> 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/sorting/merge-sort.md).

# 归并排序

### 分治思想

归并排序采用**分治**：不断把数组二分，直到每段只有一个元素，再把有序的两段**合并**成一个有序段。

### 时间复杂度

无论最好、最坏、平均都是 **O(n log n)**；需要额外 O(n) 空间；**稳定**。

```cpp
int tmp[100005];

void mergeSort(int a[], int l, int r) {
    if (l >= r) return;
    int mid = (l + r) / 2;
    mergeSort(a, l, mid);      // 分
    mergeSort(a, mid + 1, r);  // 分

    int i = l, j = mid + 1, k = l;   // 治：合并
    while (i <= mid && j <= r) {
        if (a[i] <= a[j]) tmp[k++] = a[i++];  // <= 保证稳定
        else              tmp[k++] = a[j++];
    }
    while (i <= mid) tmp[k++] = a[i++];
    while (j <= r)   tmp[k++] = a[j++];
    for (int t = l; t <= r; t++) a[t] = tmp[t];
}
```

### 应用

* 求逆序对：合并时若右边元素先取，则左边剩余元素都与它构成逆序对。
* 归并排序是稳定排序的典型代表。


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