# Review 1

* Hajin Ju, 2024062806

## Problem 1

1
Fill in the blank entries when the numbers are sorted by insertion sort in non-decreasing order.

```python { cmd, output='html' hide }
from bs4 import BeautifulSoup as bs

data = [
[(7, False), (4, False), (3, False), (6, False), (8, False), (1, False), (2, False)],
[(7, True), (4, False), (3, False), (6, False), (8, False), (1, False), (2, False)],
[(4, True), (7, True), None, None, None, None, None],
]

soup = bs("<div></div>", 'lxml')
td_style = "border: 1px solid black; padding: 10px; text-align: center; font-size: 1.2rem; min-width: 3rem;"
td_style_high = "border: 1px solid black; padding: 10px; text-align: center; font-size: 1.2rem; min-width: 3rem; background-color: lightgray;"

for i in range(3):
    table = soup.new_tag("table")
    table["style"] = "display: flex; justify-content: center;"
    tr = soup.new_tag("tr")
    for d in data[i]:
        td = soup.new_tag("td")
        if d:
            td['style'] = td_style_high if d[1] else td_style
            td.string = str(d[0]) if d else " "
        else:
            td["style"] = td_style
            td.string = " "
        tr.append(td)
    table.append(tr)
    soup.div.append(table)

print(soup.prettify())
```

### Solution 1

```python { cmd, output='html', hide }
from bs4 import BeautifulSoup as bs

data = [
[(7, False), (4, False), (3, False), (6, False), (8, False), (1, False), (2, False)],
]

# insertion sorting process

for i in range(len(data[0])):
    cur =  data[-1].copy()
    cur[i] = (cur[i][0], True)
    key = cur[i][0]
    while i > 0:
        i -= 1
        if key >= cur[i][0]:
            break
        else:
            temp = cur[i + 1]
            cur[i + 1] = cur[i]
            cur[i] = temp
        
    data.append(cur)
        


soup = bs("<div></div>", 'lxml')
td_style = "border: 1px solid black; padding: 10px; text-align: center; font-size: 1.2rem; min-width: 3rem;"
td_style_high = "border: 1px solid black; padding: 10px; text-align: center; font-size: 1.2rem; min-width: 3rem; background-color: lightgray;"

for i in range(len(data)):
    table = soup.new_tag("table")
    table["style"] = "display: flex; justify-content: center;"
    tr = soup.new_tag("tr")
    for d in data[i]:
        td = soup.new_tag("td")
        if d:
            td['style'] = td_style_high if d[1] else td_style
            td.string = str(d[0]) if d else " "
        else:
            td["style"] = td_style
            td.string = " "
        tr.append(td)
    table.append(tr)
    soup.div.append(table)

print(soup.prettify())
```


## Problem 2

Fill in the blanks with proper number of iterations.

### Solution 2

```
// INSERTION-SORT(A)             // cost // number of iterations
for j = 2 to n                   // c1   // n - 1
    key = A[j]                   // c2   // n - 1
    i = j - 1                    // c3   // n - 1
    while i > 0 and A[i] > key   // c4   // tj
        A[i + 1] = A[i]          // c5   // tj - 1
        i = i - 1                // c6   // tj - 1
    A[i + 1] = key               // c7   // n - 1
```

## Problem 3

What is the running time of insertion sort when the input size is $n$?

### Solution 3

* best case: $\theta(n)$ $t_j = 1$

* worst case: $\theta(n^2)$ $t_j = j$