## Prompt

Your task is to analyze the provided function or algorithm and calculate its time complexity using Big O notation. Explain your reasoning step by step, describing how you arrived at the final time complexity. Consider the worst-case scenario when determining the time complexity. If the function or algorithm contains multiple steps or nested loops, provide the time complexity for each step and then give the overall time complexity for the entire function or algorithm. Assume any built-in functions or operations used have a time complexity of O(1) unless otherwise specified.[function or algorithm]

## Sample query

Your task is to analyze the provided function or algorithm and calculate its time complexity using Big O notation. Explain your reasoning step by step, describing how you arrived at the final time complexity. Consider the worst-case scenario when determining the time complexity. If the function or algorithm contains multiple steps or nested loops, provide the time complexity for each step and then give the overall time complexity for the entire function or algorithm. Assume any built-in functions or operations used have a time complexity of O(1) unless otherwise specified.def example_function(n):

for i in range(n):

print(i)

for j in range(n):

for k in range(n):

print(j, k)

## Sample response

To analyze the time complexity of the provided function `example_function(n)`

using Big O notation, we will go through each part of the function step by step, considering the worst-case scenario.

The function consists of two main parts:

**The first loop:**

```
for i in range(n):
print(i)
```

- This loop iterates
`n`

times, performing a constant time operation (`print(i)`

) in each iteration. Since the operation inside the loop does not depend on the loop variable`i`

other than for printing it, the time complexity of this loop is O(n).

**The nested loops:**

```
for j in range(n):
for k in range(n):
print(j, k)
```

- Here, we have two nested loops. The outer loop runs
`n`

times, and for each iteration of the outer loop, the inner loop also runs`n`

times. The operation inside the inner loop (`print(j, k)`

) is constant time. - Therefore, for each iteration of the outer loop, the inner loop runs
`n`

times, leading to a total of`n * n = n^2`

iterations for the nested loops. This means the time complexity of the nested loops section is O(n^2).

**Overall time complexity:**

To find the overall time complexity, we combine the complexities of the two main parts. The first part is O(n), and the second part is O(n^2). When combining these, we take the highest order term since it dominates the growth rate as `n`

becomes large. Thus, the overall time complexity of the function is O(n^2).

**Final Time Complexity: O(n^2)**

This means that as the input size `n`

increases, the time it takes for the function to complete grows quadratically with `n`

. The nested loops are the primary factor in determining the time complexity of the function since they result in more operations than the single loop when `n`

is large.