{"id":76,"date":"2026-04-06T16:31:42","date_gmt":"2026-04-06T08:31:42","guid":{"rendered":"https:\/\/www.sjll.top\/index.php\/2026\/04\/06\/%e6%b4%9b%e8%b0%b7-p1503-%e9%ac%bc%e5%ad%90%e8%bf%9b%e6%9d%91%ef%bc%9a%e6%a0%91%e7%8a%b6%e6%95%b0%e7%bb%84%e4%b8%8e%e4%ba%8c%e5%88%86%e6%9f%a5%e6%89%be%e7%9a%84%e7%bb%8f%e5%85%b8%e5%ba%94%e7%94%a8\/"},"modified":"2026-04-06T16:31:42","modified_gmt":"2026-04-06T08:31:42","slug":"%e6%b4%9b%e8%b0%b7-p1503-%e9%ac%bc%e5%ad%90%e8%bf%9b%e6%9d%91%ef%bc%9a%e6%a0%91%e7%8a%b6%e6%95%b0%e7%bb%84%e4%b8%8e%e4%ba%8c%e5%88%86%e6%9f%a5%e6%89%be%e7%9a%84%e7%bb%8f%e5%85%b8%e5%ba%94%e7%94%a8","status":"publish","type":"post","link":"https:\/\/www.sjll.top\/?p=76","title":{"rendered":"\u6d1b\u8c37 P1503 \u9b3c\u5b50\u8fdb\u6751\uff1a\u6811\u72b6\u6570\u7ec4\u4e0e\u4e8c\u5206\u67e5\u627e\u7684\u7ecf\u5178\u5e94\u7528"},"content":{"rendered":"<h2>\u9898\u76ee\u63cf\u8ff0<\/h2>\n<p>\u53bf\u57ce\u91cc\u6709 n \u4e2a\u7528\u5730\u9053\u76f8\u8fde\u7684\u623f\u5b50\u6392\u6210\u4e00\u6761\u76f4\u7ebf\uff0c\u7b2c i \u4e2a\u623f\u5b50\u53ea\u4e0e\u7b2c i-1 \u548c i+1 \u4e2a\u76f8\u8fde\u3002\u7279\u522b\u5730\uff0c\u7b2c 1 \u4e2a\u623f\u5b50\u53ea\u548c\u7b2c 2 \u4e2a\u8054\u901a\uff0c\u7b2c n \u4e2a\u623f\u5b50\u53ea\u548c\u7b2c n-1 \u4e2a\u8054\u901a\u3002<\/p>\n<p>\u6709 m \u6761\u6d88\u606f\u4f9d\u6b21\u4f20\u6765\uff1a<\/p>\n<ul>\n<li><strong>D x<\/strong>\uff1a\u9b3c\u5b50\u5c06 x \u53f7\u623f\u5b50\u6467\u6bc1\u4e86\uff0c\u5730\u9053\u88ab\u5835\u4e0a<\/li>\n<li><strong>R<\/strong>\uff1a\u6751\u6c11\u4eec\u5c06\u9b3c\u5b50\u4e0a\u4e00\u4e2a\u6467\u6bc1\u7684\u623f\u5b50\u4fee\u590d\u4e86<\/li>\n<li><strong>Q x<\/strong>\uff1a\u6709\u4e00\u540d\u58eb\u5175\u88ab\u56f4\u5835\u5728 x \u53f7\u623f\u5b50\u4e2d\uff0c\u8be2\u95ee\u80fd\u5230\u8fbe\u7684\u623f\u5b50\u6570\u91cf<\/li>\n<\/ul>\n<p>\u6570\u636e\u8303\u56f4\uff1a1 \u2264 n, m \u2264 5\u00d710\u2074<\/p>\n<h2>\u95ee\u9898\u5206\u6790<\/h2>\n<h3>\u6838\u5fc3\u89c2\u5bdf<\/h3>\n<p>\u5bf9\u4e8e\u672a\u88ab\u6467\u6bc1\u7684 x \u53f7\u623f\u5b50\uff0c\u58eb\u5175\u80fd\u5230\u8fbe\u7684\u8303\u56f4\u662f\u8fde\u7eed\u7684\uff0c\u7531\u5de6\u53f3\u4e24\u8fb9\u6700\u8fd1\u7684\u88ab\u6467\u6bc1\u623f\u5b50\u51b3\u5b9a\u3002<\/p>\n<p>\u8bbe\u5de6\u8fb9\u6700\u8fd1\u7684\u88ab\u6467\u6bc1\u623f\u5b50\u4f4d\u7f6e\u4e3a L\uff08L < x\uff09\uff0c\u53f3\u8fb9\u6700\u8fd1\u7684\u88ab\u6467\u6bc1\u623f\u5b50\u4f4d\u7f6e\u4e3a R\uff08R > x\uff09\uff0c\u5219\u80fd\u5230\u8fbe\u7684\u8303\u56f4\u662f (L+1) \u5230 (R-1)\uff0c\u80fd\u5230\u8fbe\u7684\u623f\u5b50\u6570\u4e3a <strong>R &#8211; L &#8211; 1<\/strong>\u3002<\/p>\n<h3>\u95ee\u9898\u8f6c\u5316<\/h3>\n<p>\u539f\u95ee\u9898\u8f6c\u5316\u4e3a\uff1a<\/p>\n<ul>\n<li>\u7ef4\u62a4\u4e00\u4e2a\u96c6\u5408 S\uff0c\u5b58\u50a8\u6240\u6709\u88ab\u6467\u6bc1\u7684\u623f\u5b50<\/li>\n<li>\u5bf9\u4e8e\u67e5\u8be2 x\uff0c\u9700\u8981\u627e\u5230 S \u4e2d x \u7684\u524d\u9a71\uff08\u5c0f\u4e8e x \u7684\u6700\u5927\u503c\uff09\u548c\u540e\u7ee7\uff08\u5927\u4e8e x \u7684\u6700\u5c0f\u503c\uff09<\/li>\n<li>\u7b54\u6848 = \u540e\u7ee7 &#8211; \u524d\u9a71 &#8211; 1<\/li>\n<\/ul>\n<h2>\u7b97\u6cd5\u8bbe\u8ba1<\/h2>\n<h3>\u4e3a\u4ec0\u4e48\u4f7f\u7528\u6811\u72b6\u6570\u7ec4\uff1f<\/h3>\n<p><strong>\u6811\u72b6\u6570\u7ec4\uff08Fenwick Tree\uff09<\/strong>\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7528\u4e8e\u7ef4\u62a4\u6570\u7ec4\u524d\u7f00\u548c\u7684\u52a8\u6001\u67e5\u8be2\u548c\u66f4\u65b0\u3002<\/p>\n<ul>\n<li><strong>\u5355\u70b9\u66f4\u65b0 O(log n)<\/strong>\uff1a\u4fee\u6539\u4e00\u4e2a\u5143\u7d20\u7684\u503c<\/li>\n<li><strong>\u533a\u95f4\u67e5\u8be2 O(log n)<\/strong>\uff1a\u67e5\u8be2\u524d\u7f00\u548c<\/li>\n<li>\u53ef\u4ee5\u901a\u8fc7\u4e8c\u5206\u67e5\u627e\u5b9e\u73b0\u300c\u8fde\u7eed\u5b8c\u597d\u7684\u533a\u95f4\u957f\u5ea6\u300d\u67e5\u8be2<\/li>\n<\/ul>\n<p>\u6838\u5fc3\u6280\u5de7\uff1a\u4e0d\u76f4\u63a5\u627e\u88ab\u6467\u6bc1\u7684\u623f\u5b50\uff0c\u800c\u662f\u627e x \u5de6\u53f3\u8fde\u7eed\u5b8c\u597d\u7684\u623f\u5b50\u6570\u91cf\u3002<\/p>\n<h3>\u6838\u5fc3\u64cd\u4f5c\uff1a\u67e5\u8be2\u8fde\u7eed\u5b8c\u597d\u533a\u95f4<\/h3>\n<p>\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u5206\u522b\u5bfb\u627e\u5de6\u53f3\u4e24\u8fb9\u8fde\u7eed\u5b8c\u597d\u7684\u533a\u95f4\u957f\u5ea6\u3002<\/p>\n<h4>\u67e5\u8be2\u5de6\u8fb9\u8fde\u7eed\u5b8c\u597d\u623f\u5b50\u6570<\/h4>\n<pre><code>\/\/ \u67e5\u8be2 x \u5de6\u8fb9\u8fde\u7eed\u5b8c\u597d\u7684\u623f\u5b50\u6570\nint find_left(int x) {\n    int l = 1, r = x, ans = x;\n    while (l <= r) {\n        int mid = (l + r) >> 1;\n        \/\/ \u68c0\u67e5 [mid, x] \u533a\u95f4\u662f\u5426\u5168\u90e8\u5b8c\u597d\n        if (query(mid, x) == x - mid + 1) {\n            ans = mid;\n            r = mid - 1;  \/\/ \u5411\u5de6\u6536\u7f29\uff0c\u5bfb\u627e\u66f4\u5de6\u8fb9\u7684\u8d77\u70b9\n        } else {\n            l = mid + 1;  \/\/ \u4e0d\u5168\u90e8\u5b8c\u597d\uff0c\u5411\u53f3\u6536\u7f29\n        }\n    }\n    return x - ans + 1;  \/\/ \u5de6\u8fb9\u8fde\u7eed\u5b8c\u597d\u6570\n}<\/code><\/pre>\n<h4>\u67e5\u8be2\u53f3\u8fb9\u8fde\u7eed\u5b8c\u597d\u623f\u5b50\u6570<\/h4>\n<pre><code>\/\/ \u67e5\u8be2 x \u53f3\u8fb9\u8fde\u7eed\u5b8c\u597d\u7684\u623f\u5b50\u6570\nint find_right(int x) {\n    int l = x, r = n, ans = x;\n    while (l <= r) {\n        int mid = (l + r) >> 1;\n        \/\/ \u68c0\u67e5 [x, mid] \u533a\u95f4\u662f\u5426\u5168\u90e8\u5b8c\u597d\n        if (query(x, mid) == mid - x + 1) {\n            ans = mid;\n            l = mid + 1;  \/\/ \u5411\u53f3\u6269\u5927\uff0c\u5bfb\u627e\u66f4\u53f3\u8fb9\u7684\u7ec8\u70b9\n        } else {\n            r = mid - 1;  \/\/ \u4e0d\u5168\u90e8\u5b8c\u597d\uff0c\u5411\u5de6\u6536\u7f29\n        }\n    }\n    return ans - x + 1;  \/\/ \u53f3\u8fb9\u8fde\u7eed\u5b8c\u597d\u6570\n}<\/code><\/pre>\n<h4>\u4e8c\u5206\u67e5\u627e\u7684\u6b63\u786e\u6027<\/h4>\n<p>\u5b9a\u4e49\u51fd\u6570 f(mid) = [\u533a\u95f4[mid, x]\u662f\u5426\u5168\u90e8\u5b8c\u597d]<\/p>\n<ul>\n<li>\u82e5 f(mid) = true\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f mid&apos; > mid\uff0c\u533a\u95f4[mid&apos;, x] \u662f [mid, x] \u7684\u5b50\u533a\u95f4\uff0c\u5fc5\u7136\u4e5f\u5168\u90e8\u5b8c\u597d<\/li>\n<li>\u82e5 f(mid) = false\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f mid&apos; < mid\uff0c\u533a\u95f4[mid&apos;, x] \u5305\u542b\u533a\u95f4[mid, x]\uff0c\u5fc5\u7136\u4e5f\u5b58\u5728\u88ab\u6467\u6bc1\u623f\u5b50<\/li>\n<\/ul>\n<p>\u56e0\u6b64 f(mid) \u5177\u6709\u5355\u8c03\u6027\uff0c\u5b58\u5728\u4e00\u4e2a\u5206\u754c\u70b9 M\uff0c\u4f7f\u5f97\u5f53 mid \u2265 M \u65f6 f(mid)=true\uff0c\u5f53 mid < M \u65f6 f(mid)=false\u3002\u4e8c\u5206\u67e5\u627e\u53ef\u4ee5\u627e\u5230\u6700\u5c0f\u7684 mid \u4f7f\u5f97 f(mid)=true\u3002<\/p>\n<h2>\u5b8c\u6574\u4ee3\u7801\u5b9e\u73b0<\/h2>\n<pre><code>#include &lt;iostream&gt;\n#include &lt;cstdio&gt;\nusing namespace std;\n\nconst int N = 50010;\n\nint n, m;          \/\/ \u623f\u5b50\u6570\u3001\u6d88\u606f\u6570\nint bad[N];       \/\/ \u6807\u8bb0\u623f\u5b50\u662f\u5426\u88ab\u6467\u6bc1\uff0c1=\u5df2\u6467\u6bc1\nint tr[N];        \/\/ \u6811\u72b6\u6570\u7ec4\uff0c\u5b58\u50a8\u5b8c\u597d\u7684\u623f\u5b50\u6570\nint stk[N], top;  \/\/ \u6808\uff0c\u8bb0\u5f55\u6467\u6bc1\u987a\u5e8f\u7528\u4e8e R \u64cd\u4f5c\n\n\/\/ lowbit \u51fd\u6570\uff1a\u8fd4\u56de x \u4e8c\u8fdb\u5236\u7684\u6700\u4f4e\u4f4d 1\nint lowbit(int x) {\n    return x & -x;\n}\n\n\/\/ \u6811\u72b6\u6570\u7ec4\u5355\u70b9\u66f4\u65b0\nvoid add(int x, int c) {\n    for (; x <= n; x += lowbit(x))\n        tr[x] += c;\n}\n\n\/\/ \u6811\u72b6\u6570\u7ec4\u524d\u7f00\u548c\u67e5\u8be2\nint sum(int x) {\n    int res = 0;\n    for (; x; x -= lowbit(x))\n        res += tr[x];\n    return res;\n}\n\n\/\/ \u533a\u95f4\u67e5\u8be2\uff1a[l, r] \u533a\u95f4\u5185\u5b8c\u597d\u7684\u623f\u5b50\u6570\nint query(int l, int r) {\n    if (l > r) return 0;\n    return sum(r) - sum(l - 1);\n}\n\n\/\/ \u67e5\u8be2 x \u5de6\u8fb9\u8fde\u7eed\u5b8c\u597d\u7684\u623f\u5b50\u6570\nint find_left(int x) {\n    int l = 1, r = x, ans = x;\n    while (l <= r) {\n        int mid = (l + r) >> 1;\n        if (query(mid, x) == x - mid + 1) {\n            ans = mid;\n            r = mid - 1;\n        } else {\n            l = mid + 1;\n        }\n    }\n    return x - ans + 1;\n}\n\n\/\/ \u67e5\u8be2 x \u53f3\u8fb9\u8fde\u7eed\u5b8c\u597d\u7684\u623f\u5b50\u6570\nint find_right(int x) {\n    int l = x, r = n, ans = x;\n    while (l <= r) {\n        int mid = (l + r) >> 1;\n        if (query(x, mid) == mid - x + 1) {\n            ans = mid;\n            l = mid + 1;\n        } else {\n            r = mid - 1;\n        }\n    }\n    return ans - x + 1;\n}\n\nint main() {\n    scanf(\"%d%d\", &n, &m);\n    \n    \/\/ \u521d\u59cb\u5316\u6811\u72b6\u6570\u7ec4\uff1a\u6240\u6709\u623f\u5b50\u90fd\u662f\u5b8c\u597d\u7684\n    for (int i = 1; i <= n; i++)\n        add(i, 1);\n\n    while (m--) {\n        char op[2];\n        int x;\n        scanf(\"%s\", op);\n        \n        if (op[0] == &apos;D&apos;) {  \/\/ \u6467\u6bc1\u64cd\u4f5c\n            scanf(\"%d\", &#038;x);\n            if (!bad[x]) {\n                bad[x] = 1;\n                add(x, -1);      \/\/ \u5b8c\u597d\u6570\u51cf 1\n                stk[++top] = x;  \/\/ \u5165\u6808\n            }\n        } else if (op[0] == &apos;R&apos;) {  \/\/ \u4fee\u590d\u64cd\u4f5c\n            if (top > 0) {\n                x = stk[top--];   \/\/ \u53d6\u51fa\u4e0a\u4e00\u4e2a\u88ab\u6467\u6bc1\u7684\u623f\u5b50\n                bad[x] = 0;\n                add(x, 1);       \/\/ \u5b8c\u597d\u6570\u52a0 1\n            }\n        } else if (op[0] == &apos;Q&apos;) {  \/\/ \u67e5\u8be2\u64cd\u4f5c\n            scanf(\"%d\", &x);\n            if (bad[x]) {\n                printf(\"0\\n\");  \/\/ \u58eb\u5175\u5728\u88ab\u6467\u6bc1\u7684\u623f\u5b50\u4e2d\n            } else {\n                int l = find_left(x);   \/\/ \u5de6\u8fb9\u8fde\u7eed\u5b8c\u597d\u6570\n                int r = find_right(x);  \/\/ \u53f3\u8fb9\u8fde\u7eed\u5b8c\u597d\u6570\n                printf(\"%d\\n\", l + r - 1);  \/\/ \u51cf\u53bb\u91cd\u590d\u8ba1\u7b97\u7684 x\n            }\n        }\n    }\n    return 0;\n}<\/code><\/pre>\n<h2>\u590d\u6742\u5ea6\u5206\u6790<\/h2>\n<table>\n<tr>\n<th>\u64cd\u4f5c<\/th>\n<th>\u65f6\u95f4\u590d\u6742\u5ea6<\/th>\n<th>\u7a7a\u95f4\u590d\u6742\u5ea6<\/th>\n<\/tr>\n<tr>\n<td>\u5355\u70b9\u4fee\u6539\uff08add\uff09<\/td>\n<td>O(log n)<\/td>\n<td rowspan=\"3\">O(n)<\/td>\n<\/tr>\n<tr>\n<td>\u533a\u95f4\u67e5\u8be2\uff08query\uff09<\/td>\n<td>O(log n)<\/td>\n<\/tr>\n<tr>\n<td>\u4e8c\u5206\u67e5\u627e<\/td>\n<td>O(log\u00b2 n)<\/td>\n<\/tr>\n<tr>\n<td>\u603b\u590d\u6742\u5ea6<\/td>\n<td>O(m log\u00b2 n)<\/td>\n<td><\/td>\n<\/tr>\n<\/table>\n<p>\u5bf9\u4e8e n, m \u2264 50000\uff0clog\u208250000 \u2248 16\uff0c\u603b\u64cd\u4f5c\u6570\u7ea6\u4e3a 50000 \u00d7 256 \u2248 1.28\u00d710\u2077\uff0c\u5b8c\u5168\u53ef\u4ee5\u63a5\u53d7\u3002<\/p>\n<h2>\u5173\u952e\u8981\u70b9\u603b\u7ed3<\/h2>\n<ol>\n<li><strong>\u6808\u7684\u5e94\u7528<\/strong>\uff1a\u7ef4\u62a4\u88ab\u6467\u6bc1\u623f\u5b50\u7684\u987a\u5e8f\uff0c\u786e\u4fdd R \u64cd\u4f5c\u4fee\u590d\u7684\u662f\u300c\u6700\u8fd1\u300d\u88ab\u6467\u6bc1\u7684\u623f\u5b50<\/li>\n<li><strong>\u6811\u72b6\u6570\u7ec4<\/strong>\uff1a\u9ad8\u6548\u7ef4\u62a4\u300c\u5b8c\u597d\u623f\u5b50\u6570\u300d\uff0c\u652f\u6301 O(log n) \u7684\u5355\u70b9\u66f4\u65b0\u548c\u533a\u95f4\u67e5\u8be2<\/li>\n<li><strong>\u4e8c\u5206\u67e5\u627e<\/strong>\uff1a\u5229\u7528\u5355\u8c03\u6027\u5feb\u901f\u5b9a\u4f4d\u8fde\u7eed\u5b8c\u597d\u533a\u95f4\u7684\u8fb9\u754c<\/li>\n<li><strong>\u7b54\u6848\u516c\u5f0f<\/strong>\uff1aleft_good + right_good &#8211; 1\uff08x \u88ab\u91cd\u590d\u8ba1\u7b97\uff09<\/li>\n<\/ol>\n<h2>\u6837\u4f8b\u9a8c\u8bc1<\/h2>\n<p>\u8f93\u5165\uff1a<\/p>\n<pre><code>7 9\nD 3\nD 6\nD 5\nQ 4\nQ 5\nR\nQ 4\nR\nQ 4<\/code><\/pre>\n<p>\u5904\u7406\u8fc7\u7a0b\uff1a<\/p>\n<pre><code>\u64cd\u4f5c      \u72b6\u6001\uff08\u88ab\u6467\u6bc1\uff09  \u67e5\u8be24\u53f7\u7ed3\u679c  \u8bf4\u660e\n\u521d\u59cb      {}              -           \u6240\u6709\u623f\u5b50\u5b8c\u597d\nD 3      {3}             -           \u6467\u6bc13\u53f7\nD 6      {3,6}           -           \u6467\u6bc16\u53f7\nD 5      {3,5,6}         -           \u6467\u6bc15\u53f7\nQ 4     {3,5,6}         1           4\u53f7\u53ea\u80fd\u5230\u8fbe4\u53f7\u672c\u8eab\nQ 5     {3,5,6}         0           5\u53f7\u5df2\u88ab\u6467\u6bc1\nR        {3,6}           -           \u4fee\u590d5\u53f7\nQ 4     {3,6}           2           4\u53f7\u53ef\u5230\u8fbe4,5\nR        {3}             -           \u4fee\u590d6\u53f7\nQ 4     {3}             4           4\u53f7\u53ef\u5230\u8fbe4,5,6,7<\/code><\/pre>\n<p>\u8f93\u51fa\uff1a1 0 2 4<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6d1b\u8c37 P1503 \u9b3c\u5b50\u8fdb\u6751\u662f\u4e00\u9053\u7ecf\u5178\u7684\u7b97\u6cd5\u9898\u76ee\uff0c\u8003\u5bdf\u6811\u72b6\u6570\u7ec4\u4e0e\u4e8c\u5206\u67e5\u627e\u7684\u7ed3\u5408\u5e94\u7528\u3002\u672c\u6587\u8be6\u7ec6\u8bb2\u89e3\u5982\u4f55\u7ef4\u62a4\u88ab\u6467\u6bc1\u623f\u5b50\u7684\u96c6\u5408\uff0c\u5e76\u901a\u8fc7\u4e8c\u5206\u67e5\u627e\u5b9a\u4f4d\u8fde\u7eed\u5b8c\u597d\u533a\u95f4\u3002<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-76","post","type-post","status-publish","format-standard","hentry","category-3"],"_links":{"self":[{"href":"https:\/\/www.sjll.top\/index.php?rest_route=\/wp\/v2\/posts\/76","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sjll.top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sjll.top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sjll.top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sjll.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=76"}],"version-history":[{"count":0,"href":"https:\/\/www.sjll.top\/index.php?rest_route=\/wp\/v2\/posts\/76\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sjll.top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=76"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sjll.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=76"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sjll.top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=76"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}