{"id":547,"date":"2023-03-04T14:52:42","date_gmt":"2023-03-04T06:52:42","guid":{"rendered":"http:\/\/120.55.184.7\/?p=547"},"modified":"2025-04-16T00:44:23","modified_gmt":"2025-04-15T16:44:23","slug":"547","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=547","title":{"rendered":"\u53ef\u6301\u4e45\u5316\u7ebf\u6bb5\u6811"},"content":{"rendered":"\n<p>\u53ef\u6301\u4e45\u5316\u7ebf\u6bb5\u6811\uff08Persistent Segment Tree\uff09\uff0c\u53c8\u79f0\u4e3b\u5e2d\u6811\uff0c\u662f\u4e00\u79cd\u652f\u6301\u67e5\u8be2\u5386\u53f2\u7248\u672c\u7684\u6570\u636e\u7ed3\u6784\u3002\u5b83\u5728\u666e\u901a\u7ebf\u6bb5\u6811\u7684\u57fa\u7840\u4e0a\u901a\u8fc7\u5171\u7528\u8282\u70b9\u6765\u5b58\u50a8\u591a\u4e2a\u7248\u672c\uff0c\u4ece\u800c\u9ad8\u6548\u5730\u7ef4\u62a4\u5386\u53f2\u4fe1\u606f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6838\u5fc3\u539f\u7406<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u57fa\u672c\u6982\u5ff5<\/h3>\n\n\n\n<p>\u53ef\u6301\u4e45\u5316\u7ebf\u6bb5\u6811\u7684\u6838\u5fc3\u601d\u60f3\u662f\uff1a<strong>\u6bcf\u6b21\u4fee\u6539\u53ea\u521b\u5efa\u88ab\u4fee\u6539\u8def\u5f84\u4e0a\u7684\u65b0\u8282\u70b9\uff0c\u5176\u4f59\u8282\u70b9\u4e0e\u65e7\u7248\u672c\u5171\u4eab<\/strong>\u3002\u8fd9\u6837\u65e2\u4fdd\u7559\u4e86\u5386\u53f2\u7248\u672c\uff0c\u53c8\u907f\u514d\u4e86\u5b8c\u5168\u590d\u5236\u6574\u68f5\u6811\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u5173\u952e\u7279\u6027<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7248\u672c\u63a7\u5236<\/strong>\uff1a\u6bcf\u4e2a\u7248\u672c\u5bf9\u5e94\u4e00\u4e2a\u6839\u8282\u70b9\uff0c\u901a\u8fc7\u6839\u8282\u70b9\u53ef\u4ee5\u8bbf\u95ee\u8be5\u7248\u672c\u7684\u5b8c\u6574\u7ebf\u6bb5\u6811<\/li>\n\n\n\n<li><strong>\u8def\u5f84\u590d\u5236<\/strong>\uff1a\u4fee\u6539\u64cd\u4f5c\u53ea\u590d\u5236\u4ece\u6839\u5230\u76ee\u6807\u8282\u70b9\u7684\u8def\u5f84\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6O(log <sup>n<\/sup>)<\/li>\n\n\n\n<li><strong>\u524d\u7f00\u548c\u601d\u60f3<\/strong>\uff1a\u5e38\u7528\u4e8e\u89e3\u51b3\u533a\u95f4\u7b2ck\u5927\u95ee\u9898\uff0c\u901a\u8fc7\u7248\u672c\u76f8\u51cf\u5f97\u5230\u533a\u95f4\u4fe1\u606f<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. \u6570\u5b66\u57fa\u7840<\/h3>\n\n\n\n<p>\u5bf9\u4e8en\u6b21\u64cd\u4f5c\uff0c\u53ef\u6301\u4e45\u5316\u7ebf\u6bb5\u6811\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3aO(n log<sup>n<\/sup>)\uff0c\u6bcf\u6b21\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(log <sup>n<\/sup>)\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">C++\u5b9e\u73b0<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u57fa\u7840\u7ed3\u6784<\/h3>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\n#include &lt;vector&gt;\n#include &lt;algorithm&gt;\nusing namespace std;\n\nconst int N = 1e5 + 10; \/\/ \u6570\u636e\u89c4\u6a21\nconst int M = N * 20;   \/\/ \u8282\u70b9\u6570\u91cf\u4f30\u8ba1\n\nstruct Node {\n    int l, r;    \/\/ \u5de6\u53f3\u5b50\u8282\u70b9\u6307\u9488\n    int sum;     \/\/ \u533a\u95f4\u548c\/\u8ba1\u6570\n} tr&#x5B;M];\n\nint root&#x5B;N], idx; \/\/ \u5404\u7248\u672c\u6839\u8282\u70b9\u548c\u8282\u70b9\u8ba1\u6570\u5668\nvector&lt;int&gt; nums; \/\/ \u79bb\u6563\u5316\u6570\u7ec4\n<\/pre><\/div>\n\n\n<h3 class=\"wp-block-heading\">2. \u79bb\u6563\u5316\u5904\u7406<\/h3>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\n\/\/ \u79bb\u6563\u5316\u9884\u5904\u7406\nvoid discretize(vector&lt;int&gt;&amp; a) {\n    nums = a;\n    sort(nums.begin(), nums.end());\n    nums.erase(unique(nums.begin(), nums.end()), nums.end());\n}\n\n\/\/ \u83b7\u53d6\u79bb\u6563\u5316\u540e\u7684\u503c\nint get_id(int x) {\n    return lower_bound(nums.begin(), nums.end(), x) - nums.begin() + 1;\n}\n<\/pre><\/div>\n\n\n<h3 class=\"wp-block-heading\">3. \u5efa\u6811\u4e0e\u66f4\u65b0<\/h3>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\n\/\/ \u5efa\u7acb\u521d\u59cb\u7a7a\u6811\nint build(int l, int r) {\n    int p = ++idx;\n    if (l == r) return p;\n    int mid = (l + r) &gt;&gt; 1;\n    tr&#x5B;p].l = build(l, mid);\n    tr&#x5B;p].r = build(mid + 1, r);\n    return p;\n}\n\n\/\/ \u66f4\u65b0\u64cd\u4f5c\uff08\u521b\u5efa\u65b0\u7248\u672c\uff09\nint update(int p, int l, int r, int x) {\n    int q = ++idx;\n    tr&#x5B;q] = tr&#x5B;p]; \/\/ \u590d\u5236\u539f\u8282\u70b9\n    if (l == r) {\n        tr&#x5B;q].sum++;\n        return q;\n    }\n    int mid = (l + r) &gt;&gt; 1;\n    if (x &lt;= mid) tr&#x5B;q].l = update(tr&#x5B;p].l, l, mid, x);\n    else tr&#x5B;q].r = update(tr&#x5B;p].r, mid + 1, r, x);\n    tr&#x5B;q].sum = tr&#x5B;tr&#x5B;q].l].sum + tr&#x5B;tr&#x5B;q].r].sum;\n    return q;\n}\n<\/pre><\/div>\n\n\n<h3 class=\"wp-block-heading\">4. \u67e5\u8be2\u64cd\u4f5c<\/h3>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\n\/\/ \u67e5\u8be2\u533a\u95f4&#x5B;l,r]\u7684\u7b2ck\u5c0f\u503c\nint query(int p, int q, int l, int r, int k) {\n    if (l == r) return l;\n    int mid = (l + r) &gt;&gt; 1;\n    int cnt = tr&#x5B;tr&#x5B;q].l].sum - tr&#x5B;tr&#x5B;p].l].sum; \/\/ \u5de6\u5b50\u6811\u7684\u5143\u7d20\u4e2a\u6570\n    if (k &lt;= cnt) return query(tr&#x5B;p].l, tr&#x5B;q].l, l, mid, k);\n    else return query(tr&#x5B;p].r, tr&#x5B;q].r, mid + 1, r, k - cnt);\n}\n<\/pre><\/div>\n\n\n<h3 class=\"wp-block-heading\">5. \u4f7f\u7528\u793a\u4f8b\uff08\u533a\u95f4\u7b2ck\u5c0f\uff09<\/h3>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; title: ; notranslate\" title=\"\">\nint main() {\n    int n, m;\n    scanf(&quot;%d%d&quot;, &amp;n, &amp;m);\n\n    vector&lt;int&gt; a(n);\n    for (int i = 0; i &lt; n; i++) scanf(&quot;%d&quot;, &amp;a&#x5B;i]);\n\n    discretize(a);\n\n    root&#x5B;0] = build(1, nums.size());\n    for (int i = 1; i &lt;= n; i++)\n        root&#x5B;i] = update(root&#x5B;i-1], 1, nums.size(), get_id(a&#x5B;i-1]));\n\n    while (m--) {\n        int l, r, k;\n        scanf(&quot;%d%d%d&quot;, &amp;l, &amp;r, &amp;k);\n        int pos = query(root&#x5B;l-1], root&#x5B;r], 1, nums.size(), k);\n        printf(&quot;%d\\n&quot;, nums&#x5B;pos-1]);\n    }\n    return 0;\n}\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">\u5e94\u7528\u573a\u666f<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u533a\u95f4\u7b2ck\u5927\/\u5c0f\u95ee\u9898<\/strong>\uff1a\u901a\u8fc7\u7248\u672c\u76f8\u51cf\u5f97\u5230\u533a\u95f4\u7edf\u8ba1\u4fe1\u606f<\/li>\n\n\n\n<li><strong>\u53ef\u6301\u4e45\u5316\u6570\u7ec4<\/strong>\uff1a\u652f\u6301\u5386\u53f2\u7248\u672c\u7684\u5355\u70b9\u67e5\u8be2\u548c\u4fee\u6539<\/li>\n\n\n\n<li><strong>\u4e8c\u7ef4\u6570\u70b9\u95ee\u9898<\/strong>\uff1a\u7ed3\u5408\u626b\u63cf\u7ebf\u4f7f\u7528<\/li>\n\n\n\n<li><strong>\u5e26\u4fee\u6539\u7684\u533a\u95f4\u67e5\u8be2<\/strong>\uff1a\u7ed3\u5408\u6811\u72b6\u6570\u7ec4\u5b9e\u73b0<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u6027\u80fd\u4f18\u5316<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u52a8\u6001\u5f00\u70b9<\/strong>\uff1a\u907f\u514d\u9884\u5206\u914d\u5927\u6570\u7ec4\uff0c\u6309\u9700\u521b\u5efa\u8282\u70b9<\/li>\n\n\n\n<li><strong>\u5185\u5b58\u6c60<\/strong>\uff1a\u9884\u5148\u5206\u914d\u8282\u70b9\u5185\u5b58\uff0c\u51cf\u5c11\u52a8\u6001\u5206\u914d\u5f00\u9500<\/li>\n\n\n\n<li><strong>\u5783\u573e\u56de\u6536<\/strong>\uff1a\u5bf9\u4e8e\u4e0d\u518d\u4f7f\u7528\u7684\u7248\u672c\uff0c\u53ef\u4ee5\u56de\u6536\u8282\u70b9\u7a7a\u95f4<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u53ef\u6301\u4e45\u5316\u7ebf\u6bb5\u6811\uff08Persistent Segment Tree\uff09\uff0c\u53c8\u79f0\u4e3b\u5e2d\u6811\uff0c\u662f\u4e00\u79cd\u652f\u6301\u67e5\u8be2\u5386\u53f2\u7248\u672c\u7684\u6570\u636e\u7ed3\u6784 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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