> 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/discretization.md).

# 离散化

### 为什么离散化

数值很大（坐标可达 10⁹）但个数很少（10⁵）时，把数值映射成**相对大小**，用数组下标处理。

### 步骤

1. 收集出现的数值，排序、去重；
2. 用二分把每个值映射为排名（从 1 开始）。

```cpp
vector<int> xs;   // 所有需要离散化的值
sort(xs.begin(), xs.end());
xs.erase(unique(xs.begin(), xs.end()), xs.end());

int getId(int v) {
    return lower_bound(xs.begin(), xs.end(), v) - xs.begin() + 1;
}
```

### 例子

坐标 \[1000000000, 5, 42] → 排序去重 \[5, 42, 1000000000] → 映射为 1, 2, 3。

### 应用

大坐标的区间统计、前缀和、差分；值域大的计数；树状数组 / 线段树。


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