{"id":492,"date":"2010-10-25T05:20:39","date_gmt":"2010-10-24T21:20:39","guid":{"rendered":"http:\/\/comzyh.tk\/blog\/?p=492"},"modified":"2015-10-24T16:44:06","modified_gmt":"2015-10-24T08:44:06","slug":"%e7%94%a8%e4%ba%8e%e6%b1%82%e6%9c%80%e8%bf%91%e5%85%ac%e5%85%b1%e7%a5%96%e5%85%88lca%e7%9a%84-tarjan%e7%ae%97%e6%b3%95-%e4%bb%a5poj1986%e4%b8%ba%e4%be%8b","status":"publish","type":"post","link":"https:\/\/comzyh.com\/blog\/archives\/492\/","title":{"rendered":"\u7528\u4e8e\u6c42\u6700\u8fd1\u516c\u5171\u7956\u5148(LCA)\u7684 Tarjan\u7b97\u6cd5&#8211;\u4ee5POJ1986\u4e3a\u4f8b"},"content":{"rendered":"<p><a rel=\"attachment wp-att-490\" href=\":\/\/comzyh.com\/blog\/archives\/492\/lca_tarjan\">LCA_Tarjan<\/a>\u6700\u8fd1\u516c\u5171\u7956\u5148\u95ee\u9898LCA(Least Common Ancestors)\u95ee\u9898\u662f\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898<\/p>\n<p>\u7ed9\u5b9a\u6709\u5411\u65e0\u73af\u56fe(\u5c31\u662f\u6811,\u4e0d\u4e00\u5b9a\u6709\u6ca1\u6709\u6839),\u7ed9\u5b9a\u70b9U,V,\u627e\u51fa\u70b9R,\u4fdd\u8bc1\u70b9R\u662fU,V\u7684\u516c\u5171\u7956\u5148,\u4e14\u6df1\u5ea6\u6700\u6df1;\u6216\u8005\u7406\u89e3\u4e3aR\u79bb\u8fd9\u4e24\u4e2a\u70b9\u7684\u8ddd\u79bb\u4e4b\u548c\u6700\u5c0f.\u5982\u4f55\u627e\u51faR\u5462\uff1f<\/p>\n<p>\u6700\u4e00\u822c\u7684\u7b97\u6cd5\u662fDFS(DFS\u672c\u662f\u6df1\u5ea6\u4f18\u5148\u641c\u7d22,\u5728\u8fd9\u91cc\u59d1\u4e14\u628a\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e5f\u53eb\u505aDFS,\u5176\u5b9e\u662f\u4e00\u79cd\u4e0d\u4e25\u8c28\u7684\u8bf4\u6cd5).\u5148\u770b\u4e00\u9053\u8d64\u88f8\u88f8\u7684LCA:<a href=\"http:\/\/poj.org\/problem?id=1330\" target=\"_blank\">POJ 1330 Nearest Common Ancestors<\/a> \u8fd9\u9053\u9898\u7ed9\u51fa\u4e86\u6839\u8282\u70b9,\u8fd8\u4fdd\u8bc1&#8221;the first integer is the parent node of the second integer&#8221;(\u8f93\u5165\u7b2c\u4e00\u4e2a\u6570\u662f\u7b2c\u4e8c\u4e2a\u6570\u7684\u7956\u5148),\u8fd9\u662f\u8d64\u88f8\u88f8\u7684LCA,\u7b97\u6cd5\u5f88\u7b80\u5355,\u4ece\u6839\u8282\u70b9DFS\u4e00\u904d,\u6309DFS\u5c42\u6570k\u7ed9\u6bcf\u4e2a\u8282\u70b9\u6807\u4e0a\u6df1\u5ea6deep[i]=k.\u7136\u540e\u4eceU\u70b9DFS\u5230V\u70b9,\u627e\u5230\u540e\u56de\u6eaf,\u5728\u56de\u6eaf\u7684\u8def\u5f84\u4e0a\u627e\u5230\u4e00\u4e2adeep[i]\u6700\u5c0f\u7684\u8282\u70b9\u5373\u4e3aLCA.<\/p>\n<p>\u5f3a\u5927\u7684LCA Tarjan\u7b97\u6cd5\u80fd\u5728\u4e00\u904d\u904d\u5386\u540e\u5e94\u7b54\u5168\u90e8\u7684LCA\u67e5\u8be2,\u65f6\u95f4\u590d\u6742\u7684\u7ea6\u4e3a\\(\\Theta (N)\\)<\/p>\n<p>\u6709\u4eba\u8bf4POJ1330\u662f\u4e00\u9053LCA Tarjan,\u5728\u6211\u770b\u6765\u5b8c\u5168\u4e0d\u662f,LCA Tarjan\u7b97\u6cd5\u7684\u7528\u9014\u662f\u5904\u7406\u5927\u91cf\u8bf7\u6c42,\u5982\u679c\u53ea\u6709\u51e0\u4e2a(POJ1330\u6bcf\u4e2aCase\u53ea\u6709\u4e00\u4e2a)\u8be2\u95ee\u5927\u53ef\u4e0d\u5fc5\u5199Tarjan\u7b97\u6cd5,\u4e0d\u8fc7,1986\u7684\u7f16\u7a0b\u96be\u5ea6\u9ad8,\u5982\u679c\u53ea\u662f\u60f3\u5148\u5b66LCA Tarjan, \u75281330\u9a8c\u8bc1\u6b63\u786e\u6027\u4e5f\u4e0d\u662f\u4e0d\u53ef\u4ee5.<\/p>\n<h3>LCA Tarjan\u7b97\u6cd5<\/h3>\n<p>\u518d\u6765\u770b\u4e00\u9053\u9898:<a href=\"http:\/\/poj.org\/problem?id=1986\" target=\"_blank\">POJ1986 Distance Queries<\/a> \u8fd9\u9053\u9898\u624d\u662f\u771f\u6b63\u7684LCA Tarjan,\u53ea\u7ed9\u4e00\u4e2a\u6709\u5411\u65e0\u73af\u56fe,\u6709\u6d77\u91cf\u8be2\u95ee;(\u6ce8\u610f,\u8f93\u5165\u683c\u5f0f\u4e0e<a href=\"http:\/\/poj.org\/problem?id=1984\" target=\"_blank\">POJ 1984 Navigation Nightmare<\/a> \u4e00\u6837,\u9700\u8981\u53c2\u80031984\u7684\u8f93\u5165\u683c\u5f0f)<\/p>\n<p>\u8f93\u5165\u683c\u5f0f\u5927\u610f:<\/p>\n<ul>\n<li>\u7b2c1\u884c\uff1a\u8282\u70b9\u6570N,\u8fb9\u6570M<\/li>\n<li>\u7b2c2&#8230;M+1\u884c\uff1a\u8d77\u59cb\u8282\u70b9,\u76ee\u6807\u8282\u70b9,\u8def\u5f84\u957f\u5ea6,\u65b9\u5411(\u65e0\u610f\u4e49\u5b57\u7b26,\u672c\u9898\u76f4\u63a5\u5ffd\u7565)<\/li>\n<li>\u7b2cM+2\u884c:\u8be2\u95ee\u4e2a\u6570K(1 &lt;= K &lt;= 10,000)<\/li>\n<li>\u7b2cN+3&#8230;2+M+K\u884c:\u67e5\u8be2 U,V<\/li>\n<\/ul>\n<p>\u8fd9\u9053\u9898\u7528DFS\u505a\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(\\Theta (K \\times N) \\) \u663e\u7136\u5f88\u4e0d\u7406\u60f3,\u8fd9\u4e2a\u65f6\u5019\u4f1f\u5927\u7684Tarjan\u6765\u4e86,\u95ee\u9898\u8fce\u5203\u800c\u89e3.<\/p>\n<p>\u9996\u5148,LCA Tarjan \u662f\u4e00\u79cd\u79bb\u7ebf\u7b97\u6cd5,\u8981\u6c42\u4e00\u6b21\u8bfb\u5165\u6240\u6709\u8be2\u95ee,\u4e00\u6b21\u6027\u8f93\u51fa,\u8fd9\u6b63\u662fLCA Tarjan \u7b97\u6cd5\u7684\u7cbe\u9ad3<\/p>\n<p>\u4ee5\u4e0b\u5927\u91cf\u5f15\u7528Sideman\u795e\u725b\u7684\u8bdd\uff1a<\/p>\n<p><strong>LCA Tarjan\u57fa\u672c\u6846\u67b6\uff1a<\/strong><\/p>\n<ul>\n<li>\u5148\u7528\u968f\u4fbf\u4e00\u79cd\u6570\u636e\u7ed3\u6784(\u94fe\u8868\u5c31\u884c),\u628a\u5173\u4e8e\u67d0\u4e2a\u70b9\u7684\u6240\u6709\u8be2\u95ee\u6807\u5728\u8282\u70b9\u4e0a,\u4fdd\u8bc1\u904d\u5386\u5230\u4e00\u4e2a\u70b9,\u80fd\u5f97\u5230\u6240\u6709\u6709\u5173\u8fd9\u4e2a\u8282\u70b9LCA \u67e5\u8be2<\/li>\n<li>\u5efa\u7acb\u5e76\u67e5\u96c6.<strong>\u6ce8\u610f<\/strong>:\u8fd9\u4e2a\u5e76\u67e5\u96c6\u53ea\u53ef\u4ee5\u628a\u53f6\u5b50\u8282\u70b9\u5e76\u5230\u6839\u8282\u70b9,\u5373getf(x)\u5f97\u5230\u7684\u603b\u662fx\u7684\u7956\u5148<\/li>\n<li>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6574\u68f5\u6811,\u7528\u4e00\u4e2aVisited\u6570\u7ec4\u6807\u8bb0\u904d\u5386\u8fc7\u7684\u8282\u70b9,\u6bcf\u904d\u5386\u5230\u4e00\u4e2a\u8282\u70b9\u5c06Visite[i]\u8bbe\u6210True \u5904\u7406\u5173\u4e8e\u8fd9\u4e2a\u8282\u70b9(\u4e0d\u59a8\u8bbe\u4e3aA)\u7684\u8be2\u95ee,\u82e5\u53e6\u4e00\u8282\u70b9(\u8bbe\u4e3aB)\u7684Visited[B]==True,\u5219\u56de\u5e94\u8fd9\u4e2a\u8be2\u95ee,\u8fd9\u4e2a\u8be2\u95ee\u7684\u7ed3\u679c\u5c31\u662f<strong>getf(B)<\/strong>. \u5426\u5219\u4ec0\u4e48\u90fd\u4e0d\u505a<\/li>\n<li>\u5f53A\u6240\u6709\u5b50\u6811\u90fd\u5df2\u7ecf\u904d\u5386\u8fc7\u4e4b\u540e,\u5c06\u8fd9\u4e2a\u8282\u70b9\u7528\u5e76\u67e5\u96c6\u5e76\u5230\u4ed6\u7684\u7236\u8282\u70b9(\u5176\u5b9e\u8fd9\u4e00\u6b65\u5e94\u8be5\u8bf4\u5f53\u53f6\u5b50\u8282\u70b9\u56de\u6eaf\u56de\u6765\u4e4b\u540e\u5c06\u53f6\u5b50\u8282\u70b9\u5e76\u5230\u81ea\u5df1,\u5e76DFS\u53e6\u4e00\u5b50\u6811)<\/li>\n<li>\u5f53\u4e00\u9897\u5b50\u6811\u904d\u5386\u5b8c\u65f6,\u8fd9\u68f5\u5b50\u6811\u7684\u5185\u90e8\u67e5\u8be2(\u5373LCA\u5728\u8fd9\u68f5\u5b50\u6811\u5185\u90e8)\u90fd\u5df2\u7ecf\u5904\u7406\u4e86<\/li>\n<\/ul>\n<p><!--more--><br \/>\n<figure id=\"attachment_490\" aria-describedby=\"caption-attachment-490\" style=\"width: 230px\" class=\"wp-caption alignright\"><br \/>\n<object classid=\"clsid:d27cdb6e-ae6d-11cf-96b8-444553540000\" width=\"230\" height=\"300\" codebase=\"http:\/\/download.macromedia.com\/pub\/shockwave\/cabs\/flash\/swflash.cab#version=6,0,40,0\"><param name=\"quality\" value=\"high\" \/><param name=\"src\" value=\"\/\/comzyh.com\/upload\/wordpress\/2010\/10\/LCA_Tarjan.swf\" \/><\/object><br \/>\n<figcaption id=\"caption-attachment-490\" class=\"wp-caption-text\">LCA Tarjan \u52a8\u753b\u6f14\u793a<\/figcaption><\/figure><\/p>\n<p>LCA Tarjan \u7b97\u6cd5\u6f14\u793a<\/p>\n<p>\u5047\u8bbe\u6211\u4eec\u8981\u67e5\u8be2<\/p>\n<p>(3,4) (3,5) (5,6) (6,7) (1,8)<\/p>\n<p>\u65b9\u6cd5\u89c1\u53f3<\/p>\n<p>\u4ee5(3,4)\u4e3a\u4f8b,\u8bf4\u4e0bTarjan\u662f\u5982\u4f55\u5de5\u4f5c\u7684:<\/p>\n<p>\u5f53DFS\u52303\u65f6,\u53d1\u73b0\u67e5\u8be2(3,4),\u67e5\u770b4\u662f\u5426\u88abDFS\u8fc7,\u663e\u7136\u8fd9\u662f\u4e0d\u53ef\u80fd\u7684.<\/p>\n<p>\u56de\u6eaf\u52302,\u5c063\u5e76\u51652.<\/p>\n<p>DFS\u8282\u70b94,\u53d1\u73b0\u67e5\u8be2(3,4),\u67e5\u770bvisited[3],\u53d1\u73b0\u88ab\u8bbf\u95ee\u8fc7,\u5e94\u7b54\u67e5\u8be2(3,4),\u5e94\u7b54getf(3)=2;<\/p>\n<p>LCA Tarjan \u7b97\u6cd5\u904d\u5386\u6bcf\u4e2a\u70b9\u4e00\u904d,\u5904\u7406\u6240\u6709\u8be2\u95ee,\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(\\Theta(N+2M) \\)<\/p>\n<p>\u4e0b\u9762\u8d34\u51faPOJ1986\u7684\u9898\u89e3<\/p>\n<p>\u9996\u5148LCA Tarjan \u6ca1\u7684\u8bf4,\u4f46\u662f\u9898\u76ee\u8981\u6c42\u56de\u5e94\u7684\u4e0d\u662fLCA,\u800c\u662f\u4e24\u8282\u70b9\u95f4\u8ddd\u79bb,\u53ef\u4ee5\u8fd9\u6837\u505a<\/p>\n<ul>\n<li>\u6539\u9020\u5e76\u67e5\u96c6,\u5b9a\u4e49dis[i]\u6570\u7ec4,\u4fdd\u5b58i\u5230getf(i)\u7684\u8ddd\u79bb<\/li>\n<li>\u5b9a\u4e49Deep[i]\u6570\u7ec4,\u8868\u793ai\u8282\u70b9\u7684\u6df1\u5ea6,DFS\u65f6\u987a\u4fbf\u66f4\u65b0depp[i];<\/li>\n<li>\u5b9a\u4e49Sum[I]\u6570\u7ec4,\u8868\u793a\u4ece\u6839\u8282\u70b9\u5230I\u6df1\u5ea6\u8282\u70b9\u7684\u8ddd\u79bb.\u56e0\u4e3a\u5728LCA Tarjan\u7b97\u6cd5\u4e2d ,LCA(\u8bbe\u4e3aX) \u5fc5\u7136\u5728DFS\u8def\u5f84\u4e0a,\u6240\u4ee5X\u5230I\u7684\u8ddd\u79bb\u4e3asum[deep[I]]-sum[Deep[X]]<\/li>\n<li>\u54cd\u5e94\u65f6,\u8fd4\u56de\u503c\u4e3a\uff1adis[A]+sum[deep[getf(A)]]-sum[Deep[B]];<\/li>\n<\/ul>\n<p>\u8d34\u4e4b\u4ee3\u7801:<\/p>\n<pre lang=\"Cpp\" file=\"POJ1986.cpp\">\r\n#include <iostream>\r\n#include <cstring>\r\n#include <cstdio>\r\nusing namespace std;\r\nstruct link{\/\/\u94fe\u8868\u7ed3\u6784\u4f53;\u76ee\u6807,\u957f\u5ea6 ,\u4e0b\u4e00\u4e2a \r\n       int t,l,next;\r\n       };\r\nstruct node_request{\/\/node request\r\n       int ind,t;\r\n       int next;\r\n       };\r\n\/\/Map\r\nint lhead[40005];\/\/\u90bb\u63a5\u8868\u5934\u6307\u9488        \r\nlink lin[100000];\/\/\u94fe\u8868(\u65e0\u5411\u56fe)\r\n\/\/Tarjan\r\nint rhead[40005];\r\nnode_request nqs[100000];\/\/\u94fe\u8868(\u8bf7\u6c42) \r\nint sum[40005],level[40005];\/\/\u7c7b\u4f3c\u6808\u7684\u90e8\u5206\u548c(\u88abDFS\u66f4\u65b0),\u6df1\u5ea6 \r\nint f[40005];\/\/\u5e76\u67e5\u96c6 \r\nint dis[40005];\/\/Distance about Root \r\nint ans[40005];\r\nint visited[40005];\r\n\/\/Other\r\nint N,M,K;\r\n\/\/Sub\r\nint getf(int x);\r\ninline void add(int x,int y);\/\/Add X to Y\r\nvoid DFS(int x,int k);\/\/\u8282\u70b9,\u5c42\u6570 \r\nint main(){\r\n    int i,j; \r\n    int fr,t,l;\r\n    int R=0;\/\/\u94fe\u8868\u5c3e\u6307\u9488 \r\n    char rubbish;\r\n    scanf (\"%d%d\",&N,&M);\r\n    lin[0].next=0;\r\n    memset(lhead,0,sizeof(lhead));\r\n    for (i=1;i<=M;i++) {\/\/\u8bfb\u5165 \r\n        scanf (\"%d%d%d %c\",&#038;fr,&#038;t,&#038;l,&#038;rubbish);\r\n        R++;\r\n        lin[R].next=lhead[fr];\r\n        lin[R].t=t;\r\n        lin[R].l=l;\r\n        lhead[fr]=R;\r\n        R++;\r\n        lin[R].next=lhead[t];\r\n        lin[R].t=fr;\r\n        lin[R].l=l;\r\n        lhead[t]=R;\r\n        }\r\n    \/\/Finish Build Map ...\r\n    \/\/Begin Read Request ...\r\n    scanf(\"%d\",&#038;K);\r\n    R=0;\r\n    memset(rhead,0,sizeof(rhead));\r\n    for (i=1;i<=K;i++){\r\n        scanf(\"%d%d\",&#038;fr,&#038;t);\r\n        if (fr==t){\r\n           ans[i]=0;\r\n           continue;\r\n           }\r\n        R++;\r\n        nqs[R].ind=i;\r\n        nqs[R].t=t;\r\n        nqs[R].next=rhead[fr];\r\n        rhead[fr]=R;\r\n        R++;\r\n        nqs[R].ind=i;\r\n        nqs[R].t=fr;\r\n        nqs[R].next=rhead[t];\r\n        rhead[t]=R;\r\n        }\r\n    \/\/End reading...\r\n    for (i=1;i<=N;i++){\r\n        f[i]=i;\r\n        }\r\n    memset(dis,0,sizeof(dis));\r\n    memset(visited,0,sizeof(visited));\r\n    \/\/Begin\r\n    DFS(1,1);\r\n    for (i=1;i<=K;i++)\/\/\u6309\u987a\u5e8f\u8f93\u51fa\u7ed3\u679c \r\n        printf(\"%d\\n\",ans[i]); \r\n    system(\"pause\");\r\n    }\r\nint getf(int x){\r\n    if (f[x]==x)return x;\r\n    int ret=getf(f[x]);\/\/\u6700\u7ec8\u7956\u5148 \r\n    dis[x]+=dis[f[x]];\/\/x\u5230\u7956\u5148\u7684\u8ddd\u79bb=x\u5230f[x]\u7684\u8ddd\u79bb+f[x]\u5230\u7956\u5148\u7684\u8ddd\u79bb \r\n    return f[x]=ret;\r\n    }\r\ninline void add(int x,int y){\r\n       \/\/\u8fd9\u91cc\u7684\u5e76\u67e5\u96c6\u5e76\u4e0d\u9700\u8981getf \r\n       if (getf(x)==getf(y))return ;\r\n       f[x]=y;\r\n       }\r\nvoid DFS(int x,int k){\/\/Tarjan \u7b97\u6cd5\u4e3b\u51fd\u6570 \r\n    level[x]=k;\r\n    int i;\r\n    \/\/\u5904\u7406\u8bf7\u6c42 \r\n    i=rhead[x];\/\/\u4ece\u94fe\u8868\u91cc\u8bfb\u8bf7\u6c42 \r\n    while (i!=0){\r\n          if (visited[nqs[i].t]){\r\n             \/\/printf(\"Get visited node %4d \\n\",nqs[i].t);\r\n             getf(nqs[i].t);\/\/\u505a\u8def\u5f84\u538b\u7f29\uff1b \u7136\u540e\u5c31\u53ef\u4ee5\u5f00\u5fc3\u7684\u4f7ff[x]\u800c\u4e0d\u662fgetf(x)\u4e86 \r\n             \/\/printf(\"Request to %d ,target is %4d away from its Root(%4d )\\n\",nqs[i].t,dis[nqs[i].t],f[nqs[i].t]);\r\n             ans[nqs[i].ind]=dis[nqs[i].t]+sum[k]-sum[level[getf(nqs[i].t)]];\r\n             }\r\n          i=nqs[i].next;\r\n          }\r\n    visited[x]=1;\r\n    \/\/\u6269\u5c55\r\n    i=lhead[x];\r\n    while (i!=0){\r\n          if (!visited[lin[i].t]){\r\n             sum[k+1]=sum[k]+lin[i].l;\r\n             DFS(lin[i].t,k+1);\/\/DFS\r\n             f[lin[i].t]=x;\/\/\u5f52\u5165\u5e76\u67e5\u96c6 \r\n             dis[lin[i].t]=lin[i].l;\r\n             }\r\n          i=lin[i].next;\r\n          }\r\n    }\r\n\r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>LCA_Tarjan\u6700\u8fd1\u516c\u5171\u7956\u5148\u95ee\u9898LCA(Least Common Ancestors)\u95ee\u9898\u662f\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898  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