可持久化线段树学习笔记

可持久化线段树，即主席树。

每次修改的时候不修改原来的节点，暴力建新节点，充分运用了函数式编程的思想。

模板题：给定一个数列，\(m\) 次询问求区间 \([l,r]\) 内的第 \(k\) 大。

利用前缀和思想：

#include <bits/stdc++.h>

using namespace std;

const int MAXN = 2e5 + 5;

struct node {
    int val;
    node *lchild, *rchild;
} *rt[MAXN];
int a[MAXN], subA[MAXN], n, m, cnt = 0;

node *newNode(int val, node *lc, node *rc) {
    node *ptr = new node;
    ptr->lchild = lc; ptr->rchild = rc; ptr->val = val;
    return ptr;
}

void build(node *&cur, int l, int r) {
    if(l < r) {
        cur = newNode(0, NULL, NULL);
        int mid = (l + r) >> 1;
        build(cur->lchild, l, mid);
        build(cur->rchild, mid + 1, r);
    } else cur = newNode(0, NULL, NULL);
}

void modify(node *&cur, node *fa, int l, int r, int x) {
    cur = newNode(fa->val + 1, fa->lchild, fa->rchild);
    if(l != r) {
        int mid = (l + r) >> 1;
        if(x <= mid) modify(cur->lchild, cur->lchild, l, mid, x);
        else modify(cur->rchild, cur->rchild, mid + 1, r, x);
    }
}

int query(node *u, node *v, int l, int r, int k) {
    if(l == r) return l;
    int mid = (l + r) >> 1, lessSize = v->lchild->val - u->lchild->val;
    if(lessSize >= k)
        return query(u->lchild, v->lchild, l, mid, k);
    else return query(u->rchild, v->rchild, mid + 1, r, k - lessSize);
}

int main() {
    scanf("%d%d", &n, &m);
    for(int i = 0; i < n; i++) {
        scanf("%d", a + i);
        subA[i] = a[i];
    }
    sort(subA, subA + n);
    int size = unique(subA, subA + n) - subA;
    for(int i = 0; i < n; i++)
        a[i] = lower_bound(subA, subA + size, a[i]) - subA + 1;
    build(rt[cnt++], 1, n);
    for(int i = 0; i < n; i++) {
        modify(rt[cnt], rt[cnt - 1], 1, n, a[i]);
        cnt++;
    }
    while(m--) {
        int x, y, k;
        scanf("%d%d%d", &x, &y, &k);
        printf("%d\n", subA[query(rt[x - 1], rt[y], 1, n, k) - 1]);
    }
    return 0;
}