> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/data-structures/prefix-sum.md).

# 前缀和数组

### 什么是前缀和

前缀和用于**快速求区间和**。定义 `s[i]` 为数组前 i 个元素之和：

```cpp
s[i] = s[i - 1] + a[i];   // i 从 1 开始，s[0] = 0
```

### 区间和查询 O(1)

求 `a[l] + ... + a[r]`：

```cpp
sum = s[r] - s[l - 1];
```

```cpp
#include <iostream>
using namespace std;

int a[100005], s[100005];

int main() {
    int n, q;
    cin >> n >> q;
    for (int i = 1; i <= n; i++) {
        cin >> a[i];
        s[i] = s[i - 1] + a[i];      // 预处理前缀和
    }
    while (q--) {
        int l, r;
        cin >> l >> r;
        cout << s[r] - s[l - 1] << endl;   // O(1) 查询
    }
    return 0;
}
```

### 复杂度

* 预处理：O(n)
* 每次查询：O(1)
* 若不用前缀和，每次查询要 O(n)。

### 二维前缀和

```cpp
s[i][j] = s[i-1][j] + s[i][j-1] - s[i-1][j-1] + a[i][j];
// 求 (x1,y1) 到 (x2,y2) 的矩形和：
sum = s[x2][y2] - s[x1-1][y2] - s[x2][y1-1] + s[x1-1][y1-1];
```

### 与差分的关系

前缀和与**差分**是一对逆运算：前缀和用于快速**求和**，差分用于快速进行**区间加减**。


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