{"id":197,"date":"2025-02-10T18:29:19","date_gmt":"2025-02-10T10:29:19","guid":{"rendered":"https:\/\/blog.seekrealthing.net\/blog\/?p=197"},"modified":"2025-02-12T12:50:05","modified_gmt":"2025-02-12T04:50:05","slug":"e-the-road-to-berland-is-paved-with-good-intentions-%e8%a7%a3%e6%9e%90%e6%80%9d%e7%b6%ad%e3%80%812-sat%e3%80%81scc%e3%80%81%e6%8b%93%e6%a8%b8%e6%8e%92%e5%ba%8f","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=197","title":{"rendered":"E. The Road to Berland is Paved With Good Intentions \u89e3\u6790(\u601d\u7dad\u30012-SAT\u3001SCC\u3001\u62d3\u6a38\u6392\u5e8f)"},"content":{"rendered":"\n<div class=\"wp-block-jetpack-markdown\"><p>\u4eca\u5929\u6211\u5011\u4f86\u770b\u770bCF228E\n<a href=\"https:\/\/codeforces.com\/contest\/228\/problem\/E\">\u984c\u76ee\u9023\u7d50<\/a><\/p>\n<h3>\u984c\u76ee\u63cf\u8ff0<\/h3>\n<p>\u7d66\u5b9a\u4e00\u500b\u5305\u542b $n$ \u500b\u57ce\u5e02\u548c $m$ \u689d\u9053\u8def\u7684\u7121\u5411\u5716\uff0c\u6bcf\u689d\u9053\u8def\u90fd\u6709\u4e00\u500b\u521d\u59cb\u72c0\u614b\uff1a<\/p>\n<ul>\n<li><strong>0<\/strong> \u8868\u793a\u8a72\u9053\u8def\u5c1a\u672a\u92ea\u8a2d\uff1b<\/li>\n<li><strong>1<\/strong> \u8868\u793a\u8a72\u9053\u8def\u5df2\u92ea\u8a2d\u3002<\/li>\n<\/ul>\n<p>\u6bcf\u5929\uff0c\u570b\u738b Valera II \u53ef\u4ee5\u9078\u64c7 <strong>\u6070\u597d\u4e00\u500b\u57ce\u5e02<\/strong>\uff0c\u4e26\u6307\u6d3e\u5de5\u4eba\u5c0d\u8a72\u57ce\u5e02<strong>\u76f8\u9023\u7684\u6240\u6709\u9053\u8def<\/strong>\u9032\u884c\u53d6\u53cd\u64cd\u4f5c\uff08\u5373\u5c07 0 \u8b8a\u70ba 1\uff0c1 \u8b8a\u70ba 0\uff09\u3002<br>\n\u7531\u65bc\u5c0d\u540c\u4e00\u57ce\u5e02\u91cd\u8907\u64cd\u4f5c\u6c92\u6709\u984d\u5916\u6548\u679c\uff08\u53d6\u53cd\u5169\u6b21\u6062\u5fa9\u539f\u72c0\uff09\uff0c\u6240\u4ee5\u6bcf\u500b\u57ce\u5e02\u6700\u591a\u53ea\u9700\u64cd\u4f5c\u4e00\u6b21\u3002<br>\n\u4f60\u7684\u4efb\u52d9\u662f\u5224\u65b7\u662f\u5426\u5b58\u5728\u4e00\u500b\u64cd\u4f5c\u5e8f\u5217\uff08\u64cd\u4f5c\u5929\u6578\u4e0d\u8d85\u904e $n$ \u5929\uff09\uff0c\u4f7f\u5f97\u6700\u7d42\u6240\u6709\u9053\u8def\u5747\u8b8a\u70ba\u92ea\u8a2d\u72c0\u614b\uff08\u5373\u72c0\u614b\u5747\u70ba 1\uff09\u3002\u82e5\u5b58\u5728\uff0c\u8acb\u8f38\u51fa\u4e00\u7d44\u5408\u6cd5\u7684\u64cd\u4f5c\u65b9\u6848\uff1b\u5426\u5247\uff0c\u8f38\u51fa <code>Impossible<\/code>\u3002<\/p>\n<hr>\n<h3><strong>\u8f38\u5165\u683c\u5f0f<\/strong><\/h3>\n<ul>\n<li>\u7b2c\u4e00\u884c\u5305\u542b\u5169\u500b\u6574\u6578 $n$ \u548c $m$\uff0c\u5206\u5225\u4ee3\u8868\u57ce\u5e02\u6578\u91cf\u548c\u9053\u8def\u6578\u91cf\u3002<\/li>\n<li>\u63a5\u4e0b\u4f86 $m$ \u884c\uff0c\u6bcf\u884c\u5305\u542b\u4e09\u500b\u6574\u6578 $a_i, b_i, c_i$\uff1a\n<ul>\n<li>$a_i$ \u548c $b_i$ \u8868\u793a\u4e00\u689d\u9023\u63a5\u9019\u5169\u500b\u57ce\u5e02\u7684\u9053\u8def\uff08\u4fdd\u8b49 $a_i \\neq b_i$\uff09\uff1b<\/li>\n<li>$c_i$ \u70ba\u8a72\u9053\u8def\u7684\u521d\u59cb\u72c0\u614b\uff08$0$ \u8868\u793a\u672a\u92ea\u8a2d\uff0c$1$ \u8868\u793a\u5df2\u92ea\u8a2d\uff09\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr>\n<h3><strong>\u8f38\u51fa\u683c\u5f0f<\/strong><\/h3>\n<ul>\n<li>\u5982\u679c\u5b58\u5728\u5408\u6cd5\u64cd\u4f5c\u5e8f\u5217\uff1a\n<ul>\n<li>\u7b2c\u4e00\u884c\u8f38\u51fa\u64cd\u4f5c\u5929\u6578 $x$\uff08\u6eff\u8db3 $0 \\le x \\le n$\uff09\uff1b<\/li>\n<li>\u7b2c\u4e8c\u884c\u8f38\u51fa $x$ \u500b\u6574\u6578\uff0c\u4f9d\u6b21\u8868\u793a\u6bcf\u5929\u9078\u64c7\u64cd\u4f5c\u7684\u57ce\u5e02\u7de8\u865f\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\u5982\u679c\u4e0d\u5b58\u5728\u5408\u6cd5\u65b9\u6848\uff0c\u5247\u8f38\u51fa <code>Impossible<\/code>\u3002<\/li>\n<\/ul>\n<h2>\u524d\u8a00<\/h2>\n<p>\u60f3\u4e0d\u51fa\u4f86\u7684\u6642\u5019\u9084\u662f\u591a\u770b\u89e3\u7b54\u597d\u4e86\uff0c\u4ee5\u514d\u6d6a\u8cbb\u6642\u9593\u5728\u6211\u6c92\u5b78\u904e\u7684\u6f14\u7b97\u6cd5\u4e0a\u9762<\/p>\n<h3>\u60f3\u6cd5<\/h3>\n<p>\u9019\u984c\u770b\u8d77\u4f86\u6709\u4e09\u7a2e\u89e3\u6cd5\uff1a<\/p>\n<ol>\n<li>\u9ad8\u65af\u6d88\u53bb\u6cd5<\/li>\n<li>2-SAT<\/li>\n<li>Greedy<\/li>\n<\/ol>\n<p>\u9019\u6b21\u6211\u4f7f\u7528\u7b2c\u4e8c\u7a2e\uff0c2-SAT\u7684\u65b9\u6cd5\u3002\u525b\u597d\u628a\u4ee5\u524d\u6c92\u5b78\u61c2\u7684BCC\u3001SCC\u3001DFS\u56de\u908a\u3001\u524d\u5411\u908a\u3001\u6a39\u908a\u3001\u4ea4\u932f\u908a\u5b78\u8d77\u4f86\u3002<br>\n\u9019\u984c\u5c0d\u6211\u4f86\u8aaa\u5c31\u662f2-SAT\u7684\u6a21\u677f\u984c\uff1a<br>\n\u958b\u4e00\u5f35\u65b0\u7684\u5716\uff0c\u5982\u679c\u539f\u5716\u4e2d$x,y$\u7684\u9023\u7dda\u662f$0$\uff0c\u90a3\u9ebc\u908f\u8f2f\u95dc\u4fc2\u5c31\u662f:<br>\n\\begin{align*}\n&amp;x\\rightarrow\\neg y \\\n&amp;\\neg x\\rightarrow y \\\n&amp;\\neg y\\rightarrow x \\\n&amp;y\\rightarrow\\neg x\n\\end{align*}\n\u53cd\u4e4b\u5982\u679c\u539f\u5716\u4e2d$x,y$\u7684\u9023\u7dda\u662f$1$\uff0c\u90a3\u9ebc\u908f\u8f2f\u95dc\u4fc2\u5c31\u662f:<br>\n\\begin{align*}\n&amp;x\\rightarrow y \\\n&amp;\\neg x\\rightarrow\\neg y \\\n&amp;\\neg y\\rightarrow\\neg x \\\n&amp;y\\rightarrow x\n\\end{align*}\n\u800c\u6211\u5011\u65b0\u7684\u5716\u88e1\u9762\u53ef\u4ee5\u7528$x*2$\u7576\u6210False\u7684\u9ede\uff0c$x*2+1$\u7576\u6210True\u7684\u9ede\uff0c\u7528\u9019\u500b\u898f\u5247\u9020\u4e00\u5f35\u65b0\u7684\uff0c\u5305\u542b$2n$\u500b\u7bc0\u9ede\u7684\u65b0\u5716\u3002<br>\n\u63a5\u8457\u4f7f\u7528Tarjan\u2019s Algorithm\u627eSCC\u3002<br>\nTarjan\u627eBCC\u548c\u627eSCC\u9664\u4e86\u7121\u5411\u5716\u548c\u6709\u5411\u5716\u7684\u5dee\u7570\u4ee5\u5916\uff0c\u5c31\u5728<\/p>\n<pre><code class=\"language-cpp\">if (!scc[u]) low[v] = min(low[v], low[u]);\n<\/code><\/pre>\n<p>\u9019\u4e00\u884c\u800c\u5df2\u3002<br>\n\u5982\u679c\u8981\u627eBCC\uff0c\u6211\u5011\u7684<code>dfs<\/code>\u51fd\u6578\u5c31\u9700\u8981\u6709<code>int father<\/code>\u9019\u500b\u53c3\u6578\uff0c\u5224\u65b7\u63a5\u4e0b\u4f86\u8981dfs\u7684\u9ede\u662f\u4e0d\u662f\u7236\u7bc0\u9ede(\u6709\u5411\u5716\u88e1\u9762\u5f80\u7236\u7bc0\u9ede\u7684\u908a\u4e5f\u662f\u5408\u7406\u7684\u56de\u908a)\u3002\u4e26\u4e14\u7531\u65bc\u7121\u5411\u5716\u6c92\u6709\u4ea4\u932f\u908a\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5224\u65b7<code>!scc[u]<\/code><br>\n\u6ce8\u610f\u5230Tarjan SCC\u88e1\u9762\uff0c\u6211\u5011\u6307\u6d3eSCC Number\u662f\u5728dfs\u7684\u5f8c\u5e8f\u7684\u5730\u65b9\u3002\u800c\u56de\u61b6\u8d77\u62d3\u6a38\u6392\u5e8f\u7684\u5176\u4e2d\u4e00\u7a2e\u4f5c\u6cd5\u5c31\u662f\u7d00\u9304dfs\u7684\u5f8c\u5e8f\uff0c\u5c31\u662f\u53cd\u904e\u4f86\u7684\u62d3\u6a38\u9806\u5e8f\u4e86\u3002<br>\n\u800cSCC\u7e2e\u9ede\u4ee5\u5f8c\uff0c\u6703\u662f\u4e00\u500b\u6709\u5411\u7121\u74b0\u5716(DAG)\uff0c\u56e0\u6b64SCC Number\u5c0f\u7684SCC\uff0c\u6703\u5728\u62d3\u6a38\u6392\u5e8f\u7684\u5f8c\u9762\u3002<br>\n\u7576\u7136\uff0c\u6211\u5011\u8981\u5148\u78ba\u8a8d\u6709\u6c92\u6709\u908f\u8f2f\u77db\u76fe\uff0c\u4e5f\u5c31\u662f\u6703\u4e0d\u6703\u6709$x\\rightarrow\\neg x$\u7684\u60c5\u6cc1\u767c\u751f\uff0c\u4e5f\u5c31\u662f$x$\u548c$\\neg x$\u7684\u7bc0\u9ede\u5728\u5716\u7684\u540c\u4e00\u500bSCC\u4e2d\u3002<br>\n\u518d\u4f86\u6211\u5011\u5c31\u53ef\u4ee5\u5f9e\u5f8c\u9762\u958b\u59cb\u6307\u6d3e\u5e03\u6797\u8b8a\u6578\u7684\u503c\u4e86\u3002\u56e0\u70ba\u5982\u679c\u6211\u5011\u5f9e\u5f8c\u9762\u958b\u59cb\u6307\u6d3e\uff0c\u5c31\u4e0d\u6703\u6709\u908f\u8f2f\u95dc\u4fc2\u53ef\u4ee5\u5361\u5230\u6211\u5011\u4e86\uff0c\u4f8b\u5982\uff0c$x\\rightarrow y$\u5728SCC Number=2\u7684SCC\u4e2d\uff0c\u800c$\\neg x\\rightarrow\\neg y$\u5728SCC Number=1\u7684SCC\u4e2d\u3002<br>\n\u5982\u679c\u6211\u5011\u5148\u6307\u5b9a$y=true$\uff0c\u56e0\u70baSCC=2\u6703\u6709\u908f\u8f2f\u95dc\u4fc2\u6307\u5411SCC=1\uff0c\u56e0\u6b64\u5c31\u6703\u767c\u751f$y\\rightarrow\u2026\\rightarrow\\neg x\\rightarrow\\neg y$\uff0c\u767c\u751f\u77db\u76fe\u3002<br>\n\u56e0\u6b64\u6211\u5011\u5fc5\u9808\u6307\u5b9a$\\neg y=true$\uff0c\u4e5f\u5c31\u662f$y=false$\u3002\u5982\u6b64\u4e00\u4f86\u524d\u9762\u7684\n$$\ny\\rightarrow\u2026\\rightarrow\\neg x\\rightarrow\\neg y\n$$\n\u5c31\u6703\u56e0\u70ba\u524d\u63d0$y$\u6839\u672c\u4e0d\u6210\u7acb\uff0c\u800c\u4e0d\u6703\u6709\u77db\u76fe\u3002<br>\n(\u9019\u500b\u8aaa\u660e\u65b9\u6cd5\u6211\u4e5f\u4e0d\u662f\u90a3\u9ebc\u78ba\u5b9a\u662f\u4e0d\u662f\u5c0d\u7684\uff0c\u5982\u679c\u6709\u4eba\u77e5\u9053\u6b63\u78ba\u7684\u8b1b\u6cd5\uff0c\u9084\u8acb\u8cdc\u6559)<\/p>\n<h3>\u7a0b\u5f0f\u78bc:<\/h3>\n<pre><code class=\"language-cpp\">const int _n = 100 + 10;\nint t, n, m, T = 0;\nVI G[_n * 2];\nint id[_n * 2], low[_n * 2], scc[_n * 2];\nstack&lt;int&gt; st;\nvoid dfs(int v) {        \/\/ Tarjan SCC\n  id[v] = low[v] = ++T;  \/\/ \u524d\u5e8f\n  st.push(v);\n  for (int u : G[v]) {\n    if (!id[u]) dfs(u);\n    if (!scc[u]) low[v] = min(low[v], low[u]);  \/\/ \u5982\u679c\u662f\u8981\u6c42BCC \u5c31\u8981\u5224\u65b7u!=father \u7136\u5f8c\u4e0d\u7528\u5224\u65b7!scc[u]\n  }\n  if (low[v] == id[v]) {\n    int temp;\n    do {\n      scc[temp = st.top()] = v;  \/\/ SCC\u662fdfs\u7684\u5f8c\u5e8f\uff0c\u525b\u597d\u662f\u7e2e\u9ede\u5f8c\u7684DAG\u7684\u62d3\u6a38\u6392\u5e8f\n      st.pop();\n    } while (temp != v);\n  }\n}\nmain(void) {\n  ios_base::sync_with_stdio(0);\n  cin.tie(0);\n  cout.tie(0);\n  cin &gt;&gt; n &gt;&gt; m;\n  rep(i, 0, m) {\n    int a, b, c;\n    cin &gt;&gt; a &gt;&gt; b &gt;&gt; c;\n    a--, b--;  \/\/ \u9019\u8ddf\u984c\u76ee\u6709\u95dc \u984c\u76ee\u7684vertex\u662f1-indexed\n    if (c) {\n      G[a * 2 + 1].pb(b * 2 + 1);  \/\/ x*2 false, x*2+1 true\n      G[b * 2 + 1].pb(a * 2 + 1);\n      G[a * 2].pb(b * 2);\n      G[b * 2].pb(a * 2);\n    } else {\n      G[a * 2 + 1].pb(b * 2);\n      G[b * 2 + 1].pb(a * 2);\n      G[b * 2].pb(a * 2 + 1);\n      G[a * 2].pb(b * 2 + 1);\n    }\n  }\n  rep(i, 0, n) {\n    if (!id[i * 2]) dfs(i * 2);\n    if (!id[i * 2 + 1]) dfs(i * 2 + 1);\n  }\n  \/\/ tarjan \u7b97\u51fa\u4f86\u7684scc \u5be6\u969b\u4e0a\u662f\u53cd\u904e\u4f86\u7684topo order\n  \/\/ \u800c\u9019\u525b\u597d\u662f\u6211\u5011\u627e2-SAT\u6700\u5f8c\u7684\u89e3\u8981\u7684\uff1a\u53cd\u904e\u4f86\u7684topo order\n\n  \/\/ \u6aa2\u67e5\u540c\u500bscc\u88e1\u9762\u6709\u6c92\u6709 p and ~p\uff0c\u5982\u679c\u6709\uff0c\u6709\u77db\u76fe\uff0c\u4e0d\u53ef\u80fd\n  rep(i, 0, n) {\n    if (scc[i * 2] == scc[i * 2 + 1]) {\n      cout &lt;&lt; &quot;Impossible\\n&quot;;\n      return 0;\n    }\n  }\n  VI ans;\n  rep(i, 0, n) {\n    \/\/ scc\u6bd4\u8f03\u5c0f\u7684 \u51fa\u73fe\u5728\u62d3\u6a38\u6392\u5e8f\u7684\u6bd4\u8f03\u5f8c\u9762\n    if (scc[i * 2 + 1] &lt; scc[i * 2]) {\n      ans.pb(i + 1);  \/\/ \u9019\u8ddf\u984c\u76ee\u6709\u95dc \u984c\u76ee\u7684vertex\u662f1-indexed\n    }\n  }\n  cout &lt;&lt; ans.size() &lt;&lt; '\\n';\n  for (int x : ans) {\n    cout &lt;&lt; x &lt;&lt; ' ';\n  }\n  cout &lt;&lt; '\\n';\n  return 0;\n}\n<\/code><\/pre>\n<p>\u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b\n<a href=\"https:\/\/codeforces.com\/contest\/228\/submission\/305357778\">Submission<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[3],"tags":[50,10,9,44,8,38],"class_list":["post-197","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-2-sat","tag-codeforce","tag-ojcode","tag-scc","tag-8","tag-38"],"jetpack_publicize_connections":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>E. The Road to Berland is Paved With Good Intentions \u89e3\u6790(\u601d\u7dad\u30012-SAT\u3001SCC\u3001\u62d3\u6a38\u6392\u5e8f) - Logic Formatting<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blog.seekrealthing.net\/blog\/?p=197\" \/>\n<meta property=\"og:locale\" content=\"zh_TW\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"E. 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