Algorithm Analysis: Asymptotic Complexity

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Questions and Answers

Sort the functions in increasing order of asymptotic (big O) complexity.

f3(n), f4(n), f2(n), f1(n)

f2(n), f4(n), f1(n), f3(n)

f1(n), f2(n), f4(n), f3(n) (correct)

None of these

Sort the function in decreasing order of asymptotic (big O) complexity.

f2(n), f1(n), f3(n) (correct)

f3(n), f1(n), f2(n)

f1(n), f3(n), f2(n)

None of these

For which value of n₀ does f(n) = O(g(n)) hold true with f(n) = 10n² + 100, g(n) = 2ⁿ, and c = 0.125?

13 (correct)

Which of the following statements is true based on the relationships between f₁(n), f₂(n), and f₃(n)?

X3 is slower than X1 and X2 for very large input Signup and view all the answers

Which function grows the fastest among f1, f2, f3, f4, and f5?

f3 Signup and view all the answers

Which of the following statements about functions f(n) = log log log √n and g(x) = 2³³³₃₀ is true?

(i), (ii) and (iii) only Signup and view all the answers

What is the correct time complexity of the provided C++ code with nested loops?

O(n) Signup and view all the answers

Which statement about the growth of functions f2, f3, f4, and f1 is false?

f1(n) grows faster than f2(n) Signup and view all the answers

What is the highest asymptotic worst case time complexity of the given code fragment?

O(n²) Signup and view all the answers

Which function has the fastest growth rate: f(n) = n, g(n) = n log n, h(n) = n², or k(n) = 2ⁿ?

k(n) = 2ⁿ Signup and view all the answers

How many of the following statements are false regarding asymptotic notations?

2 Signup and view all the answers

What is the time complexity of the provided recursion function P(n)?

O(n) Signup and view all the answers

If f(n) is O(n²) and g(n) is O(n⁴), which of the following is correct?

g(n) grows faster than f(n) Signup and view all the answers

Which of the following correctly represents the relationship between f2 and f5?

f2(n) = Θ(f5(n)) Signup and view all the answers

Which function is determined to be constant in terms of its growth rate?

f1 Signup and view all the answers

Which of these factors primarily influences the time complexity of a nested loop implementation?

The number of iterations in the innermost loop Signup and view all the answers

Which function has a growth rate slower than both f1 and f2?

f3 Signup and view all the answers

What can be concluded about the relationship between f1 and f3 based on their growth rates?

f3 is greater than f1 Signup and view all the answers

In terms of growth, how does f(n) relate to g(n) when f(n) = log log log√n and g(n) = 2^202020?

f(n) grows faster than g(n) Signup and view all the answers

Which of the following statements about the total work done in an algorithm with an outer loop and an inner loop is correct?

The total work is O(n²) Signup and view all the answers

Which of the following growth rate comparisons is true?

f5 > f2 > f4 Signup and view all the answers

What is the correct conclusion about the complexities stated regarding n, √n, and log n?

log n < n Signup and view all the answers

Which of the following is a possible misinterpretation of f(n) = Ω(g(n)) for f(n) = log log log√n?

f(n) < g(n) Signup and view all the answers

Based on the given complexities, which of the following is true about f2 and f5?

f5 grows faster than f2 Signup and view all the answers

Study Notes

Asymptotic Complexity

Sorting functions in increasing order of asymptotic complexity involves arranging them by how quickly they grow as input size increases.
Exponential functions grow much faster than polynomial functions, which grow faster than logarithmic and constant functions.
To determine the correct order, observe the growth rates of individual functions and compare them.

Big O Notation

Big O notation provides an upper bound on the growth rate of a function.
It indicates that the function's growth is no faster than a specific function, often the upper bound.
To verify f(n) = O(g(n)), find constants c and n₀ where 0 ≤ f(n) ≤ c * g(n) for all n ≥ n₀.

Analyzing Code Complexity

Code complexity refers to the resources (time or space) required to execute a code fragment.
Asymptotic worst-case time complexity indicates the maximum time required for the code to run.
For nested loops, multiply the complexity of each loop to determine the overall complexity.

Time Complexity of Recursion

Recursive functions often exhibit complexities based on the number of recursive calls.
To analyze the complexity, consider the worst-case scenario where n is odd, leading to n-1 or n-2 recursive calls.
The time complexity can be linear, logarithmic, or exponential depending on the algorithm.

Code Examples

Example 1:

int a = 0;
for(int x = 0; x < n; x++) {
   if (x%5==0){
      for (int y = 0; y < n; y ++){
        if (x == y)
          a+ = x * y)
      }
   }
}

The code's complexity is O(n²) due to the nested loop.

Example 2:

for (a = 0; a < n- 2; a ++)
{
    for (b = 0; b < 100; b = b+2)
    {
        for (c = 1; c < 8*n; c ++)
        {            
        }
    }
}

This code exhibits O(n²) complexity due to the innermost loop iterating 8n times within the outer loop's O(n) iterations.

Example 3:
```
{
 if (n <= 0)
 return 1;
 else if (n% 2 == 0)
 return P(n-1);
 else
 return P(n-2);
}
```
- This recursive function exhibits O(n) complexity as it makes approximately 'n' recursive calls in the worst-case scenario.

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Description

This quiz focuses on understanding asymptotic complexity and Big O notation, essential concepts for analyzing algorithm efficiency. It covers sorting functions by their growth rates and how to determine the upper bounds using Big O notation. Test your knowledge on code complexity and its implications on resource usage.