[Time complexity and space complexity]

1. Time complexity

1. What is time complexity?

• 算法的执行效率
• 算法的执行时间与算法的输入值之间的关系

2. What is big Onotation?

• O(N), where Nrefers to 例1in num,N = num

【Example 1:】

def test(num):
    total = 0             # 执行时间为：a；
    for i in range(num):
        total += i        # 执行时间为：b；num次为：num * b；
    return total          # 执行时间为：c；
# 所以test类的时间复杂度 = a + num * b + c

So num很大at that time , then testthe time complexity of the class is mainly based on num * b. So this is what 例1affects the time complexity . Finally, define the time complexity of Example 1 as .最大for循环O(N)

1.1. Common time complexity cases

Common time complexity: $O(1)$、$O(logn)$、$O(n)$、$O\left(nlogn\right)$、$O\left(n^2\right)$
[Example 2:$O(1)$】

def o1(num):
	i = num         # 执行时间为：a；
	j = num * 2     # 执行时间为：b；
	return i + j    # 执行时间为：c；
# 所以o1类的时间复杂度 = a + b + c，为常量，和num的个数无关

Define the time complexity of Example 2 as O(1).
[Example 3: $O\left(N\right)$】

def ON(num):
    total = 0             # 执行时间为：a；
    for i in range(num):
        total += i        # 执行时间为：b；num次为：num * b；
    return total          # 执行时间为：c；
# 所以test类的时间复杂度 = a + num * b + c，和num的个数有关

Define the time complexity of Example 3 as O(N).
[Example 4: $O(logN)$】

def OlogN(num):
    i = 1             # 执行时间为：a；
    while (i < num):
        total = i * 2        # 执行时间为：b；X次为：num = i*2^X，X = log2(num/i)
    return i         # 执行时间为：c；
# 所以test类的时间复杂度 = log2(num/i)，和num的个数有关，因为i为常数，可忽略

Define the time complexity of Example 4 as O(logN).
[Example 5: $O(N+M)$】

def ON(num1, num2):
    total = 0             # 执行时间为：a；
    for i in range(num1):
        total += i        # 执行时间为：b；num次为：num1 * b；
    for i in range(num2):
        total += i        # 执行时间为：b；num次为：num2 * b；
    return total          # 执行时间为：c；
# 所以test类的时间复杂度 = a + num1 * b + num2 * b + c，和num的个数有关

Define the time complexity of Example 5 as O(N + M).

【例6：$O(NlogN)$】

def OlogN(num):
    i = 1             # 执行时间为：a；
    for i in range(num1):         # 执行 num1次
    	while (i < num2):
        	total += i + j        # 执行时间为：b；X次为：num2 = i*2^X，X = log2(num2/i)
        	j = j * 2
    return i         # 执行时间为：c；
# 所以test类的时间复杂度 = (num1)*log2(num2/i)，和num的个数有关，因为i为常数，可忽略

Define the time complexity of Example 4 as NO(logN).
[Example 5: $O(N+M)$】

def ON(num1, num2):
    total = 0             # 执行时间为：a；
    for i in range(num1):
    	for i in range(num2):
        	total += i + j        # 执行时间为：b；num1 * num2次为：num1 * num2 * b；
    return total          # 执行时间为：c；
# 所以test类的时间复杂度 = a + num1 * num2 * b + c，和num的个数有关

Define the time complexity of Example 4 as $O(N^2)$。

1.2. Time complexity comparison chart

Time complexity in descending order: $O(N!)$ > $O(2^N)$ > $O(N^2)$ > $O(NlogN)$ > $O(N)$ > $O(logN)$ > $O(1)$

space complexity

1. What is space complexity?

• 算法的存储空间与算法的输入值之间的关系(The space is occupied by the variables we declare)

2. Big Orepresentation?

• O(N), where Nrefers to 例1in num,N = num

1.1. Common space complexity cases

There are two commonly used space complexities.

understand

Space complexity generally refers to disk space, so time complexity is generally given priority in work.