强连通分量 ZQUOJ 21467&&POJ 2553 The Bottom of a Graph
Description
We will use the following (standard) definitions from graph theory. Let
V
be a nonempty and finite set, its elements being called vertices (or nodes). Let
E
be a subset of the Cartesian product
V x V
, its elements being called edges. Then
G=(V,E)
is called a directed graph.
Let n be a positive integer, and let p=(e1 , ... , en ) be a sequence of length n of edges ei E such that ei = (vi ,vi+1 ) for a sequence of vertices ( v1 ,... , vn+1 ) . Then p is called a path from vertex v1 to vertex vn+1 in G and we say that vn+1 is reachable from v1 , writing ( v1 vn+1 ) .
Here are some new definitions. A node v in a graph G=( V, E ) is called a sink, if for every node w in G that is reachable from v , v is also reachable from w . The bottom of a graph is the subset of all nodes that are sinks, i.e., bottom(G) = { vV | w∈V: (vw) (wv) } . You have to calculate the bottom of certain graphs.
Let n be a positive integer, and let p=(e1 , ... , en ) be a sequence of length n of edges ei E such that ei = (vi ,vi+1 ) for a sequence of vertices ( v1 ,... , vn+1 ) . Then p is called a path from vertex v1 to vertex vn+1 in G and we say that vn+1 is reachable from v1 , writing ( v1 vn+1 ) .
Here are some new definitions. A node v in a graph G=( V, E ) is called a sink, if for every node w in G that is reachable from v , v is also reachable from w . The bottom of a graph is the subset of all nodes that are sinks, i.e., bottom(G) = { vV | w∈V: (vw) (wv) } . You have to calculate the bottom of certain graphs.
Input
The input contains several test cases, each of which corresponds to a directed graph
G
. Each test case starts with an integer number
v
, denoting the number of vertices of
G=( V, E )
, where the vertices will be identified by the integer numbers in the set
V={ 1 , ... , v }
. You may assume that
1 <= v <= 5000
. That is followed by a non-negative integer
e
and, thereafter,
e
pairs of vertex identifiers
v1 , w1 , ... , ve , we
with the meaning that
(vi , wi ) E
. There are no edges other than specified by these pairs. The last test case is followed by a zero.
Output
For each test case output the bottom of the specified graph on a single line. To this end, print the numbers of all nodes that are sinks in sorted order separated by a single space character. If the bottom is empty, print an empty line.
Sample Input
3 3 1 3 2 3 3 1 2 1 1 2 0
Sample Output
1 3
2
题目大意，给定一个有向图，按顺序输出“自己可达的顶点都可到达自己”的顶点。
解题思路：先求出图的强连通分量，缩点，求出度为0的强连通分量中的顶点，并按按顺序输出。
有关强强连通分量的算法请参考：http://www.byvoid.com/blog/scc-tarjan/（Tarjan算法）
http://www.cnblogs.com/luweiseu/archive/2012/07/14/2591370.html（Tarjan，Kosaraju，Garbow）
http://blog.csdn.net/liguanxing/article/details/5665520（Kosaraju算法）
http://hi.baidu.com/yy17yy/item/8f4142dee291e6ee3cc2cb2f（Garbow算法）
AC代码（Tarjan）：
View Code
1 #include<stdio.h> 2 #include<string.h> 3 typedef struct 4 { 5 int v,next; 6 }Node; 7 Node e[50000]; 8 int n,m,index,top,count; 9 int first[5001],DFN[5001],Low[5001],ver[5001]; 10 int belong[5001],stack[5001],instack[5001],val[5001],degree[5001]; 11 void Tarjan(int i) 12 { 13 int k,j,v; 14 DFN[i]=Low[i]=index++; 15 instack[i]=true; 16 stack[top++]=i; 17 for(j=first[i];j!=-1;j=e[j].next) 18 { 19 k=e[j].v; 20 if(!DFN[k]) 21 { 22 Tarjan(k); 23 if(Low[k]<Low[i]) 24 Low[i]=Low[k]; 25 } 26 else if(instack[k]&&DFN[k]<Low[i]) 27 Low[i]=DFN[k]; 28 } 29 if(Low[i]==DFN[i]) 30 { 31 count++; 32 do{ 33 v=stack[--top]; 34 instack[v]=false; 35 belong[v]=count; 36 val[count]++; 37 }while(i!=v); 38 } 39 } 40 int main() 41 { 42 int g,i,j,u,v; 43 while(scanf("%d",&n)&&n) 44 { 45 scanf("%d",&m); 46 g=0; index=0; top=0; count=0; 47 memset(first,-1,sizeof(first)); 48 memset(DFN,0,sizeof(DFN)); 49 memset(Low,0,sizeof(Low)); 50 memset(degree,0,sizeof(degree)); 51 for(i=0;i<m;i++) 52 { 53 scanf("%d%d",&u,&v); 54 e[g].v=v; 55 e[g].next=first[u]; 56 first[u]=g++; 57 } 58 for(i=1;i<=n;i++) 59 if(!DFN[i]) 60 Tarjan(i); 61 for(i=1;i<=n;i++) 62 for(j=first[i];j!=-1;j=e[j].next) 63 { 64 v=e[j].v; 65 if(belong[i]!=belong[v]) 66 degree[belong[i]]++; 67 } 68 for(i=1;i<=n;i++) 69 { 70 if(!degree[belong[i]]) 71 printf("%d ",i); 72 } 73 printf("\n"); 74 } 75 return 0; 76 }
AC代码（Kosaraju）：
View Code
1 #include<stdio.h> 2 #include<string.h> 3 typedef struct 4 { 5 int u,v,next1,next2; 6 }Node; 7 Node e[50000]; 8 int n,m,index; 9 int first1[5001],first2[5001],vis[5001],belong[5001],num[5001],degree[5001]; 10 void dfs1(int i) 11 { 12 int j,k; 13 vis[i]=1; 14 for(j=first1[i];j!=-1;j=e[j].next1) 15 { 16 k=e[j].v; 17 if(!vis[k]) 18 dfs1(k); 19 } 20 num[index++]=i; 21 } 22 void dfs2(int i) 23 { 24 int j,k; 25 vis[i]=1; 26 belong[i]=index; 27 for(j=first2[i];j!=-1;j=e[j].next2) 28 { 29 k=e[j].u; 30 if(!vis[k]) 31 dfs2(k); 32 } 33 } 34 void kosaraju() 35 { 36 int i; 37 index=1; 38 memset(vis,0,sizeof(vis)); 39 for(i=1;i<=n;i++) 40 if(!vis[i]) 41 dfs1(i); 42 index=1; 43 memset(vis,0,sizeof(vis)); 44 for(i=n;i>=1;i--) 45 if(!vis[num[i]]) 46 { 47 dfs2(num[i]); 48 index++; 49 } 50 } 51 int main() 52 { 53 int g,i,j,u,v; 54 while(scanf("%d",&n)&&n) 55 { 56 g=0; 57 memset(first1,-1,sizeof(first1)); 58 memset(first2,-1,sizeof(first2)); 59 memset(degree,0,sizeof(degree)); 60 scanf("%d",&m); 61 for(i=1;i<=m;i++) 62 { 63 scanf("%d%d",&u,&v); 64 e[g].v=v; 65 e[g].next1=first1[u]; 66 first1[u]=g; 67 e[g].u=u; 68 e[g].next2=first2[v]; 69 first2[v]=g++; 70 } 71 kosaraju(); 72 for(i=1;i<=n;i++) 73 for(j=first1[i];j!=-1;j=e[j].next1) 74 { 75 v=e[j].v; 76 if(belong[i]!=belong[v]) 77 degree[belong[i]]++; 78 } 79 for(i=1;i<=n;i++) 80 if(!degree[belong[i]]) 81 printf("%d ",i); 82 printf("\n"); 83 } 84 return 0; 85 }
AC代码（Garbow)：
View Code
1 #include<stdio.h> 2 #include<string.h> 3 typedef struct 4 { 5 int v,next; 6 }Node; 7 Node e[50000]; 8 int n,m,index,top1,top2,count; 9 int first[5001],low[5001],stack1[5001],stack2[5001]; 10 int belong[5001],degree[5001]; 11 void Garbow(int i) 12 { 13 int j,k; 14 stack1[++top1]=stack2[++top2]=i; 15 low[i]=index++; 16 for(j=first[i];j!=-1;j=e[j].next) 17 { 18 k=e[j].v; 19 if(!low[k]) 20 Garbow(k); 21 else if(!belong[k]) 22 { 23 while(low[stack2[top2]]>low[k]) 24 top2--; 25 } 26 } 27 if(stack2[top2]==i) 28 { 29 top2--; 30 count++; 31 do{ 32 k=stack1[top1--]; 33 belong[k]=count; 34 }while(k!=i); 35 } 36 } 37 int main() 38 { 39 int g,i,j,u,v; 40 while(scanf("%d",&n)&&n) 41 { 42 scanf("%d",&m); 43 g=0; top1=top2=-1; index=1; count=0; 44 memset(first,-1,sizeof(first)); 45 memset(low,0,sizeof(low)); 46 memset(belong,0,sizeof(belong)); 47 memset(degree,0,sizeof(degree)); 48 for(i=1;i<=m;i++) 49 { 50 scanf("%d%d",&u,&v); 51 e[g].v=v; 52 e[g].next=first[u]; 53 first[u]=g++; 54 } 55 for(i=1;i<=n;i++) 56 if(!low[i]) 57 Garbow(i); 58 for(i=1;i<=n;i++) 59 for(j=first[i];j!=-1;j=e[j].next) 60 { 61 v=e[j].v; 62 if(belong[i]!=belong[v]) 63 degree[belong[i]]++; 64 } 65 for(i=1;i<=n;i++) 66 if(!degree[belong[i]]) 67 printf("%d ",i); 68 printf("\n"); 69 } 70 return 0; 71 }