[POJ3694]Network（tarjan+链剖）

题目描述

传送门
题意：首先给出一个无向图。每一次询问向这个图里加一条边，求加了这条边之后图中的桥的个数。

题解

首先tarjan将所有的双连通分量缩成一个点，形成一棵树，每一条树边都是桥边。每连一条边实际上是将这条树链上的所有边都变成非桥边。可以用树链剖分来实现。

代码

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define N 200005

int n,m,q,x,y,dfs_clock,dcc,ans,Case;
int tot,point[N],v[N*2],nxt[N*2];
int dfn[N],low[N],stack[N],tmp,belong[N];
int father[N],h[N],size[N],num[N],son[N],top[N];
int sum[N*4],delta[N*4];
bool vis[N];
struct hp{int x,y;}e[N];

void clear()
{
    q=x=y=dfs_clock=dcc=ans=0;
    tot=0;memset(point,0,sizeof(point));memset(nxt,0,sizeof(nxt));memset(v,0,sizeof(v));
    memset(dfn,0,sizeof(dfn));memset(low,0,sizeof(low));memset(belong,0,sizeof(belong));tmp=0;
    memset(father,0,sizeof(father));memset(h,0,sizeof(h));memset(size,0,sizeof(size));memset(num,0,sizeof(num));
    memset(son,0,sizeof(son));memset(top,0,sizeof(top));memset(sum,0,sizeof(sum));memset(delta,0,sizeof(delta));
    memset(vis,0,sizeof(vis));
}
void add(int x,int y)
{
    ++tot; nxt[tot]=point[x]; point[x]=tot; v[tot]=y;
    ++tot; nxt[tot]=point[y]; point[y]=tot; v[tot]=x;
}
void tarjan(int x,int fa)
{
    dfn[x]=low[x]=++dfs_clock;vis[x]=true;stack[++tmp]=x;
    bool flag=false;
    for (int i=point[x];i;i=nxt[i])
    {
        if (v[i]==fa&&!flag)
        {
            flag=true;
            continue;
        }
        if (!dfn[v[i]])
        {
            tarjan(v[i],x);
            low[x]=min(low[x],low[v[i]]);
        }
        else if (vis[v[i]]) low[x]=min(low[x],low[v[i]]);
    }
    if (dfn[x]==low[x])
    {
        dcc++;int now=0;
        while (now!=x)
        {
            now=stack[tmp--];
            vis[now]=false;
            belong[now]=dcc;
        }
    }
}
void rebuild()
{
    tot=0;memset(point,0,sizeof(point));
    for (int i=1;i<=m;++i)
        if (belong[e[i].x]!=belong[e[i].y])
            add(belong[e[i].x],belong[e[i].y]);

}
void dfs_1(int x,int fa,int dep)
{
    father[x]=fa;h[x]=dep;size[x]=1;
    for (int i=point[x];i;i=nxt[i])
        if (v[i]!=fa)
        {
            dfs_1(v[i],x,dep+1);
            size[x]+=size[v[i]];
            if (size[v[i]]>size[son[x]]) son[x]=v[i];
        }
}
void dfs_2(int x,int fa)
{
    if (son[fa]==x) top[x]=top[fa];
    else top[x]=x;
    num[x]=++dfs_clock;
    if (son[x]) dfs_2(son[x],x);
    for (int i=point[x];i;i=nxt[i])
        if (v[i]!=fa&&v[i]!=son[x])
            dfs_2(v[i],x);
}
void update(int now)
{
    sum[now]=sum[now<<1]+sum[now<<1|1];
}
void pushdown(int now,int l,int r,int mid)
{
    if (delta[now]!=-1)
    {
        sum[now<<1]=sum[now<<1|1]=delta[now<<1]=delta[now<<1|1]=0;
        delta[now]=-1;
    }
}
void build(int now,int l,int r)
{
    int mid=(l+r)>>1;
    delta[now]=-1;
    if (l==r)
    {
        sum[now]=1;
        return;
    }
    build(now<<1,l,mid);
    build(now<<1|1,mid+1,r);
    update(now);
}
void cover(int now,int l,int r,int lrange,int rrange)
{
    int mid=(l+r)>>1;
    if (lrange<=l&&r<=rrange)
    {
        sum[now]=delta[now]=0;
        return;
    }
    pushdown(now,l,r,mid);
    if (lrange<=mid) cover(now<<1,l,mid,lrange,rrange);
    if (mid+1<=rrange) cover(now<<1|1,mid+1,r,lrange,rrange);
    update(now);
}
int query(int now,int l,int r,int lrange,int rrange)
{
    int mid=(l+r)>>1,ans=0;
    if (lrange<=l&&r<=rrange) return sum[now];
    pushdown(now,l,r,mid);
    if (lrange<=mid) ans+=query(now<<1,l,mid,lrange,rrange);
    if (mid+1<=rrange) ans+=query(now<<1|1,mid+1,r,lrange,rrange);
    return ans;
}
void Cover(int u,int t)
{
    int f1=top[u],f2=top[t];
    while (f1!=f2)
    {
        if (h[f1]<h[f2])
        {
            swap(f1,f2);
            swap(u,t);
        }
        cover(1,1,dcc,num[f1],num[u]);
        u=father[f1];
        f1=top[u];
    }
    if (num[u]==num[t]) return;
    if (num[u]>num[t]) swap(u,t);
    cover(1,1,dcc,num[u]+1,num[t]);
}
int Query(int u,int t)
{
    int f1=top[u],f2=top[t],ans=0;
    while (f1!=f2)
    {
        if (h[f1]<h[f2])
        {
            swap(f1,f2);
            swap(u,t);
        }
        ans+=query(1,1,dcc,num[f1],num[u]);
        u=father[f1];
        f1=top[u];
    }
    if (num[u]==num[t]) return ans;
    if (num[u]>num[t]) swap(u,t);
    ans+=query(1,1,dcc,num[u]+1,num[t]);
    return ans;
}

int main()
{
    while (~scanf("%d%d",&n,&m))
    {
        if (!n&&!m) break;
        printf("Case %d:\n",++Case);
        clear();
        for (int i=1;i<=m;++i)
        {
            scanf("%d%d",&e[i].x,&e[i].y);
            add(e[i].x,e[i].y);
        }
        for (int i=1;i<=n;++i)
            if (!dfn[i]) tarjan(i,0);
        rebuild();ans=dcc-1;dfs_clock=0;
        dfs_1(1,0,1);
        dfs_2(1,0);
        build(1,1,dcc);
        scanf("%d",&q);
        for (int i=1;i<=q;++i)
        {
            scanf("%d%d",&x,&y);
            if (belong[x]==belong[y]) {printf("%d\n",ans);continue;}
            ans-=Query(belong[x],belong[y]);
            Cover(belong[x],belong[y]);
            printf("%d\n",ans);
        }
        puts("");
    }
}

手工栈

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define N 200005

int n,m,q,x,y,dfs_clock,dcc,ans,Case;
int tot,point[N],v[N*2],nxt[N*2];
int dfn[N],low[N],stack[N],tmp,belong[N];
int father[N],h[N],size[N],num[N],son[N],top[N];
int sum[N*4],delta[N*4];
bool vis[N];
struct hp{int x,y;}e[N];
int cur[N];bool use[N];

void clear()
{
    q=x=y=dfs_clock=dcc=ans=0;
    tot=0;memset(point,0,sizeof(point));memset(nxt,0,sizeof(nxt));memset(v,0,sizeof(v));
    memset(dfn,0,sizeof(dfn));memset(low,0,sizeof(low));memset(belong,0,sizeof(belong));tmp=0;
    memset(father,0,sizeof(father));memset(h,0,sizeof(h));memset(size,0,sizeof(size));memset(num,0,sizeof(num));
    memset(son,0,sizeof(son));memset(top,0,sizeof(top));memset(sum,0,sizeof(sum));memset(delta,0,sizeof(delta));
    memset(vis,0,sizeof(vis));
    memset(cur,0,sizeof(cur));memset(use,0,sizeof(use));
}
void add(int x,int y)
{
    ++tot; nxt[tot]=point[x]; point[x]=tot; v[tot]=y;
    ++tot; nxt[tot]=point[y]; point[y]=tot; v[tot]=x;
}
void tarjan(int x,int fa)
{
    dfn[x]=low[x]=++dfs_clock;vis[x]=true;stack[++tmp]=x;
    bool flag=false;
    for (int i=point[x];i;i=nxt[i])
    {
        if (v[i]==fa&&!flag)
        {
            flag=true;
            continue;
        }
        if (!dfn[v[i]])
        {
            tarjan(v[i],x);
            low[x]=min(low[x],low[v[i]]);
        }
        else if (vis[v[i]]) low[x]=min(low[x],low[v[i]]);
    }
    if (dfn[x]==low[x])
    {
        dcc++;int now=0;
        while (now!=x)
        {
            now=stack[tmp--];
            vis[now]=false;
            belong[now]=dcc;
        }
    }
}
void rebuild()
{
    tot=0;memset(point,0,sizeof(point));
    for (int i=1;i<=m;++i)
        if (belong[e[i].x]!=belong[e[i].y])
            add(belong[e[i].x],belong[e[i].y]);

}
void dfs_1()
{
    int tmp=0;
    stack[++tmp]=1;h[1]=1;size[1]=1;
    for (int i=1;i<=n;++i) cur[i]=point[i];
    while (tmp)
    {
        int x=stack[tmp];
        if (cur[x]&&v[cur[x]]==father[x]) cur[x]=nxt[cur[x]];
        if (!cur[x])
        {
            --tmp;
            if (father[x])
            {
                size[father[x]]+=size[x];
                if (size[x]>size[son[father[x]]]) son[father[x]]=x;
            }
            continue;
        }
        int vt=v[cur[x]];
        stack[++tmp]=vt;
        size[vt]=1;h[vt]=h[x]+1;father[vt]=x;
        cur[x]=nxt[cur[x]];
    }
}
void dfs_2()
{
    int tmp=0;
    stack[++tmp]=1;num[1]=++dfs_clock;top[1]=1;
    for (int i=1;i<=n;++i) cur[i]=point[i];
    while (tmp)
    {
        int x=stack[tmp];
        if (!use[x])
        {
            use[x]=1;
            int vt=son[x];
            if (vt)
            {
                stack[++tmp]=vt;
                top[vt]=top[x];
                num[vt]=++dfs_clock;
            }
            continue;
        }
        while (cur[x]&&(v[cur[x]]==father[x]||v[cur[x]]==son[x])) cur[x]=nxt[cur[x]];
        if (!cur[x])
        {
            --tmp;
            continue;
        }
        else
        {
            int vt=v[cur[x]];
            stack[++tmp]=vt;
            num[vt]=++dfs_clock;
            top[vt]=vt;
            cur[x]=nxt[cur[x]];
        }
    }
}
void update(int now)
{
    sum[now]=sum[now<<1]+sum[now<<1|1];
}
void pushdown(int now,int l,int r,int mid)
{
    if (delta[now]!=-1)
    {
        sum[now<<1]=sum[now<<1|1]=delta[now<<1]=delta[now<<1|1]=0;
        delta[now]=-1;
    }
}
void build(int now,int l,int r)
{
    int mid=(l+r)>>1;
    delta[now]=-1;
    if (l==r)
    {
        sum[now]=1;
        return;
    }
    build(now<<1,l,mid);
    build(now<<1|1,mid+1,r);
    update(now);
}
void cover(int now,int l,int r,int lrange,int rrange)
{
    int mid=(l+r)>>1;
    if (lrange<=l&&r<=rrange)
    {
        sum[now]=delta[now]=0;
        return;
    }
    pushdown(now,l,r,mid);
    if (lrange<=mid) cover(now<<1,l,mid,lrange,rrange);
    if (mid+1<=rrange) cover(now<<1|1,mid+1,r,lrange,rrange);
    update(now);
}
int query(int now,int l,int r,int lrange,int rrange)
{
    int mid=(l+r)>>1,ans=0;
    if (lrange<=l&&r<=rrange) return sum[now];
    pushdown(now,l,r,mid);
    if (lrange<=mid) ans+=query(now<<1,l,mid,lrange,rrange);
    if (mid+1<=rrange) ans+=query(now<<1|1,mid+1,r,lrange,rrange);
    return ans;
}
void Cover(int u,int t)
{
    int f1=top[u],f2=top[t];
    while (f1!=f2)
    {
        if (h[f1]<h[f2])
        {
            swap(f1,f2);
            swap(u,t);
        }
        cover(1,1,dcc,num[f1],num[u]);
        u=father[f1];
        f1=top[u];
    }
    if (num[u]==num[t]) return;
    if (num[u]>num[t]) swap(u,t);
    cover(1,1,dcc,num[u]+1,num[t]);
}
int Query(int u,int t)
{
    int f1=top[u],f2=top[t],ans=0;
    while (f1!=f2)
    {
        if (h[f1]<h[f2])
        {
            swap(f1,f2);
            swap(u,t);
        }
        ans+=query(1,1,dcc,num[f1],num[u]);
        u=father[f1];
        f1=top[u];
    }
    if (num[u]==num[t]) return ans;
    if (num[u]>num[t]) swap(u,t);
    ans+=query(1,1,dcc,num[u]+1,num[t]);
    return ans;
}

int main()
{
    while (~scanf("%d%d",&n,&m))
    {
        if (!n&&!m) break;
        printf("Case %d:\n",++Case);
        clear();
        for (int i=1;i<=m;++i)
        {
            scanf("%d%d",&e[i].x,&e[i].y);
            add(e[i].x,e[i].y);
        }
        for (int i=1;i<=n;++i)
            if (!dfn[i]) tarjan(i,0);
        rebuild();ans=dcc-1;dfs_clock=0;
        dfs_1();
        dfs_2();
        build(1,1,dcc);
        scanf("%d",&q);
        for (int i=1;i<=q;++i)
        {
            scanf("%d%d",&x,&y);
            if (belong[x]==belong[y]) {printf("%d\n",ans);continue;}
            ans-=Query(belong[x],belong[y]);
            Cover(belong[x],belong[y]);
            printf("%d\n",ans);
        }
        puts("");
    }
}