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krahets
2023-03-12 18:46:03 +08:00
parent 209d82a8cc
commit f6b9a75c8f
23 changed files with 406 additions and 359 deletions
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chapter_computational_complexity/time_complexity.md
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@@ -52,7 +52,7 @@ $$
```python title=""
# 在某运行平台下
def algorithm(n):
def algorithm(n: int) -> None:
a = 2 # 1 ns
a = a + 1 # 1 ns
a = a * 2 # 10 ns
@@ -217,14 +217,14 @@ $$
```python title=""
# 算法 A 时间复杂度：常数阶
def algorithm_A(n):
def algorithm_A(n: int) -> None:
print(0)
# 算法 B 时间复杂度：线性阶
def algorithm_B(n):
def algorithm_B(n: int) -> None:
for _ in range(n):
print(0)
# 算法 C 时间复杂度：常数阶
def algorithm_C(n):
def algorithm_C(n: int) -> None:
for _ in range(1000000):
print(0)
```
@@ -420,8 +420,8 @@ $$
=== "Python"
```python title=""
def algorithm(n):
a = 1 # +1
def algorithm(n: int) -> None:
a: int = 1 # +1
a = a + 1 # +1
a = a * 2 # +1
# 循环 n 次
@@ -613,9 +613,9 @@ $$
=== "Python"
```python title=""
def algorithm(n):
a = 1 # +0（技巧 1）
a = a + n # +0（技巧 1）
def algorithm(n: int) -> None:
a: int = 1 # +0（技巧 1）
a = a + n # +0（技巧 1）
# +n（技巧 2）
for i in range(5 * n + 1):
print(0)
@@ -821,10 +821,10 @@ $$
=== "Python"
```python title="time_complexity.py"
def constant(n):
def constant(n: int) -> int:
""" 常数阶 """
count = 0
size = 100000
count: int = 0
size: int = 100000
for _ in range(size):
count += 1
return count
@@ -949,9 +949,9 @@ $$
=== "Python"
```python title="time_complexity.py"
def linear(n):
def linear(n: int) -> int:
""" 线性阶 """
count = 0
count: int = 0
for _ in range(n):
count += 1
return count
@@ -1075,9 +1075,9 @@ $$
=== "Python"
```python title="time_complexity.py"
def array_traversal(nums):
def array_traversal(nums: List[int]) -> int:
""" 线性阶（遍历数组）"""
count = 0
count: int = 0
# 循环次数与数组长度成正比
for num in nums:
count += 1
@@ -1215,9 +1215,9 @@ $$
=== "Python"
```python title="time_complexity.py"
def quadratic(n):
def quadratic(n: int) -> int:
""" 平方阶 """
count = 0
count: int = 0
# 循环次数与数组长度成平方关系
for i in range(n):
for j in range(n):
@@ -1391,16 +1391,16 @@ $$
=== "Python"
```python title="time_complexity.py"
def bubble_sort(nums):
def bubble_sort(nums: List[int]) -> int:
""" 平方阶（冒泡排序）"""
count = 0 # 计数器
count: int = 0 # 计数器
# 外循环：待排序元素数量为 n-1, n-2, ..., 1
for i in range(len(nums) - 1, 0, -1):
# 内循环：冒泡操作
for j in range(i):
if nums[j] > nums[j + 1]:
# 交换 nums[j] 与 nums[j + 1]
tmp = nums[j]
tmp: int = nums[j]
nums[j] = nums[j + 1]
nums[j + 1] = tmp
count += 3 # 元素交换包含 3 个单元操作
@@ -1604,9 +1604,10 @@ $$
=== "Python"
```python title="time_complexity.py"
def exponential(n):
def exponential(n: int) -> int:
""" 指数阶（循环实现）"""
count, base = 0, 1
count: int = 0
base: int = 1
# cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)
for _ in range(n):
for _ in range(base):
@@ -1768,7 +1769,7 @@ $$
=== "Python"
```python title="time_complexity.py"
def exp_recur(n):
def exp_recur(n: int) -> int:
""" 指数阶（递归实现）"""
if n == 1: return 1
return exp_recur(n - 1) + exp_recur(n - 1) + 1
@@ -1884,9 +1885,9 @@ $$
=== "Python"
```python title="time_complexity.py"
def logarithmic(n):
def logarithmic(n: float) -> int:
""" 对数阶（循环实现）"""
count = 0
count: int = 0
while n > 1:
n = n / 2
count += 1
@@ -2017,7 +2018,7 @@ $$
=== "Python"
```python title="time_complexity.py"
def log_recur(n):
def log_recur(n: float) -> int:
""" 对数阶（递归实现）"""
if n <= 1: return 0
return log_recur(n / 2) + 1
@@ -2133,11 +2134,11 @@ $$
=== "Python"
```python title="time_complexity.py"
def linear_log_recur(n):
def linear_log_recur(n: float) -> int:
""" 线性对数阶 """
if n <= 1: return 1
count = linear_log_recur(n // 2) + \
linear_log_recur(n // 2)
count: int = linear_log_recur(n // 2) + \
linear_log_recur(n // 2)
for _ in range(n):
count += 1
return count
@@ -2290,10 +2291,10 @@ $$
=== "Python"
```python title="time_complexity.py"
def factorial_recur(n):
def factorial_recur(n: int) -> int:
""" 阶乘阶（递归实现）"""
if n == 0: return 1
count = 0
count: int = 0
# 从 1 个分裂出 n 个
for _ in range(n):
count += factorial_recur(n - 1)
@@ -2480,15 +2481,15 @@ $$
=== "Python"
```python title="worst_best_time_complexity.py"
def random_numbers(n):
def random_numbers(n: int) -> List[int]:
""" 生成一个数组，元素为: 1, 2, ..., n ，顺序被打乱 """
# 生成数组 nums =: 1, 2, 3, ..., n
nums = [i for i in range(1, n + 1)]
nums: List[int] = [i for i in range(1, n + 1)]
# 随机打乱数组元素
random.shuffle(nums)
return nums
def find_one(nums):
def find_one(nums: List[int]) -> int:
""" 查找数组 nums 中数字 1 所在索引 """
for i in range(len(nums)):
# 当元素 1 在数组头部时，达到最佳时间复杂度 O(1)