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区间查询最值，区间和

#include<bits/stdc++.h>
using namespace std;
#define io cin.tie(0);ios::sync_with_stdio(false);
#define debug(x) cout<<#x<<"="<<x<<endl
#define lowbit(x) x&(-x)
#define pii pair<int,int>
#define mk make_pair
#define ll long long
#define rs p<<1|1
#define ls p<<1
const int maxn = 1e6 + 5;
const int mod = 1e9 + 7;
const int inf = 0x3f3f3f3f;
inline ll read(){
    ll p=0,f=1;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){p=(p<<1)+(p<<3)+(c^48),c=getchar();}
    return f*p;
}

struct node{
    int l, r;
    int lazy, sum, mx; //懒标记，区间和，区间最大值
}tree[maxn << 2];
int a[maxn];
void pushup(int p){ //向下更新
    tree[p].sum = tree[ls].sum + tree[rs].sum;
    tree[p].mx = max(tree[ls].mx, tree[rs].mx);
}
void pushdown(int p){ //向上更新
    if(tree[p].lazy){
        int num = tree[p].lazy;
        tree[ls].lazy = tree[rs].lazy = num;
        tree[ls].sum = num * (tree[ls].r - tree[ls].l + 1);
        tree[rs].sum = num * (tree[rs].r - tree[rs].l + 1);
        tree[ls].mx += num;
        tree[rs].mx += num;
        tree[p].lazy = 0;
    }
}
void build(int p, int l, int r){ //建树
    tree[p].l = l; tree[p].r = r;
    if(l == r){
        tree[p].sum = a[l];
        tree[p].mx = a[l];
        tree[p].lazy = 0;
        return;
    }
    int mid = l + r >> 1;
    build(ls, l, mid);
    build(rs, mid + 1, r);
    pushup(p);
}
void update(int p, int l, int r, int val){ //区间的每个值加val
    if(l <= tree[p].l && tree[p].r <= r){
        tree[p].sum += val * (tree[p].r - tree[p].l + 1);
        tree[p].mx += val;
        tree[p].lazy += val;
        return;
    }
    pushdown(p);
    int mid = tree[p].l + tree[p].r >> 1;
    if(mid >= l) update(ls, l, r, val);
    if(mid + 1 <= r) update(rs, l, r, val);
    pushup(p);
}
int query_sum(int p, int l, int r){ //查询区间和
    if(l <= tree[p].l && tree[p].r <= r)
        return tree[p].sum;
    pushdown(p);
    int ans = 0;
    int mid = tree[p].l + tree[p].r >> 1;
    if(mid >= l) ans += query_sum(ls, l, r);
    if(mid + 1 <= r) ans += query_sum(rs, l, r);
    return ans;
}
int query_max(int p, int l, int r){ //查询区间最大值
    if(l <= tree[p].l && tree[p].r <= r)
        return tree[p].mx;
    pushdown(p);
    int ans = -inf;
    int mid = tree[p].l + tree[p].r >> 1;
    if(mid >= l) ans = max(ans, query_max(ls, l, r));
    if(mid + 1 <= r) ans = max(ans, query_max(rs, l, r));
    return ans;
}
int main(){
    int n, q; scanf("%d%d", &n, &q);
    for(int i = 1; i <= n; i ++){
        scanf("%d", &a[i]);
    }
    build(1, 1, n);
    while(q --){
        int l, r;
        scanf("%d%d", &l, &r);
        printf("sum = %d\n", query_sum(1, l, r));
        printf("max = %d\n", query_max(1, l, r));
    }
    return 0;
}
/*
5 3
2 4 3 5 1
1 5
2 3
3 4
*/

例题

区间最值（思维）

 

先乘后加求区间和

struct node{
    int l, r;
    ll sum, add, mut;
}tree[maxn << 2];
ll a[maxn], mod;
void pushup(int p){
    tree[p].sum = (tree[ls].sum + tree[rs].sum) % mod;
}
void pushdown(int p){
    if(tree[p].add || tree[p].mut != 1){
        ll add = tree[p].add, mut = tree[p].mut;
        tree[ls].add = (tree[ls].add * mut % mod + add) % mod;
        tree[rs].add = (tree[rs].add * mut % mod + add) % mod;
        tree[ls].mut = tree[ls].mut * mut % mod;
        tree[rs].mut = tree[rs].mut * mut % mod;
        tree[ls].sum = (tree[ls].sum * mut % mod + add * (tree[ls].r - tree[ls].l + 1) % mod) % mod;
        tree[rs].sum = (tree[rs].sum * mut % mod + add * (tree[rs].r - tree[rs].l + 1) % mod) % mod;
        tree[p].add = 0, tree[p].mut = 1;
    }
}
void build(int p, int l, int r){
    tree[p].l = l, tree[p].r = r;
    tree[p].add = 0, tree[p].mut = 1;
    if(l == r){
        tree[p].sum = a[l];
        return;
    }
    int mid = l + r >> 1;
    build(ls, l, mid);
    build(rs, mid + 1, r);
    pushup(p);
}
void update(int p, int l, int r, int k, int op){
    if(l <= tree[p].l && tree[p].r <= r){
        if(op == 1) { //加
            tree[p].add = (tree[p].add + k) % mod;
            tree[p].sum = (tree[p].sum + 1ll * k * (tree[p].r - tree[p].l + 1) % mod) % mod; 
        }else if(op == 2){ //乘
            tree[p].add = tree[p].add * k % mod;
            tree[p].mut = tree[p].mut * k % mod;
            tree[p].sum = tree[p].sum * k % mod;
        }
        return;
    }
    int mid = tree[p].l + tree[p].r >> 1;
    pushdown(p);
    if(mid >= l) update(ls, l, r, k, op);
    if(mid + 1 <= r) update(rs, l, r, k, op);
    pushup(p);
}
ll query(int p, int l, int r){
    if(l <= tree[p].l && tree[p].r <= r) return tree[p].sum;
    pushdown(p);
    int mid = tree[p].l + tree[p].r >> 1;
    ll ans = 0;
    if(mid >= l) ans = (ans + query(ls, l, r)) % mod;
    if(mid + 1 <= r) ans = (ans + query(rs, l, r)) % mod;
    return ans;
}

例题

动态序列