Category Archives: Strongly Connected Components(SCC)

Light OJ : 1168 – Wishing Snake

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Problem Link : http://www.lightoj.com/volume_showproblem.php?problem=1168

Algorithm :

1. Make a graph from input data and Determine the SCC of the graph.
2. Make a new Graph from SCC. Consider Each SCC as a node of this graph.
3. If any Node of SCC_graph have more than 1 out degree and It is not possible to traverse all node from the SCC which contain’s the node 0 of the input graph then Print “NO”.
4. Otherwise Print “YES”; 


/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */

#include <bits/stdc++.h>

#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             1005
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sff(a,b)        scanf("%d%d",&a,&b)
#define sfff(a,b,c)     scanf("%d%d%d",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long

using namespace std;


const int fx[]={+1,-1,+0,+0};
const int fy[]={+0,+0,+1,-1};

vector<int>graph[MAX],rev_graph[MAX],scc_graph[MAX];
int visited[MAX];
int node[MAX],scc_num[MAX],node_num,scc;
stack<int>st;
bool test,scc_vis[MAX];

void allclear()
{
    loop(i,node_num+2)
    {
        visited[i]=0;
        graph[i].clear();
        rev_graph[i].clear();
        scc_num[i]=0;
    }
    loop(i,scc+2)
    {
        scc_graph[i].clear();
        scc_vis[i]=0;
    }
    node_num=0;
    scc=0;
}

void dfs1(int u)
{
    visited[u]++;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(visited[v]==0)
            dfs1(v);
    }
    st.push(u);
}

void dfs2(int u)
{
    visited[u]++;
    scc_num[u]=scc;
    loop(i,SZ(rev_graph[u]))
    {
        int v=rev_graph[u][i];
        if(visited[v]!=2)
            dfs2(v);
    }
}

void dfs3(int u)
{
    scc_vis[u]=1;
    if(SZ(scc_graph[u])>1)
        {test=0;return;}
    loop(i,SZ(scc_graph[u]))
    dfs3(scc_graph[u][i]);
}

int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);

    int t;
    sf(t);
    TEST_CASE(t)
    {
        ms(node,-1);
        int n,m,u,v;
        node_num=0;
        node[0]=node_num;
        sf(n);
        loop(i,n)
        {
            sf(m);
            loop(j,m)
            {
                sff(u,v);
                if(node[u]==-1) node[u]=++node_num;
                if(node[v]==-1) node[v]=++node_num;
                graph[node[u]].pb(node[v]);
                rev_graph[node[v]].pb(node[u]);
            }
        }

        loop(i,node_num)
        {
            if(visited[i]==0)
                dfs1(i);
        }

        scc=0;
        while(!st.empty())
        {
            u=st.top();
            st.pop();
            if(visited[u]!=2)
            {
                scc++;
                dfs2(u);
            }
        }

        loop(i,node_num)
        {
            loop(j,SZ(graph[i]))
            {
                u=graph[i][j];
                if(scc_num[i]!=scc_num[u])
                    scc_graph[scc_num[i]].pb(scc_num[u]);
            }
        }

        test=1;
        dfs3(scc_num[0]);
        if(test)
        {
            REP(i,1,scc+1)
            {
                if(!scc_vis[i])
                    test=0;
            }
        }
        PRINT_CASE;
        if(test)
            pf("YES\n");
        else
            pf("NO\n");
        allclear();

    }
    return 0;
}

Light OJ : 1417 – Forwarding Emails

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Problem Link : http://www.lightoj.com/volume_showproblem.php?problem=1417


/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */
 
#include <bits/stdc++.h>
 
#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             50005
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sff(a,b)        scanf("%d%d",&a,&b)
#define sfff(a,b,c)     scanf("%d%d%d",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long
 
using namespace std;
 
int n,dfsnum,dis[MAX];
vector<int> graph[MAX],rev_graph[MAX];
bool mp[MAX],mp1[MAX],mp2[MAX];
 
void allclear()
{
    loop(i,n+2)
    {
        graph[i].clear();
        rev_graph[i].clear();
        dis[i]=0;
        mp[i]=mp1[i]=mp2[i]=0;
    }
    dfsnum=0;
}
 
void dfs1(int u)
{
    mp[u]=1;
    dfsnum++;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(!mp[v])
            dfs1(v);
    }
    dis[u]=++dfsnum;
}
 
void dfs2(int u)
{
    mp1[u]=1;
    loop(i,SZ(rev_graph[u]))
    {
        int v=rev_graph[u][i];
        if(!mp1[v])
            dfs2(v);
    }
}
 
void dfs3(int u)
{
    mp2[u]=1;
    dfsnum++;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(!mp2[v])
            dfs3(v);
    }
}
 
void dfs4(int u)
{
    mp2[u]=0;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(mp2[v])
            dfs4(v);
    }
}
 
int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);
 
    int t;
    sf(t);
    TEST_CASE(t)
    {
        sf(n);
        allclear();
        int a,b;
        loop(i,n)
        {
            sff(a,b);
            graph[a].pb(b);
            rev_graph[b].pb(a);
        }
        REP(i,1,n+1)
        {
            if(!mp[i])
                dfs1(i);
        }
 
        vector<pii>top_sort;
        REP(i,1,n+1)
        {
            top_sort.pb(MP(dis[i],i));
        }
 
 
        sort(top_sort.rbegin(),top_sort.rend());
        vector<int>scc;
 
        loop(i,SZ(top_sort))
        {
            int u=top_sort[i].ss;
            if(!mp1[u])
            {
                scc.pb(u);
                dfs2(u);
            }
        }
 
        top_sort.clear();
        loop(i,SZ(scc))
        {
            int u=scc[i];
            if(!mp2[u])
            {
                dfsnum=0;
                dfs3(u);
                top_sort.pb(MP(dfsnum,u));
                dfs4(u); // this DFS is clearing only visited node.
            }
        }
 
        sort(top_sort.rbegin(),top_sort.rend());
        PRINT_CASE;
        if(SZ(top_sort)==1)
            pf("%d\n",top_sort[0].ss);
        else
        {
            loop(i,SZ(top_sort)-1)
            {
                if(top_sort[i].ff!=top_sort[i+1].ff) // This is for determine the smallest one.
                {
                    pf("%d\n",top_sort[i].ss);
                    break;
                }
            }
        }
    }
    return 0;
}

Light OJ 1210 – Efficient Traffic System

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Problem Link : http://www.lightoj.com/volume_showproblem.php?problem=1210


/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */

#include <bits/stdc++.h>

#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             20010
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sff(a,b)        scanf("%d%d",&a,&b)
#define sfff(a,b,c)     scanf("%d%d%d",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long

using namespace std;

int n,m;
vector<int>graph[MAX],reverse_graph[MAX];
int finish[MAX],start,scc,belong[MAX],indegree[MAX],outdegree[MAX];
bool visited[MAX];

bool test(pii a, pii b)
{
    return a.ff>b.ff;
}

void dfs1(int n)
{
    visited[n]=1;
    start++;
    loop(i,SZ(graph[n]))
    {
        int v=graph[n][i];
        if(!visited[v])
            dfs1(v);
    }
    finish[n]=++start;
}

void dfs2(int n)
{
    visited[n]=1;
    belong[n]=scc;
    loop(i,SZ(reverse_graph[n]))
    {
        int v=reverse_graph[n][i];
        if(visited[v]==0)
        {
            dfs2(v);
        }
    }
}

void dfs3(int u)
{
    visited[u]=1;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(belong[u]!=belong[v])
        {
            indegree[belong[v]]++;
            outdegree[belong[u]]++;
        }
        if(!visited[v])
            dfs3(v);
    }
}

void allclear()
{
    loop(i,n+2)
    {
        graph[i].clear();
        reverse_graph[i].clear();
        visited[i]=0;
        belong[i]=-1;
        indegree[i]=0;
        finish[i]=0;
        outdegree[i]=0;
    }
    start=0;
    scc=0;
}

int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);
    int t;
    sf(t);
    TEST_CASE(t)
    {
        sff(n,m);
        allclear();
        while(m--)
        {
            int u,v;
            sff(u,v);
            graph[u].pb(v);
            reverse_graph[v].pb(u);
        }

        vector<pii>v;
        for(int i=1; i<=n; i++)
            if(!visited[i])
                dfs1(i);
        for(int i=1; i<=n; i++)
        {
            v.pb(MP(finish[i],i));
        }
        sort(all(v),test);
        loop(i,n+2)
        visited[i]=0;

        loop(i,SZ(v))
        {
            int u=v[i].ss;
            if(!visited[u])
            {
                scc++;
                dfs2(u);
            }
        }

        loop(i,n+2)
        visited[i]=0;
        for(int i=1; i<=n; i++)
        {
            if(!visited[i])
                dfs3(i);
        }

        int in=0,out=0;
        for(int i=1; i<=scc && scc>1; i++)
        {
            if(!indegree[i])
                in++;
            if(!outdegree[i])
                out++;
        }
        PRINT_CASE;
        pf("%d\n",max(in,out));
    }
    return 0;
}

Light OJ 1034 – Hit the Light Switches

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Problem Link : http://www.lightoj.com/volume_showproblem.php?problem=1034


/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */

#include <bits/stdc++.h>

#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             10005
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sff(a,b)        scanf("%d%d",&a,&b)
#define sfff(a,b,c)     scanf("%d%d%d",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long

using namespace std;

int n,m;
vector<int>graph[MAX],topsort;
bool visited[MAX],test;

void allclear()
{
    loop(i,n+2)
    {
        graph[i].clear();
        visited[i]=0;
    }
    topsort.clear();
}

void dfs(int u)
{
    visited[u]=1;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(!visited[v])
            dfs(v);
    }
    if(test)
        topsort.pb(u);
}

int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);
    int t,a,b;
    sf(t);
    TEST_CASE(t)
    {
        sff(n,m);
        loop(i,m)
        {
            sff(a,b);
            graph[a].pb(b);
        }
        test=1;
        int ans=0;
        REP(i,1,n+1)
        {
            if(!visited[i])
                dfs(i);
        }

        test=0;
        loop(i,n+2) visited[i]=0;
        for(int i=SZ(topsort)-1; i>=0; i--)
        {
            int u=topsort[i];
            if(!visited[u])
            {
                dfs(u);
                ans++;
            }
        }
        PRINT_CASE;
        pf("%d\n",ans);
        allclear();
    }
    return 0;
}

Another solution of this problem using Strongly Connected components(SCC) is here…


/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */

#include <bits/stdc++.h>

#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             10005
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sff(a,b)        scanf("%d%d",&a,&b)
#define sfff(a,b,c)     scanf("%d%d%d",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long

using namespace std;

int n,m;
vector<int>graph[MAX],reverse_graph[MAX],scc;
vector<pii>v;
int indegree[MAX],topsort[MAX],finish;
bool visited[MAX];

void allclear()
{
    loop(i,n+2)
    {
        graph[i].clear();
        reverse_graph[i].clear();
        indegree[i]=0;
        visited[i]=0;
        topsort[i];
    }
    scc.clear();
    v.clear();
    finish=0;
}

void dfs1(int u)
{
    finish++;
    visited[u]=1;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(!visited[v])
            dfs1(v);
    }
    topsort[u]=++finish;
}

void dfs2(int u)
{
    visited[u]=1;
    loop(i,SZ(reverse_graph[u]))
    {
        int v=reverse_graph[u][i];
        if(!visited[v])
            dfs2(v);
    }
}

void dfs3(int u)
{
    visited[u]=1;
    loop(i,SZ(graph[u]))
    {
        int v=graph[u][i];
        if(!visited[v])
            dfs3(v);
    }
}
int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);
    int t,a,b;
    sf(t);
    TEST_CASE(t)
    {
        sff(n,m);
        loop(i,m)
        {
            sff(a,b);
            graph[a].pb(b);
            reverse_graph[b].pb(a);
            indegree[b]++;
        }

        int ans=0;
        REP(i,1,n+1)
        {
            if(!visited[i])
                dfs1(i);
        }

        REP(i,1,n+1)
        {
            v.pb(MP(topsort[i],i));
        }
        sort(all(v));
        loop(i,n+2)
        visited[i]=0;

        for(int i=SZ(v)-1; i>=0; i--)
        {
            int u=v[i].ss;
            if(!visited[u])
            {
                scc.pb(u);
                dfs2(u);
            }
        }

        loop(i,n+2)
        visited[i]=0;

        loop(i,SZ(scc))
        {
            if(!visited[scc[i]])
            {
                ans++;
                dfs3(scc[i]);
            }
        }
        PRINT_CASE;
        pf("%d\n",ans);
        allclear();
    }
    return 0;
}