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# Review 1
* Hajin Ju, 2024062806
## Problem 1
Fill in the blanks when the numbers are sorted by merge sort in non-decreasing order.
```python {cmd, output='html' hide}
from bs4 import BeautifulSoup
def create_merge_sort_html_no_arrows():
"""
BeautifulSoup를 사용하여 화살표가 없는 병합 정렬 시각화 HTML을 생성합니다.
"""
soup = BeautifulSoup("", "html.parser")
# --- 기본 스타일 정의 ---
STYLE_CONTAINER = "display: flex; flex-direction: column-reverse; align-items: center; gap: 30px; font-family: sans-serif;"
STYLE_LEVEL = "display: flex; justify-content: center; align-items: flex-end; gap: 20px; position: relative;"
STYLE_LEVEL_LABEL = "position: absolute; left: -80px; top: 50%; transform: translateY(-50%); font-size: 1.2em; color: rgb(51, 51, 51);"
STYLE_BOX_GROUP = "display: flex; position: relative;"
STYLE_BOX_BASE = "width: 40px; height: 40px; border: 2px solid rgb(85, 85, 85); display: flex; justify-content: center; align-items: center; font-size: 1.2em;"
STYLE_BOX_EMPTY = f"{STYLE_BOX_BASE} background-color: rgb(255, 255, 255);"
STYLE_BOX_FILLED = f"{STYLE_BOX_BASE} background-color: rgb(204, 204, 204); font-weight: bold;"
# --- 메인 컨테이너 생성 ---
container = soup.new_tag('div', attrs={'style': STYLE_CONTAINER})
soup.append(container)
# --- Level 0 (초기 숫자 배열) ---
level0 = soup.new_tag('div', attrs={'style': STYLE_LEVEL})
numbers = [6, 3, 5, 8, 1, 7, 4, 2]
for num in numbers:
box_group = soup.new_tag('div', attrs={'style': STYLE_BOX_GROUP})
box = soup.new_tag('div', attrs={'style': STYLE_BOX_FILLED})
box.string = str(num)
box_group.append(box)
level0.append(box_group)
container.append(level0)
# --- Level 1, 2, 3 생성 함수 ---
def create_level(level_num, groups_data):
level = soup.new_tag('div', attrs={'style': STYLE_LEVEL})
label = soup.new_tag('div', attrs={'style': STYLE_LEVEL_LABEL})
label.string = "merge"
level.append(label)
for data in groups_data:
num_boxes = data['boxes']
box_group = soup.new_tag('div', attrs={'style': STYLE_BOX_GROUP})
for i in range(num_boxes):
style = STYLE_BOX_EMPTY
if i > 0: style += " border-left: none;"
box = soup.new_tag('div', attrs={'style': style})
box_group.append(box)
level.append(box_group)
return level
# --- Level 1, 2, 3 데이터 및 생성 ---
level1_data = [{'boxes': 2}, {'boxes': 2}, {'boxes': 2}, {'boxes': 2}]
container.append(create_level(1, level1_data))
level2_data = [{'boxes': 4}, {'boxes': 4}]
container.append(create_level(2, level2_data))
level3_data = [{'boxes': 8}]
container.append(create_level(3, level3_data))
return container.prettify()
# --- 스크립트 실행 ---
if __name__ == "__main__":
final_html = create_merge_sort_html_no_arrows()
print(final_html)
```
### Solution 1
```python {cmd output=html hide}
from bs4 import BeautifulSoup
import math
# --- 실제 병합 정렬을 수행하고 중간 과정을 기록하는 함수 ---
def get_merge_sort_steps(data):
steps = {}
n = len(data)
# 원본 배열 복사하여 단계별로 정렬 진행
current_array = list(data)
# step_size는 1, 2, 4 순서로 증가 (합병되는 배열의 크기)
step_size = 1
level = 1
while step_size < n:
level_results = []
for i in range(0, n, step_size * 2):
left_start = i
left_end = i + step_size
right_start = i + step_size
right_end = min(i + step_size * 2, n)
# 두 부분 배열을 합병하고 정렬
merged = sorted(current_array[left_start:right_end])
# 정렬된 결과를 현재 배열에 다시 반영
current_array[left_start:right_end] = merged
level_results.extend(merged)
steps[f'level_{level}'] = level_results
step_size *= 2
level += 1
return steps
# --- HTML 생성을 위한 메인 함수 ---
def create_populated_merge_sort_html(data):
soup = BeautifulSoup("", "html.parser")
# 병합 정렬 실행하여 각 단계별 결과 얻기
sort_steps = get_merge_sort_steps(data)
# --- 기본 스타일 정의 (이전과 동일) ---
STYLE_CONTAINER = "display: flex; flex-direction: column-reverse; align-items: center; gap: 30px; font-family: sans-serif;"
STYLE_LEVEL = "display: flex; justify-content: center; align-items: flex-end; gap: 20px; position: relative;"
STYLE_LEVEL_LABEL = "position: absolute; left: -80px; top: 50%; transform: translateY(-50%); font-size: 1.2em; color: rgb(51, 51, 51);"
STYLE_BOX_GROUP = "display: flex; position: relative;"
STYLE_BOX_BASE = "width: 40px; height: 40px; border: 2px solid rgb(85, 85, 85); display: flex; justify-content: center; align-items: center; font-size: 1.2em;"
STYLE_BOX_EMPTY = f"{STYLE_BOX_BASE} background-color: rgb(255, 255, 255);"
STYLE_BOX_FILLED = f"{STYLE_BOX_BASE} background-color: rgb(204, 204, 204); font-weight: bold;"
container = soup.new_tag('div', attrs={'style': STYLE_CONTAINER})
soup.append(container)
# --- Level 0 (초기 데이터 배열) ---
level0 = soup.new_tag('div', attrs={'style': STYLE_LEVEL})
for num in data:
box_group = soup.new_tag('div', attrs={'style': STYLE_BOX_GROUP})
box = soup.new_tag('div', attrs={'style': STYLE_BOX_FILLED})
box.string = str(num)
box_group.append(box)
level0.append(box_group)
container.append(level0)
# --- Level 1, 2, 3 생성 및 채우기 ---
num_levels = int(math.log2(len(data)))
for i in range(1, num_levels + 1):
level = soup.new_tag('div', attrs={'style': STYLE_LEVEL})
label = soup.new_tag('div', attrs={'style': STYLE_LEVEL_LABEL})
label.string = "merge"
level.append(label)
# 해당 레벨의 정렬 결과 가져오기
level_data = sort_steps[f'level_{i}']
group_size = 2**i
# 결과를 그룹 크기에 맞게 나누어 박스 생성
for j in range(0, len(level_data), group_size):
chunk = level_data[j:j + group_size]
box_group = soup.new_tag('div', attrs={'style': STYLE_BOX_GROUP})
for k, num in enumerate(chunk):
style = STYLE_BOX_EMPTY
if k > 0: style += " border-left: none;"
box = soup.new_tag('div', attrs={'style': style})
box.string = str(num)
box_group.append(box)
level.append(box_group)
container.append(level)
return container.prettify()
# --- 스크립트 실행 ---
if __name__ == "__main__":
# 정렬할 8개의 원소를 가진 데이터 배열
data_to_sort = [6, 3, 5, 8, 1, 7, 4, 2]
final_html = create_populated_merge_sort_html(data_to_sort)
print(final_html)
```
## Problem 2
Fill in the blanks with proper asymptotic running times. Assume that `p = 1`, `r = n`
### Solution 2
```
// MERGE-SORT(A, p, r) // running time
if p < r then // theta(1)
q = floor((p + r) / 2) // theta(1)
MERGE-SORT(A, p, q) // T(n/2)
MERGE-SORT(A, q + 1, r) // T(n/2)
MERGE(A, p, q, r) // theta(n)
```
## Problem 3
Solve the recurrence of merge sort by using a recursion tree.
$$T(n) = \begin{cases}
\theta(1) &\text{if}\; n = 1\\
2T(n/2) + \theta(n) &\text{if}\; n > 1
\end{cases}$$
### Solution 3
```mermaid
graph TD
subgraph "Level 0: theta(n)"
A["T(n) | theta(n)"]
end
subgraph "Level 1: theta(n)"
B["T(n/2) | theta(n/2)"]
C["T(n/2) | theta(n/2)"]
end
subgraph "Level 2: theta(n)"
D["T(n/4) | theta(n/4)"]
E["T(n/4) | theta(n/4)"]
F["T(n/4) | theta(n/4)"]
G["T(n/4) | theta(n/4)"]
end
subgraph "Level h = log(n)"
H["T(1) = theta(1)"]
I["T(1) = theta(1)"]
end
A --> B
A --> C
B --> D
B --> E
C --> F
C --> G
D --> H
G --> I
style A fill:#f9f,stroke:#333,stroke-width:2px
style B fill:#bbf,stroke:#333,stroke-width:2px
style C fill:#bbf,stroke:#333,stroke-width:2px
style D fill:#9f9,stroke:#333,stroke-width:2px
style E fill:#9f9,stroke:#333,stroke-width:2px
style F fill:#9f9,stroke:#333,stroke-width:2px
style G fill:#9f9,stroke:#333,stroke-width:2px
```
Therefore, $T(n) = \theta(n \log n)$
## Problem 4
What is the number of multiplications to evaluate the following cubic polynomial $f(x)$
when $x=3$ if the Horner's rule is used?
$$f(x) = 4x^3 + 2x^2 + 5x + 2$$
### Solution 4
1. Applying Horner's rule
$$f(x) = ((4x + 2)x + 5)x + 2$$
2. Count the number of multiplications
1. $4x$
2. $(4x + 2)x$
3. $((4x + 2)x + 5)x$
**3**
## Problem 5
Fill in the blank entries when the numbers are sorted by selection sort in
non-decreasing order.
### Solution 5
```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)],
]
# selection sorting process
n = len(data[0])
for i in range(n):
cur = data[-1].copy()
min_idx = i
for j in range(i, n):
if cur[j][0] < cur[min_idx][0]:
min_idx = j
cur[i], cur[min_idx] = cur[min_idx], cur[i]
cur[i] = (cur[i][0], True)
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())
```