{"id":873,"date":"2021-04-30T12:27:16","date_gmt":"2021-04-30T04:27:16","guid":{"rendered":"https:\/\/beijian99.top\/?p=873"},"modified":"2025-05-06T16:15:13","modified_gmt":"2025-05-06T08:15:13","slug":"st%e8%a1%a8-sparse-table","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=873","title":{"rendered":"ST\u8868 (Sparse Table )"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"\u5b9a\u4e49\">\u5b9a\u4e49<a href=\"https:\/\/oi-wiki.org\/ds\/sparse-table\/#%E5%AE%9A%E4%B9%89\"><\/a><\/h2>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/oi-wiki.org\/ds\/images\/st.svg\" alt=\"ST \u8868\u793a\u610f\u56fe\"\/><\/figure>\n\n\n\n<p>ST \u8868\uff08Sparse Table\uff0c\u7a00\u758f\u8868\uff09\u662f\u7528\u4e8e\u89e3\u51b3&nbsp;<strong>\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898<\/strong>&nbsp;\u7684\u6570\u636e\u7ed3\u6784\u3002<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898<\/strong>\u00a0\u662f\u6307\u5bf9\u4e8e\u8fd0\u7b97\u00a0opt\uff0c\u6ee1\u8db3\u00a0x opt x = x\uff0c\u5219\u5bf9\u5e94\u7684\u533a\u95f4\u8be2\u95ee\u5c31\u662f\u4e00\u4e2a\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u6700\u5927\u503c\u6709\u00a0max(x,x) = x \uff0cgcd \u6709 gcd(x,x) = x \uff0c\u6240\u4ee5 RMQ\uff08\u533a\u95f4\u6700\u503c\u67e5\u8be2\uff09 \u548c\u533a\u95f4 GCD\uff08\u6700\u5927\u516c\u7ea6\u6570\uff09 \u5c31\u662f\u4e00\u4e2a\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898\u3002\u50cf\u533a\u95f4\u548c\u5c31\u4e0d\u5177\u6709\u8fd9\u4e2a\u6027\u8d28\uff0c\u5982\u679c\u6c42\u533a\u95f4\u548c\u7684\u65f6\u5019\u91c7\u7528\u7684\u9884\u5904\u7406\u533a\u95f4\u91cd\u53e0\u4e86\uff0c\u5219\u4f1a\u5bfc\u81f4\u91cd\u53e0\u90e8\u5206\u88ab\u8ba1\u7b97\u4e24\u6b21\uff0c\u8fd9\u662f\u6211\u4eec\u6240\u4e0d\u613f\u610f\u770b\u5230\u7684\u3002\u53e6\u5916\uff0copt \u8fd8\u5fc5\u987b\u6ee1\u8db3\u7ed3\u5408\u5f8b\u624d\u80fd\u4f7f\u7528 ST \u8868\u6c42\u89e3\u3002<\/p>\n\n\n\n<p><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"\u5f15\u5165\">\u5f15\u5165<a href=\"https:\/\/oi-wiki.org\/ds\/sparse-table\/#%E5%BC%95%E5%85%A5\"><\/a><\/h2>\n\n\n\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/P3865\" target=\"_blank\" rel=\"noreferrer noopener\">ST \u8868\u6a21\u677f\u9898<\/a><\/p>\n\n\n\n<p>\u9898\u76ee\u5927\u610f\uff1a\u7ed9\u5b9a\u00a0N\u00a0\u4e2a\u6570\uff0c\u6709 M\u4e2a\u8be2\u95ee\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u8be2\u95ee L,R \uff0c\u4f60\u9700\u8981\u56de\u7b54\u533a\u95f4\u00a0[L,R]\u4e2d\u7684\u6700\u5927\u503c\u3002<\/p>\n\n\n\n<p>\u8003\u8651\u66b4\u529b\u505a\u6cd5\u3002\u6bcf\u6b21\u90fd\u5bf9\u533a\u95f4[L,R]\u626b\u63cf\u4e00\u904d\uff0c\u6c42\u51fa\u6700\u5927\u503c\u3002<\/p>\n\n\n\n<p>\u663e\u7136\uff0c\u8fd9\u4e2a\u7b97\u6cd5\u4f1a\u8d85\u65f6\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"st-\u8868\">ST \u8868<a href=\"https:\/\/oi-wiki.org\/ds\/sparse-table\/#st-%E8%A1%A8\"><\/a><\/h2>\n\n\n\n<p>ST \u8868\u57fa\u4e8e\u500d\u589e\u601d\u60f3\uff0c\u53ef\u4ee5\u505a\u5230\u00a0O(N * log<sup> N<\/sup>)\u9884\u5904\u7406\uff0cO(1)\u00a0\u56de\u7b54\u6bcf\u4e2a\u8be2\u95ee\u3002\u4f46\u662f\u4e0d\u652f\u6301\u4fee\u6539\u64cd\u4f5c\u3002<\/p>\n\n\n\n<p>\u57fa\u4e8e\u500d\u589e\u601d\u60f3\uff0c\u6211\u4eec\u8003\u8651\u5982\u4f55\u6c42\u51fa\u533a\u95f4\u6700\u5927\u503c\u3002\u53ef\u4ee5\u53d1\u73b0\uff0c\u5982\u679c\u6309\u7167\u4e00\u822c\u7684\u500d\u589e\u6d41\u7a0b\uff0c\u6bcf\u6b21\u8df3\u00a02<sup>i<\/sup> \u6b65\u7684\u8bdd\uff0c\u8be2\u95ee\u65f6\u7684\u590d\u6742\u5ea6\u4ecd\u65e7\u662f O(log<sup>N<\/sup>)\u00a0\uff0c\u5e76\u6ca1\u6709\u6bd4\u7ebf\u6bb5\u6811\u66f4\u4f18\uff0c\u53cd\u800c\u9884\u5904\u7406\u4e00\u6b65\u8fd8\u6bd4\u7ebf\u6bb5\u6811\u6162\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u53d1\u73b0 max(x,x)=x\u00a0\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u533a\u95f4\u6700\u5927\u503c\u662f\u4e00\u4e2a\u5177\u6709\u300c\u53ef\u91cd\u590d\u8d21\u732e\u300d\u6027\u8d28\u7684\u95ee\u9898\u3002\u5373\u4f7f\u7528\u6765\u6c42\u89e3\u7684\u9884\u5904\u7406\u533a\u95f4\u6709\u91cd\u53e0\u90e8\u5206\uff0c\u53ea\u8981\u8fd9\u4e9b\u533a\u95f4\u7684\u5e76\u662f\u6240\u6c42\u7684\u533a\u95f4\uff0c\u6700\u7ec8\u8ba1\u7b97\u51fa\u7684\u7b54\u6848\u5c31\u662f\u6b63\u786e\u7684\u3002<\/p>\n\n\n\n<p>\u5982\u679c\u624b\u52a8\u6a21\u62df\u4e00\u4e0b\uff0c\u53ef\u4ee5\u53d1\u73b0\u6211\u4eec\u80fd\u4f7f\u7528\u81f3\u591a\u4e24\u4e2a\u9884\u5904\u7406\u8fc7\u7684\u533a\u95f4\u6765\u8986\u76d6\u8be2\u95ee\u533a\u95f4\uff0c\u4e5f\u5c31\u662f\u8bf4\u8be2\u95ee\u65f6\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u964d\u81f3\u00a0O(1)\uff0c\u5728\u5904\u7406\u6709\u5927\u91cf\u8be2\u95ee\u7684\u9898\u76ee\u65f6\u5341\u5206\u6709\u6548\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u5b9e\u73b0\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p>\u4ee4\u00a0f(i,j) \u8868\u793a\u533a\u95f4\u00a0[i,i+2<sup>j<\/sup> -1]\u00a0\u7684\u6700\u5927\u503c\u3002<\/p>\n\n\n\n<p>\u663e\u7136\u00a0f(i,0) = array[i]\u3002<\/p>\n\n\n\n<p>\u6839\u636e\u5b9a\u4e49\u5f0f\uff0c\u7b2c\u4e8c\u7ef4\u5c31\u76f8\u5f53\u4e8e\u500d\u589e\u7684\u65f6\u5019\u300c\u8df3\u4e86\u00a0\u00a02<sup>j<\/sup> -1\u6b65\u300d\uff0c\u4f9d\u636e\u500d\u589e\u7684\u601d\u8def\uff0c\u5199\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"339\" height=\"36\" src=\"https:\/\/beijian99.top\/wp-content\/uploads\/2025\/04\/image-50.png\" alt=\"\" class=\"wp-image-874\" srcset=\"https:\/\/beijian99.top\/wp-content\/uploads\/2025\/04\/image-50.png 339w, https:\/\/beijian99.top\/wp-content\/uploads\/2025\/04\/image-50-300x32.png 300w\" sizes=\"auto, (max-width: 339px) 100vw, 339px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/oi-wiki.org\/ds\/images\/st-preprocess-lift.svg\" alt=\"\"\/><\/figure>\n\n\n\n<p>\u4ee5\u4e0a\u5c31\u662f\u9884\u5904\u7406\u90e8\u5206\u3002\u800c\u5bf9\u4e8e\u67e5\u8be2\uff0c\u53ef\u4ee5\u7b80\u5355\u5b9e\u73b0\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6bcf\u4e2a\u8be2\u95ee[l,r]\u00a0\uff0c\u628a\u5b83\u5206\u6210\u4e24\u90e8\u5206\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"233\" height=\"37\" src=\"https:\/\/beijian99.top\/wp-content\/uploads\/2025\/04\/image-51.png\" alt=\"\" class=\"wp-image-876\"\/><\/figure>\n\n\n\n<p>\u00a0\u5176\u4e2d\u00a0<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"167\" height=\"28\" src=\"https:\/\/beijian99.top\/wp-content\/uploads\/2025\/04\/image-52.png\" alt=\"\" class=\"wp-image-878\"\/><\/figure>\n\n\n\n<p>\u4e24\u90e8\u5206\u7684\u7ed3\u679c\u7684\u6700\u5927\u503c\u5c31\u662f\u7b54\u6848\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/oi-wiki.org\/ds\/images\/st-query.svg\" alt=\"ST \u8868\u7684\u67e5\u8be2\u8fc7\u7a0b\"\/><\/figure>\n\n\n\n<p>\u6839\u636e\u4e0a\u9762\u5bf9\u4e8e\u300c\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898\u300d\u7684\u8bba\u8bc1\uff0c\u7531\u4e8e\u6700\u5927\u503c\u662f\u300c\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898\u300d\uff0c\u91cd\u53e0\u5e76\u4e0d\u4f1a\u5bf9\u533a\u95f4\u6700\u5927\u503c\u4ea7\u751f\u5f71\u54cd\u3002\u53c8\u56e0\u4e3a\u8fd9\u4e24\u4e2a\u533a\u95f4\u5b8c\u5168\u8986\u76d6\u4e86\u00a0[l,r]\uff0c\u53ef\u4ee5\u4fdd\u8bc1\u7b54\u6848\u7684\u6b63\u786e\u6027\u3002<\/p>\n\n\n\n<p>\u901a\u8fc7\u4ee3\u7801<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: cpp; gutter: true; title: ; notranslate\" title=\"\">\n#include &lt;bits\/stdc++.h&gt;\nusing namespace std;\n\nint st&#x5B;112345]&#x5B;22];\nint lg2&#x5B;112345];\n\nvoid initST(int n)\n{\n    for (int i = 2; i &lt;= n; i++)\n    {\n        lg2&#x5B;i] = lg2&#x5B;i &gt;&gt; 1] + 1;\n    }\n    for (int j = 1; j &lt;= lg2&#x5B;n]; j++)\n    {\n        for (int i = 1; i + (1 &lt;&lt; (j - 1)) &lt;= n; i++)\n        {\n            st&#x5B;i]&#x5B;j] = max(st&#x5B;i]&#x5B;j - 1], st&#x5B;i + (1 &lt;&lt; (j - 1))]&#x5B;j - 1]);\n        }\n    }\n}\n\nint RMQ(int l, int r)\n{\n    int s=lg2&#x5B;r-l+1];\n    return max(st&#x5B;l]&#x5B;s], st&#x5B;r - (1&lt;&lt;s)+1]&#x5B;s]);\n}\n\nint main()\n{\n    int n, m;\n    scanf(&quot;%d%d&quot;, &amp;n, &amp;m);\n    for (int i = 1; i &lt;= n; i++)\n    {\n        scanf(&quot;%d&quot;, &amp;st&#x5B;i]&#x5B;0]);\n    }\n    initST(n);\n    int l, r;\n    while (m--)\n    {\n        scanf(&quot;%d%d&quot;, &amp;l, &amp;r);\n        printf(&quot;%d\\n&quot;, RMQ(l, r));\n    }\n    return 0;\n}\n<\/pre><\/div>","protected":false},"excerpt":{"rendered":"<p>\u5b9a\u4e49 ST \u8868\uff08Sparse Table\uff0c\u7a00\u758f\u8868\uff09\u662f\u7528\u4e8e\u89e3\u51b3&nbsp;\u53ef\u91cd\u590d\u8d21\u732e\u95ee\u9898&nbsp;\u7684\u6570\u636e\u7ed3\u6784\u3002 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":883,"comment_status":"closed","ping_status":"closed","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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center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[4],"tags":[],"class_list":["post-873","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algorithm"],"_links":{"self":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/873","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=873"}],"version-history":[{"count":3,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/873\/revisions"}],"predecessor-version":[{"id":881,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/873\/revisions\/881"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/media\/883"}],"wp:attachment":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=873"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=873"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=873"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}