平衡树学习笔记

1.平衡树是一棵二叉查找树。

二叉排序树是一棵空树，或是具有下列性质的二叉树：

（1）若左子树不空，则左子树上所有结点的值均小于或等于它的根结点的值；

（2）若右子树不空，则右子树上所有结点的值均大于或等于它的根结点的值；

（3）左、右子树也分别为二叉排序树；

2.平衡二叉树（Balanced Binary Tree）具有以下性质：它是一棵空树或它的左右两个子树的高度差的绝对值不超过1，并且左右两个子树都是一棵平衡二叉树。

平衡二叉树的常用实现方法有红黑树、AVL、替罪羊树、Treap、伸展树等。

ref:http://blog.csdn.net/yang_yulei/article/details/46005845

例题：bzoj3224/tyvj 1728/luogu 3369 普通平衡树

ref:http://hzwer.com/1712.html

1.带旋treap

Code:

#include<cstdio>
#include<cstring>
#include<algorithm>
#define nl NULL 
using namespace std;
inline int in(){
    int x=0;bool f=0; char c;
    for (;(c=getchar())<'0'||c>'9';f=c=='-');
    for (x=c-'0';(c=getchar())>='0'&&c<='9';x=(x<<3)+(x<<1)+c-'0');
    return f?-x:x;
}
int n,opt,x;
inline int rd(){
    static int x=23333;
    x^=x<<13;x^=x>>17;x^=x<<5;return x;
}
struct treap{
    treap *ls,*rs;
    int val,pri,num,siz;
    treap(int val):val(val),pri(rd()),num(1),siz(1),ls(nl),rs(nl){}
    inline void combine(){siz=(ls?ls->siz:0)+(rs?rs->siz:0)+num;}
}*rt;
inline void lturn(treap* &x){
    treap *y=x->rs;x->rs=y->ls;y->ls=x;y->siz=x->siz;x->combine();x=y;
}
inline void rturn(treap* &x){
    treap *y=x->ls;x->ls=y->rs;y->rs=x;y->siz=x->siz;x->combine();x=y;
}
inline void ins(treap* &x,int val){
    if (x==nl){x=new treap(val);return;}
    if (val==x->val) x->num++;
    else if (val>x->val) {ins(x->rs,val);if (x->rs->pri<x->pri) lturn(x);}
    else {ins(x->ls,val);if(x->ls->pri<x->pri) rturn(x);}x->combine();
}
inline void del(treap* &x,int val){
    if (x==nl) return;
    if (val==x->val){
        if (x->num>1) x->num--,x->siz--;
        else if (x->ls==nl||x->rs==nl){
            treap *t=x;
            x=(x->ls==nl)?x->rs:x->ls;delete t;
        }else if (x->ls->pri<x->rs->pri) rturn(x),del(x,val);
        else lturn(x),del(x,val);
    }else if (val<x->val) x->siz--,del(x->ls,val);
    else x->siz--,del(x->rs,val);
}
inline int qrank(treap* &x,int val){
    if (x==nl) return 0;
    if (x->ls==nl){
        if (val==x->val) return 1;
        if (val<x->val) return 0;
        return qrank(x->rs,val)+x->num;
    }if (val==x->val) return x->ls->siz+1;
    if (val<x->val) return qrank(x->ls,val);
    return qrank(x->rs,val)+x->ls->siz+x->num;
}
inline int qnum(treap *x,int val){
    if (x==nl) return 0;
    if (x->ls==nl){
        if (val<=x->num) return x->val;
        return qnum(x->rs,val-x->num);
    }if (val<=x->ls->siz) return qnum(x->ls,val);
    if (val>x->ls->siz+x->num) return qnum(x->rs,val-x->ls->siz-x->num);
    return x->val;
}
inline int qpre(treap *x,int val,int ans){
    if (x==nl) return ans;
    if (val>x->val) return qpre(x->rs,val,x->val);
    return qpre(x->ls,val,ans);
}
inline int qpost(treap *x,int val,int ans){
    if (x==nl) return ans;
    if (val<x->val) return qpost(x->ls,val,x->val);
    return qpost(x->rs,val,ans);
}
int main()
{
    n=in();for (int i=1;i<=n;++i){
        opt=in();x=in();
        switch (opt){
            case 1:ins(rt,x);break;
            case 2:del(rt,x);break;
            case 3:printf("%d\n",qrank(rt,x));break;
            case 4:printf("%d\n",qnum(rt,x));break;
            case 5:printf("%d\n",qpre(rt,x,-1));break;
            case 6:printf("%d\n",qpost(rt,x,-1));break;
        }
    }return 0;
}

无旋treap:

Code:

#include<cstdio>
#include<cstring>
#include<algorithm>
#define nl NULL 
using namespace std;
inline int in(){
    int x=0;bool f=0; char c;
    for (;(c=getchar())<'0'||c>'9';f=c=='-');
    for (x=c-'0';(c=getchar())>='0'&&c<='9';x=(x<<3)+(x<<1)+c-'0');
    return f?-x:x;
}
inline int rd(){
    static int x=23333;
    x^=x<<13,x^=x>>17,x^=x<<5;return x;
}
int n,opt,x;
struct treap{
    treap *ls,*rs;
    int val,pri,siz;
    treap(int val):val(val),pri(rd()),siz(1),ls(nl),rs(nl){}
    inline void combine(){siz=(ls?ls->siz:0)+(rs?rs->siz:0)+1;}
}*rt;
typedef pair<treap*,treap*> droot;
inline int gsize(treap *x){return x?x->siz:0;}
treap *merge(treap *a,treap *b){
    if (!a) return b;if (!b) return a;
    if (a->pri<b->pri){a->rs=merge(a->rs,b);a->combine();return a;}
    else{b->ls=merge(a,b->ls);b->combine();return b;}
}//suppose a<=b
droot split(treap *x,int k){
    if (!x) return droot(nl,nl);droot y;
    if (gsize(x->ls)>=k){y=split(x->ls,k);x->ls=y.second;x->combine();y.second=x;}
    else{y=split(x->rs,k-gsize(x->ls)-1);x->rs=y.first;x->combine();y.first=x;}return y;
}
inline int qnum(int v){
    droot x=split(rt,v-1),y=split(x.second,1);
    treap *a=y.first;rt=merge(merge(x.first,a),y.second);return a->val;
}
inline int qrank(treap *x,int v){
    if (!x) return 0;
    if (v<x->val) return qrank(x->ls,v);
    return qrank(x->rs,v)+gsize(x->ls)+1;
}
inline void ins(int v){
    droot x=split(rt,qrank(rt,v));
    treap *a=new treap(v);rt=merge(merge(x.first,a),x.second);
}
inline void del(int v){
    droot x=split(rt,qrank(rt,v)-1),y=split(x.second,1);
    treap *a=y.first;rt=merge(x.first,y.second);delete a;a=nl;    
}
inline int qpre(int v){return qnum(qrank(rt,v-1));}
inline int qsuf(int v){return qnum(qrank(rt,v)+1);}
int main()
{
    n=in();for (int i=1;i<=n;++i){
        opt=in();x=in();switch (opt){
            case 1:ins(x);break;
            case 2:del(x);break;
            case 3:printf("%d\n",qrank(rt,x-1)+1);break;
            case 4:printf("%d\n",qnum(x));break;
            case 5:printf("%d\n",qpre(x));break;
            case 6:printf("%d\n",qsuf(x));break;
        }
    }return 0;
}

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