# Review 6

* Hajin Ju, 2024062806

## Problem 1

Illustrate the operation of `PARTITION` on the array $A = \set{2, 8, 7, 1, 3, 5, 6, 4}$

### Solution 1

```python {cmd output=html hide}
from __pyfunc__.r6 import *
from bs4 import BeautifulSoup

data = [2, 8, 7, 1, 3, 5, 6, 4]
hl = [0, 0, 0, 0, 0, 0, 1, 0]

print(
    create_array_table(data, 
    [0, 0, 0, 0, 0, 0, 1, 0]
    , 0, 0),
)

print(
    create_array_table(data, 
    [1, 0, 0, 0, 0, 0, 1, 0],
    1, 0)
)

print(
    create_array_table(
    [2, 8, 7, 1, 3, 5, 6, 4], 
    [1, 1, 0, 0, 0, 0, 1, 0],
    1, 1)
)

print(create_array_table(
    [2, 8, 7, 1, 3, 5, 6, 4],
    [1, 0, 1, 0, 0, 0, 1, 0],
    1, 2
))

print(create_array_table(
    [2, 1, 7, 8, 3, 5, 6, 4],
    [0, 1, 0, 1, 0, 0, 1, 0],
    2, 2
))

print(create_array_table(
    [2, 1, 3, 8, 7, 5, 6, 4],
    [0, 0, 1, 0, 1, 0, 1, 0],
    3, 2
))

print(create_array_table(
    [2, 1, 3, 8, 7, 5, 6, 4],
    [0, 0, 1, 0, 0, 1, 1, 0],
    3, 3
))

print(create_array_table(
    [2, 1, 3, 8, 7, 5, 6, 4],
    [0, 0, 1, 0, 0, 0, 1, 0],
    3, 4
))

print(create_array_table(
    [2, 1, 3, 4, 7, 5, 6, 8],
    [0, 0, 1, 1, 0, 0, 0, 1],
    0, 0
))

```

## Problem 2

Answer questions about the following function `SELECT(A, p, r, i)` where $r-p + 1 = n$

```py
SELECT(A, p, r, i)
    if p == r
        return A[p]
    q = PARTITION(A, p, r)
    k = q - p + 1
    if i == k
        return A[q]
    else if i < k
        return SELECT(A, p, q - 1, i)
    else
        return SELECT(A, q + 1, r, i - k)
```

* What does it do?
* What is the worst-case running time?
* What is the best-case running time?


### Solution 2-a

`SELECT` selects `i`-th smallest element in the array `A`.

### Solution 2-b

The worst-case occurs when the `PARTITION` partitiend the array unbalanced.

So, time-complexity of the case is:

$$T(n) = T(n-1) + O(n)$$

$$\therefore T(n) = O(n^2)$$

### Solution 2-c

The best-case occurs when the `PARTITION` partitiend the array balanced, which means pivot is median.

So, time-complexity of the case is:

$$T(n) = T(0) + O(n)$$

$$\therefore T(n) = O(n)$$