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;
}