Splay各种操作模板大赏

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$\ \ \ \ \ \ \ \,$关于Splay操作的复习笔记：

$\ \ \ \ \ \ \ \,$依然的，只记录模板，不讲原理，关于Splay出门左拐百度。

结构

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struct Splay{
	#define lson son[0][rt]
	#define rson son[1][rt]
	int root,cnt;
	int son[2][N],sum[N],val[N],size[N],fa[N];
	int lazy[N];
}
  1. $\tt root$ ：根；
  2. $\tt cnt$ ：下标大小；
  3. $\tt son[0/1][]$ ：左右儿子；
  4. $\tt sum[]$ ：子树大小；
  5. $\tt val[]$ ：节点权值；
  6. $\tt size[]$ ：节点大小；
  7. $\tt fa[]$ ：节点父亲；
  8. $\tt lazy[]$ ：下传标记。

基本操作

  • 标记上传 ($pushup$)
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    void pushup(int rt){
    	if(!rt)return;
    	sum[rt]=sum[lson]+sum[rson]+size[rt];
    }
  • 标记下传 ($pushdown$)，根据情况会不一样
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    void pushdown(int rt){
      	if(!lazy[rt])return;
      	lazy[lson]^=1;lazy[rson]^=1;
      	lazy[rt]=0;
      	swap(lson,rson);
      }
  • 左旋+右旋 ($rotate$)
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     void rotate(int x){
      	int y=fa[x];int z=fa[y];
      	int k=son[1][y]==x;
      	son[son[1][z]==y][z]=x;
      	fa[x]=z;
      	son[k][y]=son[k^1][x];
      	fa[son[k^1][x]]=y;
      	son[k^1][x]=y;
      	fa[y]=x;
    	pushup(y);pushup(x);
    }
  • 将$x$旋转到$goal$下 ($splay$)
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    void splay(int x,int goal){
      	while(fa[x]!=goal){
      		int y=fa[x];int z=fa[y];
      		if(z!=goal)
      		(son[1][z]==y)^(son[1][y]==x)?rotate(x):rotate(y);
      		rotate(x);
        }
        if(goal==0)root=x;
      }

修改

  • 新建节点 ($newnode$)
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    int newnode(int f,int v){
        int rt=++cnt;
        size[rt]=1;sum[rt]=1;val[rt]=v;
        fa[rt]=f;lson=rson=0;lazy[rt]=0;
        return rt;
    }
  • 清空节点 ($clean$)
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    void clean(int rt){
        lson=rson=0;fa[rt]=0;
        val[rt]=0;sum[rt]=size[rt]=0;
    }
  • 建立 ($Build$)
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    void Build(int &rt,int L,int R,int f){
      	if(L>R)return;
      	int mid=(L+R)>>1;
      	rt=newnode(f,a[mid]);
      	Build(lson,L,mid-1,rt);
      	Build(rson,mid+1,R,rt);
      	pushup(rt);
    }
  • 插入($Insert$)
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    void Insert(int v){
        if(!root){root=newnode(0,v);return;}
        int now=root,rt=0;
        pushdown(now);
        while(1){
          if(val[now]==v){
      			++size[now];
            pushup(now);pushup(rt);
            splay(now,0);
            break;
          }
          rt=now;
          val[rt]<v?now=rson:now=lson;
        	pushdown(now);
          if(!now){
            now=newnode(rt,v);
      	    val[rt]<v?rson=now:lson=now;
            pushup(rt);splay(now,0);
            break;
          }
       }
    }
  • 删除节点($delete_pos$)
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    void delete_pos(int rt){
        splay(rt,0);
        if(size[rt]>1){--size[rt];return;}
        int now;
        if(!lson||!rson){
          now=rson+lson;pushdown(now);
          if(!now){clean(rt);root=0;return;}
          clean(rt);root=now;fa[now]=0;
          pushup(now);return;
        }
        now=pre_pos(rt);
        pushdown(now);
        splay(now,rt);
        fa[now]=0;son[1][now]=rson;fa[rson]=now;
        root=now;clean(rt);
        pushup(rson);pushup(now);	
    }
  • 删除($Delete$)
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    void Delete(int v){
        v=QueryRank_pos(v);
        pushdown(v);
        delete_pos(v);
    }
  • 删除第k($Delete_QueryRank$)
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    void Delete_QueryRank(int v){
        v=Rank_pos(v);
        pushdown(v);
        delete_pos(v);
    }
  • 区间修改（翻转）($revse$)
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    void revse(int a,int b){
    	a=find(a);
      	b=find(b+2);
      	splay(a,0);splay(b,a);
      	lazy[son[0][son[1][root]]]^=1;
    }

询问

  • 查找某个值的节点编号($QueryRank_pos$)
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    int QueryRank_pos(int x){
        int rt=root;
        while(1){
        	pushdown(rt);
        	if(val[rt]==x){splay(rt,0);return rt;} 
        	if(val[rt]<x)rt=rson;
        	else if(val[rt]>x)rt=lson;
        }	
    }
  • 查找某个排名的节点编号($Rank_pos$)
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    int Rank_pos(int x){
      int rt=root;
      while(1){
      	pushdown(rt);
      	if(sum[lson]>=x){rt=lson;continue;}
      	x-=sum[lson];
        	if(size[rt]>=x){splay(rt,0);return val[rt];}
        	x-=size[rt];
        	rt=rson;
      }	
    }

  • 查找排名第k的值($QueryRank$)

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    int QueryRank(int x){
        int rt=root;
        while(1){
        	pushdown(rt);
        	if(sum[lson]>=x){rt=lson;continue;}
        	x-=sum[lson];
        	if(size[rt]>=x){splay(rt,0);return val[rt];}
        	x-=size[rt];
        	rt=rson;
      }
    }

  • 查找某个值的排名($Rank$)

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    int Rank(int num){
        int rt=root,ans=0;
        while(1){
        	pushdown(rt);
        	if(val[rt]==num){ans=ans+sum[lson]+1;splay(rt,0);return ans;}
        	if(val[rt]<num)ans+=sum[lson]+size[rt],rt=rson;
         	else if(val[rt]>num)rt=lson;
      	}
    }
  • 集合内前驱的节点编号($pre_pos$)
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    int pre_pos(int x){
        splay(x,0);int rt;
        pushdown(x);
        if(!(rt=son[0][x]))return val[x];
        while(rson)pushdown(rt),rt=rson;
        return rt;	
    }
  • 集合内后继的节点编号($nex_pos$)
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    int nex_pos(int x){
        splay(x,0);int rt;
        pushdown(x);
        if(!(rt=son[1][x]))return val[x];
        while(lson)pushdown(rt),rt=lson;
        return rt;	
    }
  • 集合内前驱的值($pre$)
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    int pre(int x){
    	splay(x,0);int rt;
        pushdown(x);
        if(!(rt=son[0][x]))return val[x];
        while(rson)pushdown(rt),rt=rson;
        return val[rt];
    }
  • 集合内后继的值($nex$)
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    int nex(int x){
        splay(x,0);int rt;
        pushdown(x);
        if(!(rt=son[1][x]))return val[x];
        while(lson)pushdown(rt),rt=lson;
        return val[rt];
    }
  • 前驱($Numpre$)
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    int Numpre(int x){
        Insert(x);
        int y=x;x=QueryRank_pos(x);
        int ls=pre(x);
        Delete(y);
        return ls;
    }
  • 后继($Numnex$)
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    int Numnex(int x){
        Insert(x);
        int y=x;x=QueryRank_pos(x);
        int ls=nex(x);
        Delete(y);
        return ls;
    }
  • 查找某个值的节点编号($find$)，不改变结构
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    int find(int k){
      	int rt=root;
      	while(1){
      		pushdown(rt);
      		if(sum[lson]>=k)rt=lson;
      		else if(sum[lson]+1==k)return rt;
      		else k-=sum[lson]+1,rt=rson;
        }
     }

完整模板

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const int N=1e5+10;
int a[N],n,m;
struct Splay{
	#define lson son[0][rt]
	#define rson son[1][rt]
	int root,cnt;
	int son[2][N],sum[N],val[N],size[N],fa[N];
  	bool lazy[N];
	void pushdown(int rt){
  		if(!lazy[rt])return;
  		lazy[lson]^=1;lazy[rson]^=1;
  		lazy[rt]=0;
  		swap(lson,rson);
  	}
	void pushup(int rt){
	  	if(!rt)return;
	  	sum[rt]=sum[lson]+sum[rson]+size[rt];
	}
  	void rotate(int x){
	  	int y=fa[x];int z=fa[y];
	  	int k=son[1][y]==x;
	  	son[son[1][z]==y][z]=x;
	  	fa[x]=z;
	  	son[k][y]=son[k^1][x];
	  	fa[son[k^1][x]]=y;
	  	son[k^1][x]=y;
	  	fa[y]=x;
	  	pushup(y);pushup(x);
	}
  	void splay(int x,int goal){
	  	while(fa[x]!=goal){
	  		int y=fa[x];int z=fa[y];
	  		if(z!=goal)
	  		(son[1][z]==y)^(son[1][y]==x)?rotate(x):rotate(y);
	  		rotate(x);
	    }
	    if(goal==0)root=x;
  	}
  	int newnode(int f,int v){
	    int rt=++cnt;
	    size[rt]=1;sum[rt]=1;val[rt]=v;
	    fa[rt]=f;lson=rson=0;lazy[rt]=0;
	    return rt;
		}
		void clean(int rt){
	    lson=rson=0;fa[rt]=0;
	    val[rt]=0;sum[rt]=size[rt]=0;
  	}
	void Build(int &rt,int L,int R,int f){
	  	if(L>R)return;
	  	int mid=(L+R)>>1;
	  	rt=newnode(f,a[mid]);
	  	Build(lson,L,mid-1,rt);
	  	Build(rson,mid+1,R,rt);
	  	pushup(rt);
	}
  	void Insert(int v){
    	if(!root){root=newnode(0,v);return;}
    	int now=root,rt=0;
    	pushdown(now);
   		while(1){
      		if(val[now]==v){
	  			++size[now];
	        	pushup(now);pushup(rt);
	        	splay(now,0);
	        	break;
	      	}
	      	rt=now;
	      	val[rt]<v?now=rson:now=lson;
	    	pushdown(now);
	      	if(!now){
	        	now=newnode(rt,v);
	  	    	val[rt]<v?rson=now:lson=now;
	        	pushup(rt);splay(now,0);
	        	break;
	      	}
   	 	}
  	}
  	int QueryRank_pos(int x){
    	int rt=root;
    	while(1){
    		pushdown(rt);
      		if(val[rt]==x){splay(rt,0);return rt;} 
      		if(val[rt]<x)rt=rson;
      		else if(val[rt]>x)rt=lson;
    	}	
  }
  int Rank_pos(int x){
    int rt=root;
    while(1){
    	pushdown(rt);
      if(sum[lson]>=x){rt=lson;continue;}
      x-=sum[lson];
      if(size[rt]>=x){splay(rt,0);return val[rt];}
      x-=size[rt];
      rt=rson;
    }	
  }
  	int QueryRank(int x){
    	int rt=root;
    	while(1){
    		pushdown(rt);
     		if(sum[lson]>=x){rt=lson;continue;}
     		x-=sum[lson];
      		if(size[rt]>=x){splay(rt,0);return val[rt];}
      		x-=size[rt];
      		rt=rson;
    	}
  	}
  	int Rank(int num){
    	int rt=root,ans=0;
    	while(1){
    		pushdown(rt);
      		if(val[rt]==num){ans=ans+sum[lson]+1;splay(rt,0);return ans;}
     		if(val[rt]<num)ans+=sum[lson]+size[rt],rt=rson;
      		else if(val[rt]>num)rt=lson;
    	}
  	}
  	int pre_pos(int x){
    	splay(x,0);int rt;
    	pushdown(x);
    	if(!(rt=son[0][x]))return val[x];
    	while(rson)pushdown(rt),rt=rson;
    	return rt;	
  	}
  	int nex_pos(int x){
    	splay(x,0);int rt;
    	pushdown(x);
    	if(!(rt=son[1][x]))return val[x];
    	while(lson)pushdown(rt),rt=lson;
    	return rt;	
  	}
  	int pre(int x){
    	splay(x,0);int rt;
    	pushdown(x);
    	if(!(rt=son[0][x]))return val[x];
    	while(rson)pushdown(rt),rt=rson;
    	return val[rt];
  	}
  	int nex(int x){
    	splay(x,0);int rt;
    	pushdown(x);
    	if(!(rt=son[1][x]))return val[x];
    	while(lson)pushdown(rt),rt=lson;
    	return val[rt];
  	}
  	void delete_pos(int rt){
    	splay(rt,0);
    	if(size[rt]>1){--size[rt];return;}
    	int now;
    	if(!lson||!rson){
      		now=rson+lson;pushdown(now);
      		if(!now){clean(rt);root=0;return;}
      		clean(rt);root=now;fa[now]=0;
      		pushup(now);return;
    	}
    	now=pre_pos(rt);
    	pushdown(now);
    	splay(now,rt);
    	fa[now]=0;son[1][now]=rson;fa[rson]=now;
    	root=now;clean(rt);
    	pushup(rson);pushup(now);	
  	}
  	void Delete(int v){
    	v=QueryRank_pos(v);
    	pushdown(v);
    	delete_pos(v);
  	}
  	void Delete_QueryRank(int v){
    	v=Rank_pos(v);
    	pushdown(v);
    	delete_pos(v);
  	}
  	int Numpre(int x){
    	Insert(x);
    	int y=x;x=QueryRank_pos(x);
    	int ls=pre(x);
    	Delete(y);
    	return ls;
  	}
  	int Numnex(int x){
    	Insert(x);
    	int y=x;x=QueryRank_pos(x);
    	int ls=nex(x);
    	Delete(y);
    	return ls;
  	}
  	int find(int k){
  		int rt=root;
  		while(1){
  			pushdown(rt);
  			if(sum[lson]>=k)rt=lson;
  			else if(sum[lson]+1==k)return rt;
  			else k-=sum[lson]+1,rt=rson;
    	}
  	}
	void revse(int a,int b){
		a=find(a);
  		b=find(b+2);
  		splay(a,0);splay(b,a);
  		lazy[son[0][son[1][root]]]^=1;
	}
}Spl;

两个模板题：

  • P3369 【模板】普通平衡树
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    #include<algorithm>
    #include<iostream>
    #include<cstdlib>
    #include<cstring>
    #include<cctype>
    #include<cstdio>
    #include<vector>
    #include<string>
    #include<queue>
    #include<stack>
    #include<cmath>
    #include<map>
    #include<set>
    using namespace std;
    const int inf=0x7fffffff;
    const double eps=1e-10;
    const double pi=acos(-1.0);
    //char buf[1<<15],*S=buf,*T=buf;
    //char getch(){return S==T&&(T=(S=buf)+fread(buf,1,1<<15,stdin),S==T)?0:*S++;}
    inline int read(){
    	int x=0,f=1;char ch;ch=getchar();
    	while(ch<'0'||ch>'9'){if(ch=='-') f=0;ch=getchar();}
    	while(ch>='0'&&ch<='9'){x=(x<<1)+(x<<3)+(ch&15);ch=getchar();}
    	if(f)return x;else return-x;
    }
    const int N=1e5+10;
    int a[N],n,m;
    struct Splay{
    	
    }Spl;
    int main()
    {
      	int n=read();
      	while(n--){
        	int opt=read(),x=read();
        	if(opt==1){Spl.Insert(x);continue;}
        	if(opt==2){Spl.Delete(x);continue;}
        	if(opt==3){printf("%d\n",Spl.Rank(x));continue;}
        	if(opt==4){printf("%d\n",Spl.QueryRank(x));continue;}
        	if(opt==5){printf("%d\n",Spl.Numpre(x));continue;}
        	if(opt==6){printf("%d\n",Spl.Numnex(x));continue;}
      	}
      	return 0;
    }
  • P3391 【模板】文艺平衡树（Splay）
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    #include<algorithm>
    #include<iostream>
    #include<cstdlib>
    #include<cstring>
    #include<cctype>
    #include<cstdio>
    #include<vector>
    #include<string>
    #include<queue>
    #include<stack>
    #include<cmath>
    #include<map>
    #include<set>
    using namespace std;
    const int inf=0x7fffffff;
    const double eps=1e-10;
    const double pi=acos(-1.0);
    //char buf[1<<15],*S=buf,*T=buf;
    //char getch(){return S==T&&(T=(S=buf)+fread(buf,1,1<<15,stdin),S==T)?0:*S++;}
    inline int read(){
    	int x=0,f=1;char ch;ch=getchar();
    	while(ch<'0'||ch>'9'){if(ch=='-') f=0;ch=getchar();}
    	while(ch>='0'&&ch<='9'){x=(x<<1)+(x<<3)+(ch&15);ch=getchar();}
    	if(f)return x;else return-x;
    }
    const int N=1e5+10;
    int a[N],n,m;
    struct Splay{
    	
    }Spl;
    void write(int rt){
      	Spl.pushdown(rt);
      	if(Spl.lson)write(Spl.lson);
      	if(Spl.val[rt]>1&&Spl.val[rt]<n+2)
      	printf("%d ",Spl.val[rt]-1);
      	if(Spl.rson)write(Spl.rson); 
    }
    int main()
    {
    	n=read();m=read();
     	for(int i=1;i<=n+2;i++)a[i]=i;
     	Spl.Build(Spl.root,1,n+2,0);
     	while(m--){
        	int l=read(),r=read();
      		Spl.revse(l,r);
      	}
      	write(Spl.root);
      	printf("\n");
    	return 0;
    }