> 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/algorithm-techniques/difference-array.md).

# 差分（一维 / 二维）

### 差分的用途

差分用于**快速进行区间加 / 减**，与前缀和互为逆运算。

定义差分数组 `d[i] = a[i] - a[i-1]`。对区间 `[l, r]` 全部加 x，只需：

```cpp
d[l] += x;
d[r+1] -= x;
```

最后对 d 求前缀和即可还原 a。

```cpp
// 区间 [l, r] 加 x
d[l] += x; d[r + 1] -= x;

// 还原
for (int i = 1; i <= n; i++) {
    d[i] += d[i - 1];
    a[i] = d[i];
}
```

### 一维示例

原数组 a = \[0,0,0,0,0]，操作「2\~4 加 3」：

```
d[2] += 3, d[5] -= 3
前缀和后：0 3 3 3 3 0
```

### 二维差分

对矩形 `(x1,y1)~(x2,y2)` 加 x：

```cpp
d[x1][y1] += x;
d[x1][y2+1] -= x;
d[x2+1][y1] -= x;
d[x2+1][y2+1] += x;
// 还原时做二维前缀和
```

### 对比前缀和

|     | 前缀和     | 差分      |
| --- | ------- | ------- |
| 作用  | 区间求和    | 区间加减    |
| 预处理 | O(n)    | O(n)    |
| 操作  | 查询 O(1) | 修改 O(1) |

两者互为逆运算，常配合使用。


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