> For the complete documentation index, see [llms.txt](https://garychang.gitbook.io/data-structure/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://garychang.gitbook.io/data-structure/3-search/3.2-sorting/3.2.1-insertion-sort.md).

# 3.2.1 - Insertion Sort

### 1. 演算法

將第$$i$$筆資料插入到前面$$i-1$$筆已經排好的資料當中，形成$$i$$筆排好的資料。

```
void Insert(char *arr, int index){
    int former = index-1;
    int insert = arr[index];
    while (insert < arr[former]){
        arr[former+1] = arr[former];
        former--;
    }
    arr[former+1] = insert;
}

void Sort(char *arr){
    for (int i = 1; i < strlen(arr); i++)
        Insert(arr, i);
}
```

演算法解析:

$$input = {5, 3, 8, 2, 6}$$&#x20;

$$i = 1,\ Insert(input,\ 1),\ insert=3\ while(3<5)\Rightarrow\underline{5,\ 5},\ 8,\ 2,\ 6\ input\[0]=insert\Rightarrow\ \underline{3,\ 5},\ 8,\ 2,\ 6$$&#x20;

$$i = 2,\ Insert(input,\ 2),\ insert=8\ while(8<5)\Rightarrow \times \ input\[2]=insert\Rightarrow\ 3,\ 5,\ \underline{8},\ 2,\ 6$$&#x20;

$$i = 3,\ Insert(input,\ 3),\ insert=2\ while(2<8)\Rightarrow 3,\ 5,\ 8,\ \underline{8},\ 6\quad ,former =1\  while(2<5)\Rightarrow 3,\ 5,\ \underline{5},\ 8,\ 6\quad ,former =0\ \ while(2<3)\Rightarrow 3,\ \underline{3},\ 5,\ 8,\ 6\quad ,former =-1\ \ input\[0]=insert\Rightarrow\ 2,\ 3,\ 5,\ 8,\ 6$$&#x20;

$$i = 4,\ Insert(input,\ 4),\ insert=6\ while(6<8)\Rightarrow 2,\ 3,\ 5,\ 8,\ \underline{8}\quad ,former =2\  while(6<5)\Rightarrow\times \ input\[3]=insert\Rightarrow\ 2,\ 3,\ 5,\ 8,\ 6$$&#x20;

### 2. 性質

* Time Complexity: Best Case O(n)
* Space Complexity: O(1)

Insertion sorting is a **stable** sorting method.

### 3. 其他

Insertion sort主要分成兩個步驟，

1. 找到正確的插入位置(linear search)⇒O(n)
2. 將 $$n-i$$筆資料往後挪動⇒O(n)

所以可以朝這兩個步驟來改善時間效率。

#### 3.1 Binary Insertion Sort

1. 使用Binary Search找到正確位置⇒O(logn)
2. 將 $$n-i$$筆資料往後挪動⇒O(n)

#### 3.2 Linear Insertion Sort

1. 找到正確的插入位置(linear search)⇒O(n)
2. 改用Linklist來挪動資料⇒O(1)


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