题解Luogu SP8791 DYNALCA - Dynamic LCA

Posted yzhang-rp-inf

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原题传送门

这道题的难点就在如何求LCA:

我们珂以先对其中一个点进行access操作,然后对另一个点进行access操作,因为LCA到根的边一定都由第一次access变为实边了,在之后的这一次access操作的最后一条从虚边变为实边的边的父亲就是两点的LCA

回求LCA后,剩下的几乎是模板,但有两种做法

1.建立虚拟点n+1,保证一直是一颗有n+1个点的树,写起来比较无脑(lct部分珂以复制),常数比较大

#include <bits/stdc++.h>
#define N 100005
#define getchar nc
using namespace std;
inline char nc(){
    static char buf[100000],*p1=buf,*p2=buf; 
    return p1==p2&&(p2=(p1=buf)+fread(buf,1,100000,stdin),p1==p2)?EOF:*p1++; 
}
inline int read()
{
    register int x=0,f=1;register char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=getchar();}
    while(ch>='0'&&ch<='9')x=(x<<3)+(x<<1)+ch-'0',ch=getchar();
    return x*f;
}
inline void write(register int x)
{
    if(!x)putchar('0');if(x<0)x=-x,putchar('-');
    static int sta[20];register int tot=0;
    while(x)sta[tot++]=x%10,x/=10;
    while(tot)putchar(sta[--tot]+48);
}
inline void Swap(register int &a,register int &b)
{
    a^=b^=a^=b;
}
struct Link_Cut_Tree{
    int c[N][2],fa[N],top,q[N],rev[N];
    inline void pushdown(register int x)
    {
        if(rev[x])
        {
            register int l=c[x][0],r=c[x][1];
            rev[l]^=1,rev[r]^=1,rev[x]^=1;
            Swap(c[x][0],c[x][1]);
        }
    }
    inline bool isroot(register int x)
    {
        return c[fa[x]][0]!=x&&c[fa[x]][1]!=x;
    }
    inline void rotate(register int x)
    {
        int y=fa[x],z=fa[y],l,r;
        l=c[y][0]==x?0:1;
        r=l^1;
        if(!isroot(y))
            c[z][c[z][0]==y?0:1]=x;
        fa[x]=z;
        fa[y]=x;
        fa[c[x][r]]=y;
        c[y][l]=c[x][r];
        c[x][r]=y;
    }
    inline void splay(register int x)
    {
        top=1;
        q[top]=x;
        for(register int i=x;!isroot(i);i=fa[i])
            q[++top]=fa[i];
        for(register int i=top;i;--i)
            pushdown(q[i]);
        while(!isroot(x))
        {
            int y=fa[x],z=fa[y];
            if(!isroot(y))
                rotate((c[y][0]==x)^(c[z][0]==y)?(x):(y));
            rotate(x);
        }
    }
    inline void access(register int x)
    {
        for(register int t=0;x;t=x,x=fa[x])
        {
            splay(x);
            c[x][1]=t;
        }
    }
    inline void makeroot(register int x)
    {
        access(x);
        splay(x);
        rev[x]^=1;
    }
    inline void split(register int x,register int y)
    {
        makeroot(x);
        access(y);
        splay(y);
    }
    inline void cut(register int x,register int y)
    {
        split(x,y);
        c[y][0]=0;
        fa[x]=0;
    }
    inline void link(register int x,register int y)
    {
        makeroot(x);
        fa[x]=y;    
    }
    inline int LCA(register int x,register int y)
    {
        access(x);
        int t;
        for(t=0;y;t=y,y=fa[y])
        {
            splay(y);
            c[y][1]=t;  
        }   
        return t;
    }   
}T;
int fa[N];
int main()
{
    int n=read(),m=read();
    for(register int i=1;i<=n;++i)
        T.link(i,n+1),fa[i]=n+1;
    while(m--)
    {
        char c=getchar();
        while(c!='k'&&c!='t'&&c!='a')
            c=getchar();
        if(c=='k')
        {
            int x=read(),y=read();
            T.cut(x,fa[x]);
            fa[x]=y;
            T.link(x,y);
        }
        else if(c=='t')
        {
            int x=read();
            T.cut(x,fa[x]);
            fa[x]=n+1;
            T.link(x,n+1);
        }
        else
        {
            int x=read(),y=read();
            T.makeroot(n+1);
            write(T.LCA(x,y)),puts("");
        }
    }
    return 0;
 } 

2.按照题目加边,删边(常数小,容易写错)

注意:cut和LCA操作是不能使用makeroot操作的,因为makeroot本身会修改树上的LCA关系,当然link操作使用makeroot是没有关系的

#include <bits/stdc++.h>
#define N 100005
#define getchar nc
using namespace std;
inline char nc(){
    static char buf[100000],*p1=buf,*p2=buf; 
    return p1==p2&&(p2=(p1=buf)+fread(buf,1,100000,stdin),p1==p2)?EOF:*p1++; 
}
inline int read()
{
    register int x=0,f=1;register char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=getchar();}
    while(ch>='0'&&ch<='9')x=(x<<3)+(x<<1)+ch-'0',ch=getchar();
    return x*f;
}
inline void write(register int x)
{
    if(!x)putchar('0');if(x<0)x=-x,putchar('-');
    static int sta[20];register int tot=0;
    while(x)sta[tot++]=x%10,x/=10;
    while(tot)putchar(sta[--tot]+48);
}
inline void Swap(register int &a,register int &b)
{
    a^=b^=a^=b;
}
struct Link_Cut_Tree{
    int c[N][2],fa[N],top,q[N],rev[N];
    inline void pushdown(register int x)
    {
        if(rev[x])
        {
            register int l=c[x][0],r=c[x][1];
            rev[l]^=1,rev[r]^=1,rev[x]^=1;
            Swap(c[x][0],c[x][1]);
        }
    }
    inline bool isroot(register int x)
    {
        return c[fa[x]][0]!=x&&c[fa[x]][1]!=x;
    }
    inline void rotate(register int x)
    {
        int y=fa[x],z=fa[y],l,r;
        l=c[y][0]==x?0:1;
        r=l^1;
        if(!isroot(y))
            c[z][c[z][0]==y?0:1]=x;
        fa[x]=z;
        fa[y]=x;
        fa[c[x][r]]=y;
        c[y][l]=c[x][r];
        c[x][r]=y;
    }
    inline void splay(register int x)
    {
        top=1;
        q[top]=x;
        for(register int i=x;!isroot(i);i=fa[i])
            q[++top]=fa[i];
        for(register int i=top;i;--i)
            pushdown(q[i]);
        while(!isroot(x))
        {
            int y=fa[x],z=fa[y];
            if(!isroot(y))
                rotate((c[y][0]==x)^(c[z][0]==y)?(x):(y));
            rotate(x);
        }
    }
    inline void access(register int x)
    {
        for(register int t=0;x;t=x,x=fa[x])
        {
            splay(x);
            c[x][1]=t;
        }
    }
    inline void makeroot(register int x)
    {
        access(x);
        splay(x);
        rev[x]^=1;
    }
    inline void cut(register int x)
    {
        access(x);
        splay(x);
        fa[c[x][0]]=0;
        c[x][0]=0;
    }
    inline void link(register int x,register int y)
    {
        makeroot(x);
        fa[x]=y;    
    }
    inline int LCA(register int x,register int y)
    {
        access(x);
        int t;
        for(t=0;y;t=y,y=fa[y])
        {
            splay(y);
            c[y][1]=t;  
        }   
        return t;
    }   
}T;
int fa[N];
int main()
{
    int n=read(),m=read();
    while(m--)
    {
        char c=getchar();
        while(c!='k'&&c!='t'&&c!='a')
            c=getchar();
        if(c=='k')
        {
            int x=read(),y=read();
            T.link(x,y);
        }
        else if(c=='t')
        {
            int x=read();
            T.cut(x);
        }
        else
        {
            int x=read(),y=read();
            write(T.LCA(x,y)),puts("");
        }
    }
    return 0;
 } 

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