Problem 13.

What is the time complexity of the following function? State your answer as a function of n using asymptotic

notation in the simplest form possible. E.g., “Θ(n)”.

from math import sqrt, log, ceil

def foo(n):

for i in range(ceil(n**3 - 10*n + sqrt(n))):

for j in range(ceil(log(n**2) + sqrt(n))):

print(i, j)

Problem 14.

What is the expected time complexity of the function below? State your answer using asymptotic notation.

import random

def foo(n):

# draw a random number uniformly from 0, 1, 2, ..., 99

# in constant time

x = random.randrange(100)

for i in range(x**2):

for j in range(n):

print("Hi!")

Let's analyze the time complexities of each function separately.

### Problem 13: Time Complexity Analysis

Given the function:

```python
from math import sqrt, log, ceil

def foo(n):
    for i in range(ceil(n**3 - 10*n + sqrt(n))):
        for j in range(ceil(log(n**2) + sqrt(n))):
            print(i, j)
```

**Step 1: Analyze the outer loop**  
The outer loop runs `ceil(n**3 - 10*n + sqrt(n))` times. For large values of \( n \), the dominant term inside the `ceil` function is \( n^3 \), so the loop will run approximately \( n^3 \) times. Thus, the time complexity of the outer loop is \( \Theta(n^3) \).

**Step 2: Analyze the inner loop**  
The inner loop runs `ceil(log(n**2) + sqrt(n))` times. The term \( log(n^2) \) simplifies to \( 2\log(n) \), which is asymptotically larger than \( \sqrt{n} \) for large \( n \). Therefore, the loop will run approximately \( \log(n) \) times. The time complexity of the inner loop is \( \Theta(\log(n)) \).

**Step 3: Combine the loops**  
The total time complexity is the product of the complexities of the two loops:

\[
\text{Total Time Complexity} = \Theta(n^3) \times \Theta(\log(n)) = \Theta(n^3 \log(n))
\]

So, the time complexity of `foo` in Problem 13 is **\( \Theta(n^3 \log(n)) \)**.

---

### Problem 14: Time Complexity Analysis

Given the function:

```python
import random

def foo(n):
    # draw a random number uniformly from 0, 1, 2, ..., 99
    # in constant time
    x = random.randrange(100)
    
    for i in range(x**2):
        for j in range(n):
            print("Hi!")
```

**Step 1: Analyze the outer loop**  
The outer loop runs `x**2` times, where `x` is a random number uniformly drawn from the range 0 to 99. The expected value of \( x \) is the average of the numbers 0 to 99, which is:

\[
\mathbb{E}[x] = \frac{0 + 99}{2} = 49.5
\]

Thus, the expected value of \( x^2 \) is:

\[
\mathbb{E}[x^2] \approx \mathbb{E}[(49.5)^2] = 2450.25
\]

Therefore, the outer loop will run on average about 2450 times, which is a constant and doesn't depend on \( n \). Thus, the time complexity of the outer loop is \( \Theta(1) \).

**Step 2: Analyze the inner loop**  
The inner loop runs \( n \) times for each iteration of the outer loop. Since the outer loop runs a constant number of times, the overall time complexity is determined by the inner loop.

**Step 3: Combine the loops**  
Since the outer loop runs \( \Theta(1) \) times and the inner loop runs \( \Theta(n) \) times, the total time complexity is:

\[
\text{Total Time Complexity} = \Theta(1) \times \Theta(n) = \Theta(n)
\]

So, the expected time complexity of `foo` in Problem 14 is **\( \Theta(n) \)**.

---

### Summary:

- **Problem 13:** The time complexity is \( \Theta(n^3 \log(n)) \).
- **Problem 14:** The expected time complexity is \( \Theta(n) \).

---

Would you like further details or have any questions?

---

Here are some related questions you might find interesting:

1. How does the choice of random numbers affect the time complexity in general?
2. What is the difference between worst-case, average-case, and expected time complexity?
3. Can you provide examples of algorithms with time complexities of \( \Theta(\log n) \)?
4. How would the time complexity change if `n**3` in Problem 13 was replaced by `n**4`?
5. What are some common pitfalls when analyzing time complexity involving logarithmic functions?

**Tip:** When analyzing time complexity, focus on the dominant terms that grow the fastest as \( n \) increases, as they will determine the overall complexity.

Understand the time complexity of Python functions through detailed analysis. Problem 13 explores nested loops with varying complexities, while Problem 14 involves random number generation and loop iterations. Learn about asymptotic notation and practical application in computational analysis.

Math Problem Statement

draw a random number uniformly from 0, 1, 2, ..., 99

in constant time

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