【模板整合计划】图论—图的连通性

【模板整合计划】图论—图的连通性


一:【欧拉路问题】

1.【欧拉回路】

(1).【逐步插入回路法 (Hierholzer)】

欧拉回路 \(\text{[Loj10105]}\)

#include<algorithm>
#include<cstring>
#include<cstdio>
#define Re register int
const int N=1e5+3,M=2e5+3;
int n,m,x,y,m_,op;
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
struct Euler_Circle__Undirected_Graph{//无向图欧拉回路
    int o,du[N],cur[N],ans[M],pan[M<<1],head[N];
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void dfs(Re x){
        for(Re i=head[x];i;i=head[x]){
            head[x]=a[i].next;//当前弧优化
            if(!pan[i]){
                pan[i]=pan[i^1]=1;
                dfs(a[i].to);
                ans[++ans[0]]=(i&1)?-i/2:i/2;//正数表示 x->y,负数表示y->x 
            }
        }
    }
    inline void sakura(){
        in(n),in(m),m_=m,o=1;
        while(m_--)in(x),in(y),add(x,y),add(y,x),++du[x],++du[y];
        for(Re i=1;i<=n;cur[i]=head[i],++i)if(du[i]&1){puts("NO");return;}//无向图点度数必须为偶数
        for(Re i=1;i<=n;++i)if(du[i]){dfs(i);break;}//任意找一个有度数的起点
        if(ans[0]!=m){puts("NO");return;}//跑不满m条边
        puts("YES");
        for(Re i=ans[0];i>=1;--i)printf("%d ",ans[i]);
    }
}T1;
struct Euler_Circle__Directed_Graph{//有向图欧拉回路
    int o,ru[N],chu[N],cur[N],ans[M],head[N];
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void dfs(Re x){
        for(Re i=head[x];i;i=head[x]){
            head[x]=a[i].next;//当前弧优化
            dfs(a[i].to);
            ans[++ans[0]]=i;
        }
    }
    inline void sakura(){
        in(n),in(m),m_=m;
        while(m_--)in(x),in(y),add(x,y),++ru[y],++chu[x];
        for(Re i=1;i<=n;cur[i]=head[i],++i)if(ru[i]!=chu[i]){puts("NO");return;}//有向图点入度必须等于出度
        for(Re i=1;i<=n;++i)if(chu[i]){dfs(i);break;}//任意找一个有出度的起点
        if(ans[0]!=m){puts("NO");return;}//跑不满m条边
        puts("YES");
        for(Re i=ans[0];i>=1;--i)printf("%d ",ans[i]);
    }
}T2;
int main(){
    in(op);
    if(op==1)T1.sakura();//无向图
    if(op==2)T2.sakura();//有向图
}

2.【欧拉路】


二:【无向图】

1.【割】

(1).【割点】

【模板】割点(割顶)\(\text{[P3388]}\)

#include<algorithm>
#include<cstring>
#include<cstdio>
#define Re register int
const int N=2e4+3,M=2e5+3;
int n,m,x,y,ans,cut[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
struct Cut_Point{
    int o,dfn_o,low[N],dfn[N],head[N];
    struct QAQ{int to,next;}a[M];
    inline int min(Re a,Re b){return a<b?a:b;}
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void tarjan(Re x,Re root){
        Re son=0;dfn[x]=low[x]=++dfn_o;
        for(Re i=head[x],to;i;i=a[i].next)
            if(!dfn[to=a[i].to]){
                tarjan(to,root),++son;
                low[x]=min(low[x],low[to]);
                cut[x]|=(x!=root&&low[to]>=dfn[x]);
            }
            else low[x]=min(low[x],dfn[to]);
        cut[x]|=(x==root&&son>1),ans+=cut[x];
    }
    inline void cut_point(){for(Re i=1;i<=n;++i)if(!dfn[i])tarjan(i,i);}
}T1;
int main(){
    in(n),in(m);
    while(m--)in(x),in(y),T1.add(x,y),T1.add(y,x);
    T1.cut_point();
    printf("%d\n",ans);
    for(Re i=1;i<=n;++i)if(cut[i])printf("%d ",i);
}

(2).【割边(桥)】

【模板】旅游航道 \(\text{[Loj10102]}\)

#include<algorithm>
#include<cstring>
#include<cstdio>
#define Re register int
using namespace std;
const int N=3e4+3,M=3e4+3;
int n,m,x,y,ans;
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
struct Tarjan{
    int o,dfn_o,dfn[N],low[N],head[N],bridge[M<<1];
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void CL(){
        memset(bridge,0,sizeof(bridge));
        memset(head,0,sizeof(head));
        memset(low,0,sizeof(low));
        memset(dfn,0,sizeof(dfn));
        dfn_o=ans=0,o=1;
    }
    inline void tarjan(Re x,Re p){
        dfn[x]=low[x]=++dfn_o;
        for(Re i=head[x],to;i;i=a[i].next)
            if(!dfn[to=a[i].to]){
                tarjan(to,i);
                low[x]=min(low[x],low[to]);
                if(low[to]>dfn[x])bridge[i]=bridge[i^1]=1,++ans;
            }
            else if(i!=(p^1))low[x]=min(low[x],dfn[to]);
    }
    inline void get_bridg(){for(Re i=1;i<=n;++i)if(!dfn[i])tarjan(i,-1);}
}T1;
int main(){
    while(scanf("%d%d",&n,&m)&&n&&m){
        T1.CL();
        while(m--)in(x),in(y),T1.add(x,y),T1.add(y,x);
        T1.get_bridg();
        printf("%d\n",ans);
    }
}

2.【双连通分量】

(1).【边双连通分量 (e-DCC)】

【模板】冗余路径 \(\text{Redundant Paths}\) \(\text{[P2860]}\)

#include<algorithm>
#include<cstdio>
#define Re register int
using namespace std;
const int N=5003,M=10003;
int n,m,x,y,Q_o,leaf,ip[N],du[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
struct Tarjan{
    int o,dfn_o,dfn[N],low[N],head[N],bridge[M<<1];
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void tarjan(Re x,Re p){
        dfn[x]=low[x]=++dfn_o;
        for(Re i=head[x],to;i;i=a[i].next)
            if(!dfn[to=a[i].to]){
                tarjan(to,i);
                low[x]=min(low[x],low[to]);
                if(low[to]>dfn[x])bridge[i]=bridge[i^1]=1;
            }
            else if(i!=(p^1))low[x]=min(low[x],dfn[to]);
    }
    inline void dfs(Re x){
        ip[x]=Q_o;
        for(Re i=head[x];i;i=a[i].next)
            if(!ip[a[i].to]&&!bridge[i])dfs(a[i].to);
    }
    inline void SuoPoint(){
        for(Re i=1;i<=n;++i)if(!dfn[i])tarjan(i,-1);
        for(Re i=1;i<=n;++i)if(!ip[i])++Q_o,dfs(i);
    }
}T1;
int main(){
    in(n),in(m),T1.o=1;
    while(m--)in(x),in(y),T1.add(x,y),T1.add(y,x);
    T1.SuoPoint();
    for(Re i=2;i<=T1.o;i+=2)
        if((x=ip[T1.a[i].to])!=(y=ip[T1.a[i^1].to]))++du[x],++du[y];
    for(Re i=1;i<=Q_o;++i)leaf+=(du[i]==1);
    printf("%d",leaf+1>>1);
}

(2).【点双连通分量 (v-DCC)】

【模板】矿场搭建 \(\text{[P3225]}\)

#include<algorithm>
#include<cstring>
#include<cstdio>
#include<vector>
#define LL long long
#define Re register int
using namespace std;
const int N=1003,M=503;
int n,m,x,y,T,Q_o,tmp,ans1,cut[N];LL ans2;
std::vector<int>dcc[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
inline int max(Re a,Re b){return a>b?a:b;}
struct Tarjan{
    int o,t,dfn_o,Q[N],dfn[N],low[N],head[N];
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void CL(){
        memset(head,0,sizeof(head));
        memset(low,0,sizeof(low));
        memset(dfn,0,sizeof(dfn));
        memset(cut,0,sizeof(cut));
        dfn_o=t=o=0;
    }
    inline void tarjan(Re x,Re rt){
        Re son=0;Q[++t]=x,dfn[x]=low[x]=++dfn_o;
        for(Re i=head[x],to;i;i=a[i].next)
            if(!dfn[to=a[i].to]){
                tarjan(to,rt),++son;
                low[x]=min(low[x],low[to]);
                if(low[to]>=dfn[x]){
                    if(x!=rt)cut[x]=1;
                    else cut[x]|=son>1;
                    ++Q_o;
                    while(1){
                        dcc[Q_o].push_back(Q[t]);
                        if(to==Q[t--])break;
                    }
                    dcc[Q_o].push_back(x);
                }
            }
            else low[x]=min(low[x],dfn[to]);
    }
    inline void SuoPoint(){for(Re i=1;i<=n;++i)if(!dfn[i])tarjan(i,i);}
}T1;
int main(){
    while(scanf("%d",&m)&&m){
        for(Re i=1;i<=Q_o;++i)dcc[i].clear();
        T1.CL(),Q_o=ans1=n=0,ans2=1;
        while(m--)in(x),in(y),T1.add(x,y),T1.add(y,x),n=max(n,max(x,y));
        T1.SuoPoint();
        for(Re i=1;i<=Q_o;++i){
            tmp=0,n=dcc[i].size();
            for(Re j=0;j<n;++j)tmp+=cut[dcc[i][j]];
            if(!tmp)ans1+=2,ans2*=(n-1)*n/2;
            else if(tmp<2)ans1+=1,ans2*=(n-1);
        }
        printf("Case %d: %d %lld\n",++T,ans1,ans2);
    }
}

三:【有向图】

1.【强连通分量 (SCC)】

(1).【Tarjan】

【模板】缩点 \(\text{[P3387]}\)

#include<cstring>
#include<cstdio>
#include<queue>
#define Re register int
const int N=1e4+3,M=1e5+3;
int n,m,x,y,Q_o,ans,A[N],ru[N],gs[N],ip[N],dp[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
inline int min(Re a,Re b){return a<b?a:b;}
inline int max(Re a,Re b){return a>b?a:b;}
struct Tarjan{
    int o,t,dfn_o,Q[N],low[N],dfn[N],pan[N],head[N];
    struct QAQ{int x,to,next;}a[M];
    inline void add(Re x,Re y){a[++o].x=x,a[o].to=y,a[o].next=head[x],head[x]=o;}
    inline void tarjan(Re x){
        dfn[x]=low[x]=++dfn_o,Q[++t]=x,pan[x]=1;
        for(Re i=head[x],to;i;i=a[i].next)
            if(!dfn[to=a[i].to])tarjan(to),low[x]=min(low[x],low[to]);
            else if(pan[to])low[x]=min(low[x],dfn[to]);
        if(low[x]==dfn[x]){
            ++Q_o;
            while(1){
                ip[Q[t]]=Q_o,gs[Q_o]+=A[Q[t]],pan[Q[t]]=0;
                if(x==Q[t--])break;
            }
        }
    }
    inline void SuoPoint(){for(Re i=1;i<=n;++i)if(!dfn[i])tarjan(i);}
}T1;
struct Tuopu{
    int o,ru[N],pan[N],head[N];std::queue<int>Q;
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void creat(){
        for(Re i=1;i<=T1.o;++i)
            if((x=ip[T1.a[i].x])!=(y=ip[T1.a[i].to]))
                ++ru[y],add(x,y);
    }
    inline void tuopu(){
        for(Re i=1;i<=Q_o;++i)if(!ru[i])dp[i]=gs[i],Q.push(i);
        while(!Q.empty()){
            Re x=Q.front();Q.pop();
            for(Re i=head[x],to;i;i=a[i].next){
                to=a[i].to;
                dp[to]=max(dp[to],dp[x]+gs[to]);
                if(!(--ru[to]))Q.push(to);
            }
        }
    }
}T2;
int main(){
    in(n),in(m);
    for(Re i=1;i<=n;++i)in(A[i]);
    while(m--)in(x),in(y),T1.add(x,y);
    T1.SuoPoint(),T2.creat(),T2.tuopu();
    for(Re i=1;i<=Q_o;++i)ans=max(ans,dp[i]);
    printf("%d",ans);
}

(2).【Kosaraju】

【模板】缩点 \(\text{[P3387]}\)

#include<cstring>
#include<cstdio>
#include<queue>
#define Re register int
const int N=1e6+3,M=1e6+3;
int n,m,Q_o,ans,X[M],Y[M],A[N],ip[N],gs[N],dp[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
inline int max(Re a,Re b){return a>b?a:b;}
struct Kosaraju{
    int o,ndfn_o,nQ[N],pan[N],head[N];
    struct QAQ{int x,to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].x=x,a[o].to=y,a[o].next=head[x],head[x]=o;}
    inline void dfs1(Re x){
        pan[x]=1;
        for(Re i=head[x];i;i=a[i].next)if(!pan[a[i].to])dfs1(a[i].to);
        nQ[++ndfn_o]=x;
    }
    inline void dfs2(Re x){
        ip[x]=Q_o,gs[Q_o]+=A[x];
        for(Re i=head[x];i;i=a[i].next)if(!ip[a[i].to])dfs2(a[i].to);
    }
    inline void kosaraju(){
        for(Re i=1;i<=m;++i)add(X[i],Y[i]);
        for(Re i=1;i<=n;++i)if(!pan[i])dfs1(i);//跑逆dfs序
        memset(head,0,sizeof(head));
        memset(pan,0,sizeof(pan));
        memset(a,0,sizeof(a));o=0;
        for(Re i=1;i<=m;++i)add(Y[i],X[i]);//建反图
        for(Re i=ndfn_o;i;--i)if(!ip[nQ[i]])++Q_o,dfs2(nQ[i]);//缩点
    }
}T1;
struct Tuopu{
    int o,x,y,ru[N],pan[N],head[N];std::queue<int>Q;
    struct QAQ{int to,next;}a[M<<1];
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void creat(){
        for(Re i=1;i<=T1.o;++i)
            if((x=ip[T1.a[i].x])!=(y=ip[T1.a[i].to]))
                ++ru[y],add(x,y);
    }
    inline void tuopu(){
        creat();
        for(Re i=1;i<=Q_o;++i)if(!ru[i])dp[i]=gs[i],Q.push(i);
        while(!Q.empty()){
            Re x=Q.front();Q.pop();
            for(Re i=head[x],to;i;i=a[i].next){
                to=a[i].to;
                dp[to]=max(dp[to],dp[x]+gs[to]);
                if(!(--ru[to]))Q.push(to);
            }
        }
    }
}T2;
int main(){
    in(n),in(m);
    for(Re i=1;i<=n;++i)in(A[i]);
    for(Re i=1;i<=m;++i)in(X[i]),in(Y[i]);
    T1.kosaraju(),T2.tuopu();
    for(Re i=1;i<=Q_o;++i)ans=max(ans,dp[i]);
    printf("%d",ans);
}

2.【2-SAT】

【模板】\(\text{2-SAT}\) 问题 \(\text{[P4782]}\)

(1).【Tarjan】

#include<algorithm>
#include<cstring>
#include<cstdio>
#define Re register int
const int N=2e6+3,M=2e6+3;
int n,m,p1,p2,x1,x2,Q_o,ip[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
struct Tarjan{
    int o,t,dfn_o,Q[N],low[N],dfn[N],pan[N],head[N];
    struct QAQ{int to,next;}a[M];
    inline int min(Re a,Re b){return a<b?a:b;}
    inline void add(Re x,Re y){a[++o].to=y,a[o].next=head[x],head[x]=o;}
    inline void tarjan(Re x){
        low[x]=dfn[x]=++dfn_o,Q[++t]=x,pan[x]=1;
        for(Re i=head[x],to;i;i=a[i].next)
            if(!dfn[to=a[i].to])tarjan(to),low[x]=min(low[x],low[to]);
            else if(pan[to])low[x]=min(low[x],dfn[to]);
        if(low[x]==dfn[x]){
            ++Q_o;
            while(1){
                ip[Q[t]]=Q_o,pan[Q[t]]=0;
                if(x==Q[t--])break;
            }
        }
    }
    inline void SuoPoint(){for(Re i=1;i<=2*n;++i)if(!dfn[i])tarjan(i);}
}T1;
int main(){
    in(n),in(m);
    while(m--)in(p1),in(x1),in(p2),in(x2),T1.add(p1+(x1^1)*n,p2+x2*n),T1.add(p2+(x2^1)*n,p1+x1*n);
    T1.SuoPoint();
    for(Re i=1;i<=n;++i)if(ip[i]==ip[i+n])return !printf("IMPOSSIBLE");
    puts("POSSIBLE");
    for(Re i=1;i<=n;++i)printf("%d ",ip[i]>ip[i+n]);
}

(2).【Kosaraju】

#include<algorithm>
#include<cstring>
#include<cstdio>
#define Re register int
const int N=2e6+3,M=2e6+3;
int n,m,mm,p1,p2,x1,x2,Q_o,X[M],Y[M],ip[N];
inline void in(Re &x){
    int f=0;x=0;char c=getchar();
    while(c<'0'||c>'9')f|=c=='-',c=getchar();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+(c^48),c=getchar();
    x=f?-x:x;
}
struct Kosaraju{
    int o,ndfn_o,nQ[N],pan[N],head[N];
    struct QAQ{int x,to,next;}a[M];
    inline void add(Re x,Re y){a[++o].x=x,a[o].to=y,a[o].next=head[x],head[x]=o;}
    inline void dfs1(Re x){
        pan[x]=1;
        for(Re i=head[x];i;i=a[i].next)if(!pan[a[i].to])dfs1(a[i].to);
        nQ[++ndfn_o]=x;
    }
    inline void dfs2(Re x){
        ip[x]=Q_o;
        for(Re i=head[x];i;i=a[i].next)if(!ip[a[i].to])dfs2(a[i].to);
    }
    inline void kosaraju(){
        for(Re i=1;i<=m;++i)add(X[i],Y[i]);
        for(Re i=1;i<=2*n;++i)if(!pan[i])dfs1(i);//跑逆dfs序
        memset(head,0,sizeof(head));
        memset(pan,0,sizeof(pan));
        memset(a,0,sizeof(a));o=0;
        for(Re i=1;i<=m;++i)add(Y[i],X[i]);//建反图
        for(Re i=ndfn_o;i;--i)if(!ip[nQ[i]])++Q_o,dfs2(nQ[i]);//缩点
    }
}T1;
int main(){
    in(n),in(mm);
    while(mm--)in(p1),in(x1),in(p2),in(x2),X[++m]=p1+(x1^1)*n,Y[m]=p2+x2*n,X[++m]=p2+(x2^1)*n,Y[m]=p1+x1*n;
    T1.kosaraju();
    for(Re i=1;i<=n;++i)if(ip[i]==ip[i+n])return !printf("IMPOSSIBLE");
    puts("POSSIBLE");
    for(Re i=1;i<=n;++i)printf("%d ",ip[i]<ip[i+n]);
}

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posted @ 2019-11-12 11:40  辰星凌  阅读(526)  评论(0编辑  收藏  举报