Splay树简单操作

---

前几天刚刚自学了一下splay，发现思路真简单实现起来好麻烦

---

先贴一下头文件

# include <stdio.h>
# include <stdlib.h>
# include <iostream>
# include <string.h>
# define ll long long
# define RG register//卡常
# define IL inline//再卡常
# define UN unsigned
# define mem(a, b) memset(a, b, sizeof(a))
# define min(a, b) ((a) < (b)) ? (a) : (b)
# define max(a, b) ((a) > (b)) ? (a) : (b)
using namespace std;

• 核心旋转操作 Splay操作（之后的操作基本上都要用到）
  1.当旋转的节点为爷爷节点的左儿子的左儿子时
  进行两次右旋操作，先转父亲，再转自己
  2.当旋转的节点为爷爷节点的左儿子的右儿子时
  进行一次左旋操作，一次右旋操作，都转自己
  3.如果要转到的点的为爷爷节点的右儿子，直接左旋
  4.剩下两种情况把以上两种情况左右互换即可
  贴一段代码

IL void Rot(RG tree *x, RG int i){ //0为左旋，1为右旋，0为左儿子，1为右儿子
    RG tree *y = x -> fa;
    y -> ch[!i] = x -> ch[i];
    if(x -> ch[i] != NULL) x -> ch[i] -> fa = y;
    x -> fa = y -> fa;
    if(y -> fa != NULL)
        if(y -> fa -> ch[0] == y) y -> fa -> ch[0] = x;
        else y -> fa -> ch[1] = x;
    x -> ch[i] = y; y -> fa = x;
    if(y == root) root = x;
}

IL void Splay(RG tree *x, RG tree *f){
    while(x -> fa != f)
        if(x -> fa -> fa == f)
            if(x == x -> fa -> ch[0]) Rot(x, 1);
            else Rot(x, 0);
        else{
            RG tree *y = x -> fa, *z = y -> fa;
            if(z -> ch[0] == y)
                if(y -> ch[0] == x) Rot(y, 1), Rot(x, 1);
                else Rot(x, 0), Rot(x, 1);
            else
                if(y -> ch[1] == x) Rot(y, 0), Rot(x, 0);
                else Rot(x, 1), Rot(x, 0);
        }
}

• 插入操作
  和二叉排序树一样，只不过弄完后把它Splay到根（不要问为什么）
  丑陋的代码

IL void Insert(RG int num){
    RG tree *x = root;
    if(x == NULL){
        x = new tree;
        x -> val = num;
        root = x;
    }
    else while(2333)
        if(num < x -> val){
            if(x -> ch[0] == NULL){
                x -> ch[0] = new tree;
                x -> ch[0] -> fa = x;
                x = x -> ch[0];
                x -> val = num;
                break;
            }
            x = x -> ch[0];
        }
        else if(num > x -> val){
            if(x -> ch[1] == NULL){
                x -> ch[1] = new tree;
                x -> ch[1] -> fa = x;
                x = x -> ch[1];
                x -> val = num;
                break;
            }
            x = x -> ch[1];
        }
    Splay(x, NULL);
}

• 查找
  与二叉排序树一样

IL void Find(RG int num){
    RG tree *x = root; 
    while(x -> val != num)
        if(num < x -> val) x = x -> ch[0];
        else x = x -> ch[1];
    Splay(x, NULL);
}

• 查找前驱和后缀
  前驱，跳到它的左儿子再不停地跳右儿子
  后继，跳到它的右儿子再不停地跳左儿子

IL void Findmx(RG tree *x, RG tree *f, RG int i){ //0表示后继，1表示前驱，f为该节点，x为它的左或右儿子1
    while(x -> ch[i] != NULL) x = x -> ch[i];
    Splay(x, f);
}

• 删除操作
  先找到数字的位置，Splay到根，删掉它，找它左子树中的最大数（前驱）Splay到它下面作为新的根，连接右子树即可
  代码

IL void Delete(RG int num){
    Find(num);
    RG tree *x = root;
    else if(x -> ch[0] == NULL || x -> ch[1] == NULL)
        if(x -> ch[0] != NULL) root = x -> ch[0], root -> fa = NULL;
        else if(x -> ch[1] != NULL) root = x -> ch[1], root -> fa = NULL;
        else root = NULL;
    else{
        Findmx(x -> ch[0], x, 1);
        root = x -> ch[0];
        root -> fa = NULL;
        root -> ch[1] = x -> ch[1];
        if(root -> ch[1] != NULL) root -> ch[1] -> fa = root;
    }
}

• 查找某数的排名
  实行查找操作，排名就是他左子树大小加一

IL int Size(RG tree *x){
    return (x == NULL) ? 0 : x -> size + 1;
}

IL int Rank(RG int num){
    Find(num);
    return Size(root -> ch[0]) + 1;
}

• 查找排名为k的数
  若大于当前的左子树大小加一，跳右儿子，k -= 左子树大小加一；
  否则跳右节点

IL int Pos(RG int num){
    RG tree *x = root;
    while(2333){
        RG int l = Size(x -> ch[0]);
        if(num == l + 1) break;
        if(num <= l) x = x -> ch[0];
        else{
            num -= (l + 1);
            x = x -> ch[1];
        }
    }
    return x -> val;
}

以上就是基本操作

• 关于更新
  如子树大小

IL int Size(RG tree *x){
    return (x == NULL) ? 0 : x -> size + 1;
}

IL void Updata(RG tree *x){
    if(x == NULL) return;
    x -> size = Size(x -> ch[0]) + Size(x -> ch[1]);
}

IL void Rot(RG tree *x, RG int i){ //0为左旋，1为右旋 
    RG tree *y = x -> fa;
    y -> ch[!i] = x -> ch[i];
    if(x -> ch[i] != NULL) x -> ch[i] -> fa = y;
    x -> fa = y -> fa;
    if(y -> fa != NULL)
        if(y -> fa -> ch[0] == y) y -> fa -> ch[0] = x;
        else y -> fa -> ch[1] = x;
    x -> ch[i] = y; y -> fa = x;
    Updata(y); //大佬说写在这里
    if(y == root) root = x;
}

IL void Splay(RG tree *x, RG tree *f){
    while(x -> fa != f)
        if(x -> fa -> fa == f)
            if(x == x -> fa -> ch[0]) Rot(x, 1);
            else Rot(x, 0);
        else{
            RG tree *y = x -> fa, *z = y -> fa;
            if(z -> ch[0] == y)
                if(y -> ch[0] == x) Rot(y, 1), Rot(x, 1);
                else Rot(x, 0), Rot(x, 1);
            else
                if(y -> ch[1] == x) Rot(y, 0), Rot(x, 0);
                else Rot(x, 1), Rot(x, 0);
        }
    Updata(x); //大佬说要写在这里
}

简单的栗子：
链接bzoj3224
请读者思考2分钟
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
。
好了直接看代码
代码调了好久QAQ

# include <stdio.h>
# include <stdlib.h>
# include <iostream>
# include <string.h>
# define ll long long
# define RG register
# define IL inline
# define UN unsigned
# define mem(a, b) memset(a, b, sizeof(a))
# define min(a, b) ((a) < (b)) ? (a) : (b)
# define max(a, b) ((a) > (b)) ? (a) : (b)
using namespace std;

IL int Get(){
    char c = '!'; int z = 1, num = 0;
    while(c != '-' && (c < '0' || c > '9'))
        c = getchar();
    if(c == '-')
        z = -1, c = getchar();
    while(c >= '0' && c <= '9')
        num = num * 10 + c - '0', c = getchar();
    return num * z;
}

struct tree{
    tree *fa, *ch[2];
    int val, tot, size;//tot用来判断重复的数
    IL tree(){
        fa = ch[0] = ch[1] = NULL;
        size = val = tot = 0;
    }
} *root;

IL int Size(RG tree *x){
    return (x == NULL) ? 0 : x -> size + x -> tot + 1;
}

IL void Updata(RG tree *x){
    if(x == NULL) return;
    x -> size = Size(x -> ch[0]) + Size(x -> ch[1]);
}

IL void Rot(RG tree *x, RG int i){ //0为左旋，1为右旋 
    RG tree *y = x -> fa;
    y -> ch[!i] = x -> ch[i];
    if(x -> ch[i] != NULL) x -> ch[i] -> fa = y;
    x -> fa = y -> fa;
    if(y -> fa != NULL)
        if(y -> fa -> ch[0] == y) y -> fa -> ch[0] = x;
        else y -> fa -> ch[1] = x;
    x -> ch[i] = y; y -> fa = x; Updata(y);
    if(y == root) root = x;
}

IL void Splay(RG tree *x, RG tree *f){
    while(x -> fa != f)
        if(x -> fa -> fa == f)
            if(x == x -> fa -> ch[0]) Rot(x, 1);
            else Rot(x, 0);
        else{
            RG tree *y = x -> fa, *z = y -> fa;
            if(z -> ch[0] == y)
                if(y -> ch[0] == x) Rot(y, 1), Rot(x, 1);
                else Rot(x, 0), Rot(x, 1);
            else
                if(y -> ch[1] == x) Rot(y, 0), Rot(x, 0);
                else Rot(x, 1), Rot(x, 0);
        }
    Updata(x);
}

IL void Insert(RG int num){
    RG tree *x = root;
    if(x == NULL){
        x = new tree;
        x -> val = num;
        root = x;
    }
    else if(num == x -> val) x -> tot++, root = x;
    else while(2333)
        if(num < x -> val){
            if(x -> ch[0] == NULL){
                x -> ch[0] = new tree;
                x -> ch[0] -> fa = x;
                x = x -> ch[0];
                x -> val = num;
                break;
            }
            x = x -> ch[0];
        }
        else if(num > x -> val){
            if(x -> ch[1] == NULL){
                x -> ch[1] = new tree;
                x -> ch[1] -> fa = x;
                x = x -> ch[1];
                x -> val = num;
                break;
            }
            x = x -> ch[1];
        }
        else{
            x -> tot++;
            break;
        }
    Splay(x, NULL);
}

IL void Find(RG int num){
    RG tree *x = root; 
    while(x -> val != num)
        if(num < x -> val) x = x -> ch[0];
        else x = x -> ch[1];
    Splay(x, NULL);
}

IL void Findmx(RG tree *x, RG tree *f, RG int i){
    while(x -> ch[i] != NULL) x = x -> ch[i];
    Splay(x, f);
}

IL void Delete(RG int num){
    Find(num);
    RG tree *x = root;
    if(root -> tot) root -> tot--;
    else if(x -> ch[0] == NULL || x -> ch[1] == NULL)
        if(x -> ch[0] != NULL) root = x -> ch[0], root -> fa = NULL;
        else if(x -> ch[1] != NULL) root = x -> ch[1], root -> fa = NULL;
        else root = NULL;
    else{
        Findmx(x -> ch[0], x, 1);
        root = x -> ch[0];
        root -> fa = NULL;
        root -> ch[1] = x -> ch[1];
        if(root -> ch[1] != NULL) root -> ch[1] -> fa = root;
    }
}

IL int Rank(RG int num){
    Find(num);
    return Size(root -> ch[0]) + 1;
}

IL int Pos(RG int num){
    RG tree *x = root;
    while(2333){
        RG int l = Size(x -> ch[0]);
        if(num > l && num <= l + x -> tot + 1) break;
        if(num <= l) x = x -> ch[0];
        else{
            num -= (l + x -> tot + 1);
            x = x -> ch[1];
        }
    }
    return x -> val;
}

int main(){
    RG int n = Get(), opt, x;
    while(n--){
        opt = Get(); x = Get();
        if(opt == 1) Insert(x);
        if(opt == 2) Delete(x);
        if(opt == 3) printf("%d
", Rank(x));
        if(opt == 4) printf("%d
", Pos(x));
        //找前驱后缀：插入数后再删除，显然有更快的（不想打其他方法了，反正能过）
        if(opt == 5){
            Insert(x);
            Findmx(root -> ch[0], NULL, 1);
            printf("%d
", root -> val);
            Delete(x);
        }
        if(opt == 6){
            Insert(x);
            Findmx(root -> ch[1], NULL, 0);
            printf("%d
", root -> val);
            Delete(x);
        }
    }
    return 0;
}

• 关于区间操作
  一张丑陋的图
  
  把l-1splay到根，r+1splay到根的右儿子，则图中那个丑陋的子树就是要求的[l，r]了。

• 删除区间
  直接断开边（显然浪费空间，自己想办法目前没遇到MLE的情况）

• 翻转区间
  用类似于线段树的懒懒的lazy标记，每次Find，splay等操作时把标记下放，更换两个子树

又一个栗子
链接bzoj3223
再思考两分钟
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
，
看代码

# include <stdio.h>
# include <stdlib.h>
# include <iostream>
# include <string.h>
# define ll long long
# define RG register
# define IL inline
# define UN unsigned
# define mem(a, b) memset(a, b, sizeof(a))
# define min(a, b) ((a) < (b)) ? (a) : (b)
# define max(a, b) ((a) > (b)) ? (a) : (b)
using namespace std;

IL int Get(){
    char c = '!'; int z = 1, num = 0;
    while(c != '-' && (c < '0' || c > '9'))
        c = getchar();
    if(c == '-')
        z = -1, c = getchar();
    while(c >= '0' && c <= '9')
        num = num * 10 + c - '0', c = getchar();
    return num * z;
}

struct tree{
    tree *fa, *ch[2];
    int size, lazy, pos;
    IL tree(){
        fa = ch[0] = ch[1] = NULL;
        pos = lazy = size = 0;
    }
    IL void Pushdown(){//下放
        if(!lazy) return;
        lazy = 0;
        if(ch[1] == NULL && ch[0] == NULL) return;
        if(ch[0] != NULL) ch[0] -> lazy ^= 1;
        if(ch[1] != NULL) ch[1] -> lazy ^= 1;
        swap(ch[0], ch[1]);
    }
} *root;
int n;

IL int Size(RG tree *x){
    return (x == NULL) ? 0 : x -> size + 1;
}

IL void Updata(RG tree *x){
    if(x == NULL) return;
    x -> size = Size(x -> ch[0]) + Size(x -> ch[1]);
}

IL void Dfs(RG tree *x){
    if(x == NULL) return;
    x -> Pushdown();
    Dfs(x -> ch[0]);
    if(x -> pos && x -> pos <= n)
        printf("%d ", x -> pos);
    Dfs(x -> ch[1]);
}

IL void Rot(RG tree *x, RG int i){ //0为左旋，1为右旋
    RG tree *y = x -> fa;
    x -> Pushdown(); y -> Pushdown();
    y -> ch[!i] = x -> ch[i];
    if(x -> ch[i] != NULL) x -> ch[i] -> fa = y;
    x -> fa = y -> fa;
    if(y -> fa != NULL)
        if(y -> fa -> ch[0] == y) y -> fa -> ch[0] = x;
        else y -> fa -> ch[1] = x;
    x -> ch[i] = y; y -> fa = x; Updata(y);
    if(y == root) root = x;
}

IL void Splay(RG tree *x, RG tree *f){
    while(x -> fa != f){
        x -> Pushdown();
        if(x -> fa -> fa == f)
            if(x == x -> fa -> ch[0]) Rot(x, 1);
            else Rot(x, 0);
        else{
            RG tree *y = x -> fa, *z = y -> fa;
            if(z -> ch[0] == y)
                if(y -> ch[0] == x) Rot(y, 1), Rot(x, 1);
                else Rot(x, 0), Rot(x, 1);
            else
                if(y -> ch[1] == x) Rot(y, 0), Rot(x, 0);
                else Rot(x, 1), Rot(x, 0);
        }
    }
     Updata(x);
}

IL tree *Build(RG int l, RG int r, RG tree *f){
    RG int mid = (l + r) >> 1;
    tree *x = new tree;
    x -> pos = mid;
    x -> fa = f;
    if(mid > l) x -> ch[0] = Build(l, mid - 1, x);
    if(mid < r) x -> ch[1] = Build(mid + 1, r, x);
    Updata(x);
    return x;
}

IL void Find(RG int num, RG tree *f){
    RG tree *x = root;
    while(2333){
        x -> Pushdown();
        RG int l = Size(x -> ch[0]);
        if(num < l) x = x -> ch[0];
        else if(num == l) break;
        else{
            num -= (l + 1);
            x = x -> ch[1];
        }
    }
    Splay(x, f);
}

IL void Turn(){
    RG int l = Get(), r = Get();
    Find(l - 1, NULL); Find(r + 1, root);
    root -> ch[1] -> ch[0] -> lazy ^= 1;
}

int main(){
    n = Get();
    RG int m = Get();
    root = Build(0, n + 1, NULL);
    while(m--) Turn();
    Dfs(root);
    printf("
");
    return 0;
}

解释或代码错误还请大佬指出，本蒟蒻一定会改