Hello, can you please help me do exercise 2 (with subparts 1, 2 and 3) please **Exercise 1:**Consider two algorithms A and B that solve the same class of problems.- The time complexity of A is 5000n. The time complexity for B is \( \text{ceiling}(1.1^n) \) or \( \lceil 1.1^n \rceil \).Find the time required to execute each for n=10, n=100, n=1000, n=100000.Tabulate your results. Which algorithm has better performance for n>1000?---**Exercise 2:**Find an upper bound and a lower bound for each of the function f(x).1. \( f(x) = x^2 + 2x + 1 \)2. \( f(x) = 3x^4 + 2x^3 + x \)3. \( f(n) = (6n+4)(1+\log n) \)---**Exercise 3:**Consider the algorithm below:1. Initialize: \( \text{i = n;} \)2. While \( \text{i ≥ 1} \): 1. \( \text{x = x+1} \) 2. \( \text{i = i/2} \)Trace the algorithm, and fill the table that indicates the number of times x=x+1 is executed for the following values of n.| **n** | **i (range)** | **#times x=x+1 executed** ||-------|---------------|--------------------------|| 8 | (8,4,2,1) | 4 || 16 | | || 32 | | |Find a theta notation in terms of n for the number of times the statement x=x+1 is executed based on generalizing the results above for n=2^k. Note that if n=2^k, then k=log2(n).

Answered: Exercise 2: Find an upper bound and a…

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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Hello, can you please help me do exercise 2 (with subparts 1, 2 and 3) please

**Exercise 1:**

Consider two algorithms A and B that solve the same class of problems.

- The time complexity of A is 5000n. The time complexity for B is \( \text{ceiling}(1.1^n) \) or \( \lceil 1.1^n \rceil \).

Find the time required to execute each for n=10, n=100, n=1000, n=100000.

Tabulate your results. Which algorithm has better performance for n>1000?

---

**Exercise 2:**

Find an upper bound and a lower bound for each of the function f(x).

1. \( f(x) = x^2 + 2x + 1 \)
2. \( f(x) = 3x^4 + 2x^3 + x \)
3. \( f(n) = (6n+4)(1+\log n) \)

---

**Exercise 3:**

Consider the algorithm below:

1. Initialize: \( \text{i = n;} \)
2. While \( \text{i ≥ 1} \):
1. \( \text{x = x+1} \)
2. \( \text{i = i/2} \)

Trace the algorithm, and fill the table that indicates the number of times x=x+1 is executed for the following values of n.

| **n** | **i (range)** | **#times x=x+1 executed** |
|-------|---------------|--------------------------|
| 8 | (8,4,2,1) | 4 |
| 16 | | |
| 32 | | |

Find a theta notation in terms of n for the number of times the statement x=x+1 is executed based on generalizing the results above for n=2^k. Note that if n=2^k, then k=log2(n).

Expert Solution

Step 1: Definition of upper and lower bound in algorithms:

Upper Bound (Big O Notation):

An upper bound is an asymptotic limit on the growth rate of an algorithm's resource utilisation, usually its time complexity. It is commonly written using Big O notation (O()).

It gives an upper bound on the amount of time or space that an algorithm will need when the input size approaches the worst-case scenario or reaches infinity.

Lower bound (Big Omega Notation):

The lower bound, which is commonly represented by Big Omega notation (Ω()), denotes an asymptotic lower bound on the rate of increase in an algorithm's resource consumption.

It indicates that an algorithm will require a minimum amount of time or space, irrespective of the input, and sets a lower bound on the optimal case for the method's temporal complexity.