luoguP3835 [模板]可持久化平衡树

https://www.luogu.org/problemnew/show/P3835

因为博主精力和实力有限，学不懂 fhq treap 了，因此只介绍 leafy tree 解法

leafy tree 的本质是一颗平衡线段树，它的根节点保存整颗树的信息，是不会变的，因此可以高效的实现可持久化

#include <bits/stdc++.h>
#define update(u) if(u -> left -> size) u -> size = u -> left -> size + u -> right -> size, u -> value = u -> right -> value
#define new_Node(a, b, c, d) (&(t[cnt++] = Node(a, b, c, d)))
#define merge(a, b) new_Node(a -> size + b -> size, b -> value, a, b)
#define ratio 4
using namespace std;

const int N = 500000 + 10;
const int logN = 20;

struct Node {
    int size, value;
    Node *left, *right;
    Node () {}
    Node (int a, int b, Node *c, Node *d) : size(a), value(b), left(c), right(d) {}
}*root[N], *null, t[N * logN * 11 / 10];

int n, cnt = 0;

Node *maintain(Node *u) {
    Node *cur = new_Node(u -> size, u -> value, u -> left, u -> right);
    if(cur -> left -> size > cur -> right -> size * ratio) cur -> left = maintain(cur -> left), cur -> right = maintain(cur -> right), cur -> right = merge(cur -> left -> right, cur -> right), cur -> left = cur -> left -> left;
    if(cur -> right -> size > cur -> left -> size * ratio) cur -> left = maintain(cur -> left), cur -> right = maintain(cur -> right), cur -> left = merge(cur -> left, cur -> right -> left), cur -> right = cur -> right -> right;
    return cur;
}

Node *ins(Node *u, int x) {
    Node *cur = new_Node(u -> size, u -> value, u -> left, u -> right);
    if(cur -> size == 1) cur -> left = new_Node(1, min(cur -> value, x), null, null), cur -> right = new_Node(1, max(cur -> value, x), null, null);
    else if(x > cur -> left -> value) cur -> right = ins(cur -> right, x); else cur -> left = ins(cur -> left, x);
    update(cur); return cur;
}

Node *earse(Node *u, int x) {
    Node *cur = new_Node(u -> size, u -> value, u -> left, u -> right);
    if(u -> size == 1 && u -> value != x) return cur;
    if(cur -> left -> size == 1 && x == cur -> left -> value) *cur = *cur -> right;
    else if(cur -> right -> size == 1 && x == cur -> right -> value) *cur = *cur -> left;
    else if(x > cur -> left -> value) cur -> right = earse(cur -> right, x); else cur -> left = earse(cur -> left, x);
    update(cur); return cur;
}

int find(Node *u, int x) {
    if(u -> size == 1) return u -> value;
    return x > u -> left -> size ? find(u -> right, x - u -> left -> size) : find(u -> left, x);
}

int Rank(Node *u, int x) {
//	printf("u -> value = %d, x = %d\n", u -> value, x);
    if(u -> size == 1) return 1;
    return x > u -> left -> value ? Rank(u -> right, x) + u -> left -> size : Rank(u -> left, x);
}

int main() {
    scanf("%d", &n);
    null = new Node(0, 0, 0, 0);
    root[0] = new Node(1, INT_MAX, null, null);
    for(int i = 1; i <= n; i++) {
        int a, t, pre;
        scanf("%d %d %d", &pre, &t, &a);
        if(t == 1) root[i] = maintain(ins(root[pre], a));
        if(t == 2) root[i] = maintain(earse(root[pre], a));
        if(t == 3) printf("%d\n", Rank(root[pre], a)), root[i] = root[pre];
        if(t == 4) printf("%d\n", find(root[pre], a)), root[i] = root[pre];
        if(t == 5) {
            int k = Rank(root[pre], a) - 1;
            if(k == 0) puts("-2147483647");
            else printf("%d\n", find(root[pre], k));
            root[i] = root[pre];
        }
        if(t == 6) {
            int k = Rank(root[pre], a + 1);
            if(k == root[pre] -> size) puts("2147483647");
            else printf("%d\n", find(root[pre], k));
            root[i] = root[pre];
        }
    }
    return 0;
}

关于新建节点时写

#define new_Node(a, b, c, d) (&(*st[cnt++] = Node(a, b, c, d)))

和

#define new_Node(a, b, c, d) (&(t[cnt++] = Node(a, b, c, d)))

的区别

leafy tree 实现可持久化平衡树的时候不能高效的垃圾回收，第一种就变成废物了，第二种在可持久化时更加高效

关于旋转的时候写

Node *maintain(Node *u) {
    Node *cur = new_Node(u -> size, u -> value, u -> left, u -> right);
    if(cur -> left -> size > cur -> right -> size * ratio) cur -> left = maintain(cur -> left), cur -> right = maintain(cur -> right), cur -> right = merge(cur -> left -> right, cur -> right), st[--cnt] = cur -> left, cur -> left = cur -> left -> left;
    else if(cur -> right -> size > cur -> left -> size * ratio) cur -> left = maintain(cur -> left), cur -> right = maintain(cur -> right), cur -> left = merge(cur -> left, cur -> right -> left), st[--cnt] = cur -> right, cur -> right = cur -> right -> right;
    return cur;
}

和

Node *maintain(Node *u) {
    Node *cur = new_Node(u -> size, u -> value, u -> left, u -> right);
    if(cur -> left -> size > cur -> right -> size * ratio) cur -> right = merge(cur -> left -> right, cur -> right), cur -> left = cur -> left -> left;
    if(cur -> right -> size > cur -> left -> size * ratio) cur -> left = merge(cur -> left, cur -> right -> left), cur -> right = cur -> right -> right;
    return cur;
}

的区别

第一种情况需要在插入和删除的时候调用 maintain，是整棵树平衡，比较正常

第二种情况在每次 update 之后 maintain，在可持久化时不能保证全局平衡，可能不太优秀？（这玩意是个玄学

关于 merge 的高效实现（因为博主太菜了就咕咕咕了