\u6848\u7b2c\u4e00\u4f4d\u662f\u5426\u4f1a\u4e3a0\uff0c \u82e5\u4e3a0\u5219\u53bb\u6389\u4e00\u4f4d\uff0c\u5faa\u73af\u5230\u7b2c\u4e00\u4f4d\u4e0d\u662f0\u6216\u8005\u53ea\u5269\u4e00\u4f4d\u6570\u4e14\u4e3a0\u7684\u60c5\u51b5\u3002<\/li>\n<\/ol>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-6\">\u4ee3\u7801<\/h4>\n\n\n\n<p><strong>\u9ad8\u7cbe\u5ea6\u6bd4\u8f83\u5927\u5c0f\uff1a<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >bool cmp(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B){\n    if (A.size() != B.size()) return A.size() - B.size();\n    for(int i = A.size() - 1; i &gt;= 0 ; i--){\n        if(A&#091;i] != B&#091;i]) return A&#091;i] - B&#091;i];\n    }\n    return true;\n}<\/code><\/pre>\n\n\n\n<p><strong>\u9ad8\u7cbe\u5ea6\u51cf\u6cd5\uff1a<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >vector&lt;int&gt; jian(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B){\n    vector&lt;int&gt; C;\n    \/\/if (cmp(A,B) == false) return jian(B,A); \u6b64\u5904\u65e0\u6cd5\u52a0\u8d1f\u53f7\uff0c\u8981\u5728\u5916\u9762\u5224\u65ad\u9650\u5236\n    int t = 0;\n    for(int i = 0; i&lt;A.size()||i&lt;B.size(); i++){\n        int w; \/\/ \u7528\u6765\u5b58B&#091;i]\u6216\u8005B\u7684\u7a7a\u4f4d\n        if(i&gt;=B.size()) w = 0; \n        else w = B&#091;i];\n        if(A&#091;i] - t &gt;= w){ \n            C.push_back(A&#091;i] - t - w);\n            t = 0;\n        }\n        else {\n            C.push_back(A&#091;i] - w - t + 10);\n            t = 1;\n        }\n    }\n    while (C.size()&gt;1 &amp;&amp; C.back() == 0) C.pop_back();\n    return C;\n}<\/code><\/pre>\n\n\n\n<p>\u8001\u5e08\u7ed9\u7684\u7248\u672c\uff08\u4f18\u96c5\u4e00\u4e9b\uff09\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >\/\/ C = A - B, \u6ee1\u8db3A &gt;= B, A &gt;= 0, B &gt;= 0\nvector&lt;int&gt; sub(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B)\n{\n    vector&lt;int&gt; C;\n    for (int i = 0, t = 0; i &lt; A.size(); i ++ )\n    {\n        t = A&#091;i] - t;\n        if (i &lt; B.size()) t -= B&#091;i];\n        C.push_back((t + 10) % 10);\n        if (t &lt; 0) t = 1;\n        else t = 0;\n    }\n\n    while (C.size() &gt; 1 &amp;&amp; C.back() == 0) C.pop_back();\n    return C;\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-7\">\u9ad8\u7cbe\u5ea6\u4e58\u4f4e\u7cbe\u5ea6<\/h3>\n\n\n\n<p>\u6838\u5fc3\u601d\u60f3\u8fd8\u662f<strong>\u5217\u7ad6\u5f0f<\/strong>\uff1a\u5bf9\u4e8e\u4e58\u6cd5 $A\\times b$ \uff0c$A$ \u7684\u6bcf\u4e00\u4f4d\u8ba1\u4e3a $a_i$ \uff0c\u8fdb\u4f4d\u8bbe\u4e3a $t$ \u3002\u5219 $C_i = (a_i \\times b + t) \\% 10$ \uff0c\u800c $t = \\frac{a_i \\times b + t}{10}$ \u3002<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u76f8\u5bf9\u7b80\u5355\uff0c\u6ca1\u6709\u4ec0\u4e48\u9700\u8981\u989d\u5916\u8003\u8651\u7684\uff0c\u9700\u6ce8\u610f\u5faa\u73af\u6761\u4ef6\u8981\u5305\u62ec $t &gt; 0$ \uff0c\u540c\u6837\u5904\u7406\u6700\u540e\u8fd8\u6709\u8fdb\u4f4d\u7684\u60c5\u51b5\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >vector&lt;int&gt; cheng(vector&lt;int&gt; &amp;A, int b){\n    vector&lt;int&gt; C;\n    int t = 0;\n    for(int i = 0; i &lt; A.size() || t &gt; 0;i++){\n        if(i &lt; A.size()) t += A&#091;i] * b;\n        C.push_back(t%10);\n        t = t\/10;\n    }\n    return C;\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-8\">\u9ad8\u7cbe\u5ea6\u9664\u4ee5\u4f4e\u7cbe\u5ea6<\/h3>\n\n\n\n<p>\u8fd8\u662f\u5217\u7ad6\u5f0f\uff0c\u4f46\u662f\u9664\u6cd5\u662f\u4ece\u6700\u9ad8\u4f4d\u5f00\u59cb\u770b\u7684\u3002\u6240\u4ee5\u521d\u59cb $i = A.size() - 1$ \u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u9664\u6cd5 $A\\div B$ \uff0c$A$ \u7684\u6bcf\u4e00\u4f4d\u8ba1\u4e3a $a_i$ \uff0c\u4e0a\u4e00\u4f4d\u7684\u9000\u4f4d\u8bbe\u4e3a $r$ \u3002\u5219 $C_i = \\frac{r \\times 10 + a_i}{b}$ \uff0c$r = (r \\times 10 + a_i)\\%b$ \u3002<\/p>\n\n\n\n<p>\u5faa\u73af\u5b8c\u5f97\u5230\u7684 $C$ \u662f\u53cd\u7740\u7684\uff0c\u6240\u4ee5\u8981\u5148\u7528 <code class=\"prettyprint linenums\" >#include &lt;algorithm&gt;<\/code>\u91cc\u9762\u7684<code class=\"prettyprint linenums\" >reverse(C.begin(), C.end())<\/code>\u51fd\u6570\u8fdb\u884c\u8c03\u8f6c\u3002\u6b64\u65f6\u4e5f\u6709\u53ef\u80fd\u5b58\u5728\u540c\u6837\u7684\u95ee\u9898\uff0c\u5373\u524d\u9762\u6709\u524d\u5bfc\u96f6\uff0c\u6240\u4ee5\u9700\u8981\u5220\u9664\u6240\u6709\u76840\uff0c\u4e0e\u51cf\u6cd5\u4e00\u6837\u3002<\/p>\n\n\n\n<p>\u4e5f\u53ef\u4ee5\u628a\u4f59\u6570\u8f93\u51fa\u51fa\u6765\uff0c\u53ef\u4ee5\u5728\u53c2\u6570\u91cc\u589e\u52a0\u4e00\u4e2a\u5f15\u7528\u7c7b\u578b <code class=\"prettyprint linenums\" >int &amp;r<\/code>\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >vector&lt;int&gt; chu(vector&lt;int&gt; &amp;A, int b){\n    vector&lt;int&gt; C;\n    int r = 0;\n    for(int i = A.size() - 1; i&gt;=0; i--){\n        r = r * 10 + A&#091;i];\n        C.push_back(r\/b);\n        r %= b;\n    }\n    reverse(C.begin(), C.end());\n    while (C.size() &gt; 1 &amp;&amp; C.back() == 0) C.pop_back();\n    return C;\n}<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"toc-9\">\u524d\u7f00\u548c<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-10\">\u4e00\u7ef4\u524d\u7f00\u548c<\/h3>\n\n\n\n<p>\u6709\u4e00\u4e2a\u957f\u5ea6\u4e3a $n$ \u7684\u6570\u7ec4\uff0c\u524d\u7f00\u548c\u6570\u7ec4 $S_i = a_1 + a_2 + a_3 +\u2026+a_i$ \uff0c\u6570\u7ec4\u6700\u597d\u4ece\u4e0b\u68071\u5f00\u59cb\u3002\u5b9a\u4e49 $a_0 = 0$ \u4ee5\u53ca $S_0 = 0$ \u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-11\">\u5982\u4f55\u6c42 $S_i$<\/h4>\n\n\n\n<p>\u9012\u5f52\uff0c$S_i = S_{i-1} + a_i$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-12\">\u4f5c\u7528<\/h4>\n\n\n\n<p>\u6c42\u6570\u7ec4\u91cc\u4e00\u6bb5\u6570\u7684\u548c\uff1a\u6bd4\u5982\u6c42 $[l,r]$ \u4e4b\u95f4\u7684\u6570\u7ec4\u548c\uff0c\u76f4\u63a5\u4f7f\u7528 $S_r - S_{l-1}$ \u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-13\">\u793a\u4f8b\u9898\u76ee<\/h4>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/2bpic.oss-cn-beijing.aliyuncs.com\/image-20250731144106651.png\" alt=\"image-20250731144106651\"\/><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-14\">\u4ee3\u7801<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >#include &lt;iostream&gt;\nusing namespace std;\n\nint n,m;\nint arr&#091;100009],s&#091;100009];\nint r,l;\n\nint main(){\n    cin &gt;&gt; n &gt;&gt; m;\n    arr&#091;0] = 0;\n    s&#091;0] = 0;\n    for(int i = 1; i &lt;= n; i++){\n        cin &gt;&gt; arr&#091;i];\n        s&#091;i] = s&#091;i-1] + arr&#091;i];\n    }\n    for(int i=0;i&lt;m;i++){\n        cin &gt;&gt; l &gt;&gt; r;\n        cout &lt;&lt; (s&#091;r] - s&#091;l-1]) &lt;&lt; endl;\n    }\n}<\/code><\/pre>\n\n\n\n<p><em>\u63d0\u5347cin\u6548\u7387\uff1a<\/em> <code class=\"prettyprint linenums\" >ios::sync_with_stdio(false);<\/code><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-15\">\u4e8c\u7ef4\u524d\u7f00\u548c<\/h3>\n\n\n\n<p>$S[i][j]$ \u8868\u793a\u4ee5$ (i, j) $\u4e3a\u53f3\u4e0b\u89d2\u7684\u77e9\u5f62\u533a\u57df\u7684\u548c\u3002\u603b\u4f53\u662f\u5bb9\u65a5\u539f\u7406\u3002<\/p>\n\n\n\n<p><strong>\u6784\u9020\u65b9\u6cd5<\/strong>:$S[i][j] = A[i][j] + S[i - 1][j] + S[i][j - 1] - S[i - 1][j - 1]$<\/p>\n\n\n\n<p>\u540c\u6837\u8fd8\u662f\u8981\u8bbe <code class=\"prettyprint linenums\" >S[0][j] = 0<\/code> \u548c <code class=\"prettyprint linenums\" >S[i][0] = 0<\/code>\uff0c <code class=\"prettyprint linenums\" >S[i][j]<\/code> \u4e2d <code class=\"prettyprint linenums\" >i<\/code> \u548c <code class=\"prettyprint linenums\" >j<\/code> \u90fd\u4ece1\u5f00\u59cb\u8ba1\u6570\u3002\u6b64\u65f6\u4f7f\u7528<code class=\"prettyprint linenums\" >vector<\/code>\u4f1a\u65b9\u4fbf\u4e00\u4e9b\uff0c\u6216\u8005\u5b9a\u4e49\u5230\u5168\u5c40\u6570\u7ec4\u3002<\/p>\n\n\n\n<p><strong>\u4f7f\u7528\u65b9\u6cd5\uff1a<\/strong><\/p>\n\n\n\n<p>\u67e5\u8be2\u5b50\u77e9\u9635\u548c\uff0c\u67e5\u8be2\u4ee5$ (x_1, y_1) $\u4e3a\u5de6\u4e0a\u89d2\u3001$(x_2, y_2) $\u4e3a\u53f3\u4e0b\u89d2\u7684\u5b50\u77e9\u9635\u548c\uff1a<\/p>\n\n\n\n<p>$sum = S[x_2][y_2]-S[x_1-1][y_2]-S[x_2][y_1-1]+S[x_1-1][y_1-1]$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"toc-16\">\u5dee\u5206<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-17\">\u4e00\u7ef4\u5dee\u5206<\/h3>\n\n\n\n<p>\u5bf9\u4e8e\u6570\u7ec4 $a_1,a_2,\u2026,a_n$ \uff0c\u6784\u9020\u6570\u7ec4 $b_1,b_2,\u2026,b_n$ \uff0c\u4f7f\u5f97 $a_i=b_1+b_2+\u2026+b_i$\uff0c\u5373\u8ba9b\u6570\u7ec4\u662fa\u7684\u524d\u7f00\u548c\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-18\">\u8ba1\u7b97b\u6570\u7ec4\uff1a<\/h4>\n\n\n\n<p>$$<br>\\begin{cases}<br>b_1 = a_1 \\<br>b_i = a_i - a_{i-1}<br>\\end{cases}<br>$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-19\">\u4f7f\u7528\u65b9\u6cd5<\/h4>\n\n\n\n<p>\u7531 $O(n)$ \u53ef\u4ee5\u7531$b$\u63a8\u51fa$a$\u3002<\/p>\n\n\n\n<p>\u82e5\u9700\u8981\u5bf9$a$\u6570\u7ec4\u7684 $[l,r]$ \u533a\u95f4\u4e0a\u6bcf\u4e00\u4e2a\u6570\u90fd\u52a0\u4e0a\u5e38\u6570$c$\uff0c\u53ea\u9700\u7ed9 $b_l + c$ \u5e76\u7ed9 $b_{r+1} - c$ \u3002\u6b64\u65f6\u7ed9$a$\u6570\u7ec4\u8fde\u7eed\u533a\u95f4\u5168\u90e8\u6dfb\u52a0\u4e0a\u6216\u8005\u51cf\u53bb\u4e00\u4e2a\u5e38\u6570\uff0c\u53ea\u9700\u8981\u6539\u53d8\u5dee\u5206\u6570\u7ec4\u7684\u4e24\u4e2a\u6570\uff0c\u6b64\u65f6\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u6709 $O(1)$ \u3002<\/p>\n\n\n\n<p>\u53ef\u4ee5\u5f00\u59cb\u628a\u6240\u6709 $a_i$ \u90fd\u770b\u4e3a0\uff0c\u7136\u540e\u8c03\u7528\u8fde\u7eed\u533a\u95f4\u52a0\u5e38\u6570\u7684\u51fd\u6570\u7528\u6765\u521d\u59cb\u5316\u8d4b\u503c\uff0c\u8fd9\u6837\u53ea\u9700\u8981\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >void insert(int*b,int l,int r,int c){\n    b&#091;l] += c;\n    b&#091;r+1] -= c;\n    return;\n}<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"toc-20\">\u4ee3\u7801<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"prettyprint linenums\" >#include &lt;iostream&gt;\nusing namespace std;\n\nint n,m;\nint arr&#091;100009],b&#091;100009];\nint r,l,c;\n\nvoid insert(int*b,int l,int r,int c){\n    b&#091;l] += c;\n    b&#091;r+1] -= c;\n    return;\n}\n\nint main(){\n    cin &gt;&gt; n &gt;&gt; m;\n    arr&#091;0] = 0;\n    b&#091;0] = 0;\n    for(int i = 1; i &lt;= n; i++){\n        scanf(&quot;%d&quot;,&amp;arr&#091;i]);\n    }\n    for(int i = 1; i &lt;= n; i++){\n        insert(b,i,i,arr&#091;i]);\n    }\n    while(m--){\n        int l,r,c;\n        scanf(&quot;%d%d%d&quot;,&amp;l,&amp;r,&amp;c);\n        insert(b,l,r,c);\n    }\n    for(int i = 1 ; i &lt;= n; i++){\n        printf(&quot;%d &quot;,b&#091;i]);\n    }\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"toc-21\">\u4e8c\u7ef4\u5dee\u5206<\/h3>\n\n\n\n<p>\u6709\u4e00\u4e2a\u539f\u77e9\u9635 $a_{ij}$ \uff0c\u6784\u9020\u5dee\u5206\u77e9\u9635 $b_{ij}$ \uff0c\u4f7f$a$\u662f$b$\u7684\u524d\u7f00\u548c\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/2bpic.oss-cn-beijing.aliyuncs.com\/image-20250731184049700.png\" alt=\"image-20250731184049700\"\/><\/figure>\n\n\n\n<p>\u82e5\u60f3\u5c06 $(x_1,y_1)$ \u5230 $(x_2,y_2)$ \u8303\u56f4\u5185\u6240\u6709\u6570\u52a0 $c$ \uff0c\u9700\u8981\u8fdb\u884c\u4ee5\u4e0b\u64cd\u4f5c\uff1a<br>$$<br>\\begin{cases}<br>b_{x_1,y_1} + c \\<br>b_{x_2+1,y1} - c \\<br>b_{x_1, y_2+1} - c \\<br>b_{x_2+1,y_2+1} + c<br>\\end{cases}<br>$$<br>\u6539\u53d8\u8fd9\u56db\u4e2a\u503c\u5373\u53ef\u3002<\/p>\n<div class=\"comment--location\"><svg version=\"1.1\" viewBox=\"0 0 368.666 368.666\"  width=\"14\" height=\"14\"><g><path d=\"M184.333,0C102.01,0,35.036,66.974,35.036,149.297c0,33.969,11.132,65.96,32.193,92.515\n\t\t\tc27.27,34.383,106.572,116.021,109.934,119.479l7.169,7.375l7.17-7.374c3.364-3.46,82.69-85.116,109.964-119.51\n\t\t\tc21.042-26.534,32.164-58.514,32.164-92.485C333.63,66.974,266.656,0,184.333,0z M285.795,229.355\n\t\t\tc-21.956,27.687-80.92,89.278-101.462,110.581c-20.54-21.302-79.483-82.875-101.434-110.552\n\t\t\tc-18.228-22.984-27.863-50.677-27.863-80.087C55.036,78.002,113.038,20,184.333,20c71.294,0,129.297,58.002,129.296,129.297\n\t\t\tC313.629,178.709,304.004,206.393,285.795,229.355z\" \/><path d=\"M184.333,59.265c-48.73,0-88.374,39.644-88.374,88.374c0,48.73,39.645,88.374,88.374,88.374s88.374-39.645,88.374-88.374\n\t\t\tS233.063,59.265,184.333,59.265z M184.333,216.013c-37.702,0-68.374-30.673-68.374-68.374c0-37.702,30.673-68.374,68.374-68.374\n\t\t\ts68.373,30.673,68.374,68.374C252.707,185.341,222.035,216.013,184.333,216.013z\" \/><\/g><\/svg>\u6765\u81ea\u5317\u4eac<\/div>","protected":false},"excerpt":{"rendered":"<p>\u672c\u8282\u5185\u5bb9\u4e3b\u8981\u4e3a\u9ad8\u7cbe\u5ea6\u8fd0\u7b97\u3001\u524d\u7f00\u548c\u4e0e\u5dee\u5206\u7684\u4e3b\u8981\u601d\u60f3\u548c\u4ee3\u7801\u3002 \u9ad8\u7cbe\u5ea6 \u57fa\u672c\u601d\u60f3 \u5e38\u8003\u7c7b\u578b\uff1a \u5982\u4f55\u5b58\u50a8 \u4f7f\u7528\u6570\u7ec4\u8fdb\u884c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,104],"tags":[],"class_list":["post-1230","post","type-post","status-publish","format-standard","hentry","category-study","category-biji"],"_links":{"self":[{"href":"https:\/\/diary.bid\/index.php?rest_route=\/wp\/v2\/posts\/1230","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/diary.bid\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/diary.bid\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/diary.bid\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/diary.bid\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1230"}],"version-history":[{"count":1,"href":"https:\/\/diary.bid\/index.php?rest_route=\/wp\/v2\/posts\/1230\/revisions"}],"predecessor-version":[{"id":1231,"href":"https:\/\/diary.bid\/index.php?rest_route=\/wp\/v2\/posts\/1230\/revisions\/1231"}],"wp:attachment":[{"href":"https:\/\/diary.bid\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1230"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/diary.bid\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1230"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/diary.bid\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1230"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}