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P10638 BZOJ4355 Play with sequence Solution

article 2025/1/30 12:12:49

Description

给定 $a=(a_1,a_2,\cdots,a_n)$ ，有 $m$ 个操作，分以下三种：

$\operatorname{assign}(l,r,k)$ ：对每个 $\in[l,r]$ 执行 $a_i \leftarrow k$ .
$\operatorname{modify}(l,r,k)$ ：对每个 $\in[l,r]$ 执行 $a_i \leftarrow max(a_i+c,0)$ .
$\operatorname{query}(l,r)$ ：求 $\sum\limits_{i=l}^r [a_i=0]$ .

Limitations

$\le n,m \le 3\times 10^5$
$\le a_i \le 10^9$
$\le k \le 10^9$ （ $\operatorname{assign}$ ）
$\le 10^9$ （ $\operatorname{modify}$ ）
$1.5\text{s},512\text{MB}$

Solution

看到区间最值，想到吉司机线段树.
考虑转化每个操作：

$\operatorname{assign}$ ：来一次 $\operatorname{chmax}$ ，再来一次 $\operatorname{chmin}$ .
$\operatorname{modify}$ ：来一次 $\operatorname{add}$ ，再来一次 $\operatorname{chmax}$ .
$\operatorname{query}$ ：由于 $a_i$ 非负，只需看最小值是否为 $0$ ，再看出现次数.

接下来只需把 P10639 代码拉过来改一下即可。

Code

$5.85\text{KB},1.66\text{s},94.47\text{MB}\;\texttt{(in total, C++ 20 with O2)}$

// Problem: P10638 BZOJ4355 Play with sequence
// Contest: Luogu
// URL: https://www.luogu.com.cn/problem/P10638
// Memory Limit: 512 MB
// Time Limit: 1500 ms
// 
// Powered by CP Editor (https://cpeditor.org)

#include <bits/stdc++.h>
using namespace std;

using i64 = long long;
using ui64 = unsigned long long;
using i128 = __int128;
using ui128 = unsigned __int128;
using f4 = float;
using f8 = double;
using f16 = long double;

template<class T>
bool chmax(T &a, const T &b){
	if(a < b){ a = b; return true; }
	return false;
}

template<class T>
bool chmin(T &a, const T &b){
	if(a > b){ a = b; return true; }
	return false;
}

const i64 inf = 3e18;
struct Node {
    int l, r, cmax, cmin;
    i64 max, max2, min, min2;
    i64 tag, tmin, tmax, sum;
};
using Tree = vector<Node>;
inline int ls(int u) { return 2 * u + 1; }
inline int rs(int u) { return 2 * u + 2; }

inline void pushup(Tree& tr, int u) {
    tr[u].sum = tr[ls(u)].sum + tr[rs(u)].sum;
    if (tr[ls(u)].max == tr[rs(u)].max) {
        tr[u].max = tr[ls(u)].max;
        tr[u].cmax = tr[ls(u)].cmax + tr[rs(u)].cmax;
        tr[u].max2 = max(tr[ls(u)].max2, tr[rs(u)].max2);
    }
    else if (tr[ls(u)].max > tr[rs(u)].max) {
        tr[u].max = tr[ls(u)].max;
        tr[u].cmax = tr[ls(u)].cmax;
		tr[u].max2 = max(tr[ls(u)].max2, tr[rs(u)].max);
    }
    else {
        tr[u].max = tr[rs(u)].max;
        tr[u].cmax = tr[rs(u)].cmax;
        tr[u].max2 = max(tr[ls(u)].max, tr[rs(u)].max2);
    }
    
    if (tr[ls(u)].min == tr[rs(u)].min) {
        tr[u].min = tr[ls(u)].min;
        tr[u].cmin = tr[ls(u)].cmin + tr[rs(u)].cmin;
        tr[u].min2 = min(tr[ls(u)].min2, tr[rs(u)].min2);
    }
    else if (tr[ls(u)].min < tr[rs(u)].min) {
        tr[u].min = tr[ls(u)].min;
        tr[u].cmin = tr[ls(u)].cmin;
    	tr[u].min2 = min(tr[ls(u)].min2, tr[rs(u)].min);
    }
    else {
        tr[u].min = tr[rs(u)].min;
        tr[u].cmin = tr[rs(u)].cmin;
        tr[u].min2 = min(tr[ls(u)].min, tr[rs(u)].min2);
    }
}

inline void build(Tree& tr, int u, int l, int r, const vector<i64>& A) {
    tr[u].l = l;
    tr[u].r = r;
    tr[u].tmin = inf;
    tr[u].tmax = -inf;
    
	if (l == r) {
		tr[u].sum = tr[u].max = tr[u].min = A[l];
		tr[u].max2 = -inf;
		tr[u].min2 = inf;
		tr[u].cmax = tr[u].cmin = 1;
		return;
	}
	
	const int mid = (l + r) >> 1;
	build(tr, ls(u), l, mid, A);
	build(tr, rs(u), mid + 1, r, A);
	pushup(tr, u);
}

inline void upd_add(Tree& tr, int u, i64 val) {
    tr[u].sum += val * (tr[u].r - tr[u].l + 1);
	tr[u].max += val;
	tr[u].min += val;
	if (tr[u].max2 != -inf) tr[u].max2 += val;
	if (tr[u].min2 != inf) tr[u].min2 += val;
	if (tr[u].tmax != -inf) tr[u].tmax += val;
	if (tr[u].tmin != inf) tr[u].tmin += val;
	tr[u].tag += val;
}

inline void upd_min(Tree& tr, int u, i64 val) {
    if (tr[u].max <= val) return;
	tr[u].sum += (val - tr[u].max) * tr[u].cmax;
	if (tr[u].min2 == tr[u].max) tr[u].min2 = val;
	if (tr[u].min == tr[u].max) tr[u].min = val;
	tr[u].tmax = min(tr[u].tmax, val);
	tr[u].max = val;
	tr[u].tmin = val;
}

inline void upd_max(Tree& tr, int u, i64 val) {
    if (tr[u].min > val) return;
	tr[u].sum += (val - tr[u].min) * tr[u].cmin;
	if (tr[u].max2 == tr[u].min) tr[u].max2 = val; 
	if (tr[u].max == tr[u].min) tr[u].max = val;
	tr[u].tmin = max(tr[u].tmin, val);
	tr[u].min = val;
	tr[u].tmax = val;
}

inline void pushdown(Tree& tr, int u) {
    if (tr[u].tag) {
        upd_add(tr, ls(u), tr[u].tag);
        upd_add(tr, rs(u), tr[u].tag);
    }
    if (tr[u].tmax != -inf) {
		upd_max(tr, ls(u), tr[u].tmax);
		upd_max(tr, rs(u), tr[u].tmax);
	}
	if (tr[u].tmin != inf) {
		upd_min(tr, ls(u), tr[u].tmin);
		upd_min(tr, rs(u), tr[u].tmin);
	}
	tr[u].tag = 0;
	tr[u].tmax = -inf;
	tr[u].tmin = inf;
}

inline void add(Tree& tr, int u, int l, int r, i64 val) {
    if (l <= tr[u].l && tr[u].r <= r) return upd_add(tr, u, val);
    const int mid = (tr[u].l + tr[u].r) >> 1;
    pushdown(tr, u);
    if (l <= mid) add(tr, ls(u), l, r, val);
    if (r > mid) add(tr, rs(u), l, r, val);
    pushup(tr, u);
}

inline void chmin(Tree& tr, int u, int l, int r, i64 val) {
    if (tr[u].max <= val) return;
    if (l <= tr[u].l && tr[u].r <= r && tr[u].max2 < val) return upd_min(tr, u, val);
    const int mid = (tr[u].l + tr[u].r) >> 1;
    pushdown(tr, u);
    if (l <= mid) chmin(tr, ls(u), l, r, val);
    if (r > mid) chmin(tr, rs(u), l, r, val);
    pushup(tr, u);
}

inline void chmax(Tree& tr, int u, int l, int r, i64 val) {
    if (tr[u].min >= val) return;
    if (l <= tr[u].l && tr[u].r <= r && tr[u].min2 > val) return upd_max(tr, u, val);
    const int mid = (tr[u].l + tr[u].r) >> 1;
    pushdown(tr, u);
    if (l <= mid) chmax(tr, ls(u), l, r, val);
    if (r > mid) chmax(tr, rs(u), l, r, val);
    pushup(tr, u);
}

inline int query(Tree& tr, int u, int l, int r) {
    if (tr[u].min > 0) return 0;
	if (l <= tr[u].l && tr[u].r <= r) return tr[u].cmin;
	const int mid = (tr[u].l + tr[u].r) >> 1;
	pushdown(tr, u);
	if (r <= mid) return query(tr, ls(u), l, r);
	else if (l > mid) return query(tr, rs(u), l, r);
	else return query(tr, ls(u), l, r) + query(tr, rs(u), l, r);
}

signed main() {
	ios::sync_with_stdio(0);
	cin.tie(0), cout.tie(0);
	
	int n, m;
	scanf("%d %d", &n, &m);
	vector<i64> a(n);
	for (int i = 0; i < n; i++) scanf("%lld", &a[i]);
	Tree tr(n << 2);
	build(tr, 0, 0, n - 1, a);
	
	for (int i = 0, op, l, r, v; i < m; i++) {
	    scanf("%d %d %d", &op, &l, &r);
	    l--, r--;
	    
	    if (op == 1) {
	        scanf("%d", &v);
	        chmax(tr, 0, l, r, v);
	        chmin(tr, 0, l, r, v);
	    }
	    if (op == 2) {
	        scanf("%d", &v);
	        add(tr, 0, l, r, v);
	        chmax(tr, 0, l, r, 0);
	    }
	    if (op == 3) {
	        printf("%d\n", query(tr, 0, l, r));
	    }
	}

	return 0;
}