{"id":742,"date":"2021-12-11T23:46:54","date_gmt":"2021-12-11T15:46:54","guid":{"rendered":"https:\/\/blog.cauchyschwarz.com\/?p=742"},"modified":"2021-12-14T00:23:25","modified_gmt":"2021-12-13T16:23:25","slug":"%e6%95%b0%e6%8d%ae%e7%bb%93%e6%9e%84%e4%b8%8e%e7%ae%97%e6%b3%95%e5%88%86%e6%9e%90c%e8%af%ad%e8%a8%80%e6%8f%8f%e8%bf%b0-%e5%b9%b6%e6%9f%a5%e9%9b%86","status":"publish","type":"post","link":"https:\/\/blog.cauchyschwarz.com\/?p=742","title":{"rendered":"\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790C++\u8bed\u8a00\u63cf\u8ff0 \u5e76\u67e5\u96c6"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-right counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69e34bcf2274a\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69e34bcf2274a\" checked aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E7%AD%89%E4%BB%B7%E5%85%B3%E7%B3%BBEquivalence_Relations\" >\u7b49\u4ef7\u5173\u7cfb(Equivalence Relations)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#The_Dynamic_Equivalence_Problem\" >The Dynamic Equivalence Problem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#Disjoint%E7%9A%84%E5%AE%9E%E7%8E%B0\" >Disjoint\u7684\u5b9e\u73b0<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E5%88%9D%E5%A7%8B%E7%8A%B6%E6%80%81\" >\u521d\u59cb\u72b6\u6001<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#Union%E6%93%8D%E4%BD%9C\" >Union\u64cd\u4f5c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#Find%E6%93%8D%E4%BD%9C\" >Find\u64cd\u4f5c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E5%A4%8D%E6%9D%82%E5%BA%A6%E5%88%86%E6%9E%90\" >\u590d\u6742\u5ea6\u5206\u6790<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E8%B7%AF%E5%BE%84%E5%8E%8B%E7%BC%A9Path_Compression\" >\u8def\u5f84\u538b\u7f29(Path Compression)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E5%88%A0%E9%99%A4%E6%93%8D%E4%BD%9C\" >\u5220\u9664\u64cd\u4f5c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E5%BA%94%E7%94%A8\" >\u5e94\u7528<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E7%94%9F%E6%88%90%E8%BF%B7%E5%AE%AB\" >\u751f\u6210\u8ff7\u5bab<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=742\/#%E7%BB%83%E4%B9%A0\" >\u7ec3\u4e60<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%AD%89%E4%BB%B7%E5%85%B3%E7%B3%BBEquivalence_Relations\"><\/span>\u7b49\u4ef7\u5173\u7cfb(Equivalence Relations)<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>An equivalence relation is a relation R that satisfies three properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>(Reflexive) a R a, for all a \\( \\in \\) S.<\/li><li>(Symmetric) a R b if and only if b R a.<\/li><li>(Transitive) a R b and b Rc implies that a R c.<\/li><\/ol>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"The_Dynamic_Equivalence_Problem\"><\/span>The Dynamic Equivalence Problem<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>The <strong>equivalence class <\/strong>of an element a \\( \\in \\) S is the subset of S that contains all the elements that are related to a.<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u6bcf\u4e2a\u5143\u7d20\u6240\u5c5e\u7684equivalence class\u5176\u5b9e\u5c31\u662f\u56fe\u8bba\u4e2d\u8be5\u5143\u7d20\u6240\u5c5e\u7684\u8fde\u901a\u5206\u91cf\uff0c\u800c\u8fde\u901a\u5173\u7cfb\u5c31\u7b49\u4e8e\u8fd9\u91cc\u6240\u8bf4\u7684\u7b49\u4ef7\u5173\u7cfb\u3002\u6bcf\u4e2a\u8fde\u901a\u5206\u91cf\u4e5f\u88ab\u79f0\u4e4b\u4e3adisjoint set\uff0c\u5bf9\u4e8e\u6574\u4e2a\u96c6\u5408S\uff0c\u6211\u4eec\u901a\u5e38\u9700\u8981\u505a2\u4ef6\u4e8b\u60c5\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>find\u64cd\u4f5c\uff0c\u67e5\u627e\u6bcf\u4e2a\u5143\u7d20\u6240\u5c5e\u7684\u8fde\u901a\u5206\u91cf\/disjoint set\uff1b<\/li><li>merge\u64cd\u4f5c\uff0c\u6bcf\u5f53\u77e5\u9053\u4e24\u4e2a\u5143\u7d20\u4e4b\u95f4\u5b58\u5728\u7b49\u4ef7\u5173\u7cfb\/\u8fde\u901a\u4e4b\u540e\uff0c\u5c06\u4e24\u4e2a\u5143\u7d20\u6240\u5c5e\u7684\u8fde\u901a\u5206\u91cf\/disjoint set\u878d\u5408\u4e3a\u4e00\u4e2a\uff1b<\/li><\/ol>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Disjoint%E7%9A%84%E5%AE%9E%E7%8E%B0\"><\/span>Disjoint\u7684\u5b9e\u73b0<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>\u75280\uff0c1\uff0c&#8230;\uff0cN-1\u4ee3\u8868\u603b\u5171\u7684N\u4e2a\u8282\u70b9\uff0c\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3aN\u7684\u6570\u7ec4\u6765\u4fdd\u5b58\u6240\u6709\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5185\u4fdd\u5b58\u5176\u7236\u8282\u70b9\u3002\u5982\u679c\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\u4e3a-1\uff0c\u90a3\u4e48\u8fd9\u4e2a\u8282\u70b9\u5c31\u662f\u6811\u7684\u6839\u3002\u8fd9\u4e2a\u6570\u7ec4\u5185\u5b58\u5728\u591a\u5c11\u4e2a\u6839\u8282\u70b9\uff0c\u90a3\u4e48\u8fd9\u4e2a\u6570\u7ec4\u5c31\u5305\u542b\u4e86\u591a\u5c11\u68f5\u6811\/\u591a\u5c11\u4e2a\u8fde\u901a\u5206\u91cf\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%88%9D%E5%A7%8B%E7%8A%B6%E6%80%81\"><\/span>\u521d\u59cb\u72b6\u6001<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u6240\u6709\u8282\u70b9\u7684\u7236\u8282\u70b9\u90fd\u662f-1\uff0c\u5373\u6bcf\u4e2a\u8282\u70b9\u90fd\u662f\u6811\u7684\u6839\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"286\" src=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetInitialState0-1024x286.png\" alt=\"\" class=\"wp-image-748\" srcset=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetInitialState0-1024x286.png 1024w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetInitialState0-300x84.png 300w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetInitialState0-768x215.png 768w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetInitialState0-670x187.png 670w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetInitialState0.png 1491w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>s[i] = -1 for i in 0, 1, 2,&#8230;, N &#8211; 1\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Union%E6%93%8D%E4%BD%9C\"><\/span>Union\u64cd\u4f5c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"630\" src=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0-1024x630.png\" alt=\"\" class=\"wp-image-749\" srcset=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0-1024x630.png 1024w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0-300x185.png 300w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0-768x472.png 768w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0-1536x945.png 1536w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0-670x412.png 670w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion0.png 1632w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"691\" src=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion1-1024x691.png\" alt=\"\" class=\"wp-image-750\" srcset=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion1-1024x691.png 1024w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion1-300x202.png 300w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion1-768x518.png 768w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion1-670x452.png 670w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetUnion1.png 1419w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\u4e0a\u9762\u663e\u793a\u4e86Union(4,5)\uff0cUnion(6,7)\u548cUnion(4,6)\u4e09\u6b21\u64cd\u4f5c\u540e\u7684\u7ed3\u679c\uff0c\u8fd9\u91cc\u7684Union\u64cd\u4f5c\u5b9e\u73b0\u662f\u76f4\u63a5\u5c06\u7b2c\u4e8c\u4e2a\u68f5\u6811\u8bbe\u7f6e\u4e3a\u7b2c\u4e00\u68f5\u6811\u7684\u5b50\u6811\u3002<\/p>\n\n\n\n<p>\u5b9e\u9645\u4e0a\u8fd8\u6709\u53e6\u5916\u4e24\u79cd\u5b9e\u73b0\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>union-by-size\uff1a\u5c31\u662f\u628a\u8282\u70b9\u6570\u5c11\u7684\u6811\u4f5c\u4e3a\u8282\u70b9\u6570\u5927\u7684\u6811\u7684\u5b50\u6811\uff1b\u8fd9\u79cd\u5b9e\u73b0\u53ef\u4ee5\u628a\u6811\u7684\u8282\u70b9\u6570\u7684\u76f8\u53cd\u6570\u4f5c\u4e3a\u6839\u8282\u70b9\u7684\u7236\u8282\u70b9\uff1b<\/li><li>union-by-height\uff1a\u5c31\u662f\u628a\u9ad8\u5ea6\u5c0f\u7684\u6811\u4f5c\u4e3a\u9ad8\u5ea6\u5927\u7684\u6811\u7684\u5b50\u6811\uff1b\u8fd9\u79cd\u5b9e\u73b0\u53ef\u4ee5\u628a\u9ad8\u5ea6\u7684\u76f8\u53cd\u6570\u4f5c\u4e3a\u6839\u8282\u70b9\u7684\u7236\u8282\u70b9\uff1b<\/li><\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Find%E6%93%8D%E4%BD%9C\"><\/span>Find\u64cd\u4f5c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u8fd4\u56de\u6bcf\u4e2a\u8282\u70b9\u6240\u5c5e\u7684\u6811\u7684\u6839\u8282\u70b9\u3002<\/p>\n\n\n\n<p>\u5bf9\u6bcf\u4e2a\u8282\u70b9\u5411\u4e0a\u56de\u6eaf\u7236\u8282\u70b9\uff0c\u6700\u540e\u4e00\u4e2a\u975e-1\u7684\u7236\u8282\u70b9\u5c31\u662f\u6811\u7684\u6839\u8282\u70b9\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%A4%8D%E6%9D%82%E5%BA%A6%E5%88%86%E6%9E%90\"><\/span>\u590d\u6742\u5ea6\u5206\u6790<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5047\u8bbe\u8fdb\u884c\u6700\u591aN-1\u6b21Union\u64cd\u4f5c\uff0cM\u6b21find\u64cd\u4f5c\uff0c\u90a3\u4e48\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u5bf9\u4e8e\u7b2c\u4e00\u79cdUnion\u5b9e\u73b0\uff0c\u5373\u65e0\u8111\u5730\u628a\u7b2c\u4e8c\u68f5\u6811\u4f5c\u4e3a\u7b2c\u4e00\u68f5\u6811\u7684\u5b50\u6811\uff0c\u6700\u574f\u60c5\u51b5\u4e0b\u4f1a\u9700\u8981\\( \\Theta (MN) \\)\u7684\u65f6\u95f4\uff1b<\/li><li>\u5bf9\u4e8e\u7b2c\u4e8c\u79cdunion-by-size\u5b9e\u73b0\uff0c\u56e0\u4e3a\u6bcf\u6b21Union\u65f6\uff0c\u751f\u6210\u7684\u6811\u7684\u8282\u70b9\u6570\u662f\u539f\u6765\u76842\u500d\uff0c\u6240\u4ee5N\u6b21Union\u64cd\u4f5c\u6700\u591aN\u4e2a\u8282\u70b9\uff0c\u6700\u591a\u652f\u6301\\( \\log N \\)\u6b21\u9ad8\u5ea6\u7684\u589e\u52a0\uff0c\u5373\u751f\u6210\u7684\u6811\u7684\u9ad8\u5ea6\u6700\u591a\u4e3a\\( \\log N\\)\uff0c\u6240\u4ee5\u6700\u574f\u60c5\u51b5\u4e0b\u9700\u8981\\(\\Theta(M\\log N)\\)\u65f6\u95f4\uff1b<\/li><li>\u7b2c\u4e09\u79cdunion-by-height\u7684\u5b9e\u73b0\uff0c\u751f\u6210\u7684\u6811\u7684\u9ad8\u5ea6\u4e5f\u662f\u6700\u591a\u4e3a\\(\\log N\\)\uff0c\u56e0\u6b64\u6700\u574f\u60c5\u51b5\u4e0b\u4e5f\u662f\u9700\u8981\\(\\Theta(M\\log N)\\)\u65f6\u95f4\u3002<\/li><\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%B7%AF%E5%BE%84%E5%8E%8B%E7%BC%A9Path_Compression\"><\/span>\u8def\u5f84\u538b\u7f29(Path Compression)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u591a\u6b21Union\u64cd\u4f5c\u53ef\u80fd\u4f1a\u5bfc\u81f4\u6811\u7684\u9ad8\u5ea6\u5f88\u9ad8\uff0c\u6211\u4eec\u53ef\u4ee5\u8fdb\u4e00\u6b65\u4f18\u5316\u964d\u4f4e\u6811\u7684\u9ad8\u5ea6\u3002Union\u64cd\u4f5c\u6bd4\u8f83\u96be\u4ee5\u6539\u8fdb\u4e86\uff0c\u53ef\u4ee5\u4ecefind\u64cd\u4f5c\u4e0b\u624b\u3002\u6bcf\u6b21\u8fdb\u884cfind\u64cd\u4f5c\u65f6\uff0c\u53ef\u4ee5\u5c06\u4ece\u8f93\u5165\u8282\u70b9\u5230\u6811\u7684\u6839\u8282\u70b9\u8fd9\u6761\u8def\u5f84\u4e0a\u7684\u6240\u6709\u8282\u70b9\u7684\u7236\u8282\u70b9\u8bbe\u7f6e\u4e3a\u6811\u7684\u6839\u8282\u70b9\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5b9e\u73b0\u4e00\u4e2a\u4e0a\u754c\u4e3a\\( O(M \\alpha (M,N) ) \\)\u7684\u7b97\u6cd5\u3002\\( \\alpha (M,N) \\)\u51fd\u6570\u7684\u8868\u8fbe\u5f0f\u5982\u4e0b\uff0c\u5bf9\u4e8e\u6211\u4eec\u5b9e\u9645\u95ee\u9898\u4e2d\u4f7f\u7528\u5230\u7684\u6570\u5b57\u8303\u56f4\u800c\u8a00\uff0c\u8fd9\u4e2a\u51fd\u6570\u7684\u503c\u4e0d\u4f1a\u8d85\u8fc75\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"247\" src=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetPathCompressionComplexity-1024x247.png\" alt=\"\" class=\"wp-image-761\" srcset=\"https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetPathCompressionComplexity-1024x247.png 1024w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetPathCompressionComplexity-300x72.png 300w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetPathCompressionComplexity-768x185.png 768w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetPathCompressionComplexity-670x161.png 670w, https:\/\/blog.cauchyschwarz.com\/wp-content\/uploads\/2021\/12\/DisjointSetPathCompressionComplexity.png 1436w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\u8def\u5f84\u538b\u7f29\u4e0eunion-by-size\u7b56\u7565\u53ef\u4ee5\u540c\u65f6\u4f7f\u7528\uff0c\u800c\u4e0eunion-by-height\u4e0d\u80fd\u540c\u65f6\u4f7f\u7528\uff0c\u56e0\u4e3a\u524d\u8005\u53ef\u4ee5\u5bb9\u6613\u5730\u66f4\u65b0\u6bcf\u68f5\u6811\u4e2d\u7684\u8282\u70b9\u6570\uff0c\u800c\u540e\u8005\u5374\u96be\u4ee5\u8ba1\u7b97\u8def\u5f84\u538b\u7f29\u4e4b\u540e\u7684\u6bcf\u68f5\u6811\u7684\u9ad8\u5ea6\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%88%A0%E9%99%A4%E6%93%8D%E4%BD%9C\"><\/span>\u5220\u9664\u64cd\u4f5c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Disjoint Set\u53ea\u80fd\u652f\u6301\u5220\u9664\u8282\u70b9\uff0c\u4e0d\u80fd\u652f\u6301\u5220\u9664\u4e00\u6761\u8fb9\u3002<\/p>\n\n\n\n<p>\u9996\u5148\u660e\u786e\u4e0b\u8fd9\u91cc\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff1a\u5c06\u8282\u70b9\u4ece\u5176\u5f53\u524d\u6240\u5c5e\u7684\u8fde\u901a\u5206\u91cf\/\u6811\u91cc\u53d6\u51fa\u6765\uff0c\u5355\u72ec\u5f62\u6210\u4e00\u4e2a\u8fde\u901a\u5206\u91cf\/\u6811\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u5b9e\u73b0\uff1a\u6bcf\u4e2a\u8282\u70b9\u7528\u6811\u91cc\u9762\u7684\u8282\u70b9\u5b9e\u73b0\uff0c\u6570\u7ec4\u7528\u6765\u4fdd\u5b58\u5bf9\u5e94\u7684\u8282\u70b9\u6307\u9488\u3002<\/p>\n\n\n\n<p>\u6bcf\u6b21\u6267\u884cremove(x)\u64cd\u4f5c\u65f6\uff0c\u5148\u6267\u884c\u6b63\u5e38\u7684\u5e26\u8def\u5f84\u538b\u7f29\u7684find\u64cd\u4f5c\uff0c\u7136\u540e\u518d\u5c06\u542b\u6709x\u7684\u90a3\u4e2a\u8282\u70b9\u6807\u8bb0\u4e3a\u65e0\u6548\uff0c\u5e76\u4e14\u65b0\u5efa\u4e00\u4e2a\u8282\u70b9\u4fdd\u5b58x\uff0c\u4ee5\u53ca\u6570\u7ec4\u4e2ds[x]\u6307\u5411\u65b0\u5efa\u7684\u8282\u70b9\u3002<\/p>\n\n\n\n<p>\u6267\u884cN\u6b21remove\u64cd\u4f5c\u540e\uff0c\u5bf9\u6240\u6709\u8282\u70b9\u6267\u884c\u5e26\u8def\u5f84\u538b\u7f29\u7684find(x)\u64cd\u4f5c\uff0c\u5982\u679c\u8fd4\u56de\u7684\u6839\u8282\u70b9\u662f\u88ab\u6807\u8bb0\u4e86\u65e0\u6548\uff0c\u90a3\u4e48\u5c31\u8ba9\u8fd9\u4e2a\u65e0\u6548\u8282\u70b9\u4fdd\u5b58x\uff0c\u5e76\u6e05\u9664\u65e0\u6548\u6807\u8bb0\uff0c\u7136\u540e\u8ba9\u539f\u672c\u4fdd\u5b58x\u7684\u8282\u70b9\u88ab\u6807\u8bb0\u4e3a\u65e0\u6548\u3002\u7ecf\u8fc7\u8fd9\u4e48\u4e00\u8f6e\u64cd\u4f5c\u540e\u6240\u6709\u88ab\u6807\u8bb0\u4e3a\u65e0\u6548\u7684\u8282\u70b9\u90fd\u4e0d\u4f1a\u662f\u5176\u4ed6\u8282\u70b9\u7684\u7236\u8282\u70b9\u4e86\uff0c\u53ef\u4ee5\u5220\u9664\u6389\u4e86\u3002\u7531\u4e8e\u662f\u5728\u8fdb\u884c\u4e86N\u6b21remove\u64cd\u4f5c\u540e\u5c31\u8fdb\u884c\u4e00\u8f6e\u8fd9\u6837\u7684\u64cd\u4f5c\uff0c\u6240\u4ee5\u6574\u4e2a\u68ee\u6797\u4e2d\u65e0\u6548\u8282\u70b9\u6570\u4e0d\u4f1a\u8d85\u8fc7N\u3002\u800c\u4e14\u8fd9\u4e00\u8f6e\u6e05\u7406\u64cd\u4f5c\u7684\u65f6\u95f4\u53ef\u4ee5\u5747\u644a\u5230\u8fd9N\u6b21remove\u4e0a\u3002\u56e0\u6b64\u6574\u4e2a\u5e76\u67e5\u96c6\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8fd8\u662f\u7ef4\u6301\u5728\u4e86\\( O(M\\alpha (M,N)) \\)\u3002<\/p>\n\n\n\n<p>\u5982\u679c\u60f3\u652f\u6301\u5220\u9664\u8fb9\u7684\u64cd\u4f5c\uff0c\u90a3\u4e48\u6211\u4eec\u9700\u8981\u53e6\u5916\u4e00\u79cd\u6570\u636e\u7ed3\u6784<a href=\"https:\/\/stackoverflow.com\/a\/12689835\/5357784\" data-type=\"URL\" data-id=\"https:\/\/stackoverflow.com\/a\/12689835\/5357784\">link\/cut tree<\/a>\u3002\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u65b9\u4fbf\u7684\u4ee5\\( O(\\log N) \\)\u7684\u590d\u6742\u5ea6\u8fdb\u884cUnion\uff0cisConnected\uff0cUn-union\u4e09\u79cd\u64cd\u4f5c\uff0c\u4f46\u662f\u9650\u5236\u662f\u6574\u4e2a\u56fe\u4e2d\u5fc5\u987b\u65e0\u73af\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%BA%94%E7%94%A8\"><\/span>\u5e94\u7528<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%94%9F%E6%88%90%E8%BF%B7%E5%AE%AB\"><\/span>\u751f\u6210\u8ff7\u5bab<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u521d\u59cb\u72b6\u6001\uff1a\u6240\u6709\u5bab\u683c\u4e4b\u95f4\u4e0d\u8fde\u901a\u3002<\/p>\n\n\n\n<p>\u4e0d\u505c\u5730\u968f\u673a\u8fde\u901a2\u4e2a\u76f8\u90bb\u7684\u5bab\u683c\uff0c\u76f4\u5230\u8d77\u70b9\u548c\u7ec8\u70b9\u4e4b\u95f4\u8fde\u901a\u3002<\/p>\n\n\n\n<p>\u5224\u65ad\u8d77\u70b9\u548c\u7ec8\u70b9\u4e4b\u95f4\u662f\u5426\u8fde\u901a\u5c31\u4f7f\u7528\u8fd9\u91cc\u7684\u5e76\u67e5\u96c6\u6765\u8fdb\u884c\u5224\u65ad\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%BB%83%E4%B9%A0\"><\/span>\u7ec3\u4e60<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ol class=\"wp-block-list\"><li><a href=\"https:\/\/www.hackerrank.com\/challenges\/components-in-graph\/problem?isFullScreen=false\">Components in a graph<\/a><\/li><li><a href=\"https:\/\/www.hackerrank.com\/challenges\/merging-communities\/problem?isFullScreen=false\">Merging Communities<\/a><\/li><li><a href=\"https:\/\/www.hackerrank.com\/challenges\/maximum-cost-queries\/problem?isFullScreen=true\" data-type=\"URL\" data-id=\"https:\/\/www.hackerrank.com\/challenges\/maximum-cost-queries\/problem?isFullScreen=true\">Super Maximum Cost Queries<\/a> \u6309\u7167\u8fb9\u4ece\u5c0f\u5230\u5927\u6784\u9020\u5e76\u67e5\u96c6\uff0c\u6bcf\u6b21union\u65f6\uff0c\u65b0\u589e\u7684\u4ee5\u5f53\u524d\u8fb9weight\u4e3acost\u7684\u8def\u5f84\u6570\u4e3aunion\u7684\u4e24\u68f5\u6811\u7684\u8282\u70b9\u6570\u4e58\u79ef<\/li><li><a href=\"https:\/\/www.hackerrank.com\/challenges\/kundu-and-tree\/problem?isFullScreen=true\" data-type=\"URL\" data-id=\"https:\/\/www.hackerrank.com\/challenges\/kundu-and-tree\/problem?isFullScreen=true\">Kundu and Tree<\/a> \u53ea\u8003\u8651\u9ed1\u8272\u7684\u8fb9\uff0c\u9ed1\u8272\u7684\u8fb9\u5c06\u6240\u6709\u8282\u70b9\u5206\u6210\u4e86\u82e5\u5e72\u4e2a\u8fde\u901a\u5206\u91cf\uff0c\u4ece\u4e2d\u4efb\u53d63\u4e2a\u8fde\u901a\u5206\u91cf\uff0c\u6bcf\u4e2a\u8fde\u901a\u5206\u91cf\u4e2d\u4efb\u53d61\u4e2a\u8282\u70b9\u5373\u53ef\u5f62\u6210\u4e00\u4e2a\u95ee\u9898\u4e2d\u6240\u6c42\u7684\u4e09\u5143\u7ec4\u3002<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u7b49\u4ef7\u5173\u7cfb(Equivalence Relations) An equivalence relation is a relation R that satisfies three properties:&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[26,78,9],"class_list":["post-742","post","type-post","status-publish","format-standard","hentry","category-10","tag-algorithm","tag-disjointset","tag-9"],"_links":{"self":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/742","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=742"}],"version-history":[{"count":18,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/742\/revisions"}],"predecessor-version":[{"id":777,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/742\/revisions\/777"}],"wp:attachment":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=742"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=742"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}