> 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/algorithms/binary-search.md).

# 二分查找与二分答案

### 二分查找

在**有序数组**中快速查找目标值，每次比较中间元素，排除一半。时间复杂度 O(log n)。

```cpp
int binarySearch(int a[], int n, int x) {
    int l = 0, r = n - 1;
    while (l <= r) {
        int mid = (l + r) / 2;
        if (a[mid] == x) return mid;
        else if (a[mid] < x) l = mid + 1;
        else r = mid - 1;
    }
    return -1;  // 未找到
}
```

### 二分答案（二分枚举法）

当答案具有**单调性**（越大越容易/越难满足）时，二分枚举答案，再把答案代入判断是否可行。

模板（求满足条件的最大值）：

```cpp
int l = 0, r = 1e9, ans = 0;
while (l <= r) {
    int mid = l + (r - l) / 2;
    if (check(mid)) { ans = mid; l = mid + 1; }  // 可行，尝试更大
    else r = mid - 1;
}
cout << ans;
```

### 经典场景

* 最小值最大 / 最大值最小
* 分割问题、砍树问题、跳石头问题

### 要点

* 二分的前提是**单调有序**。
* `mid` 写法 `l + (r - l) / 2` 可防溢出。
* 注意边界收缩方向与最终答案的取值。


---

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