Asymptotic Notations in Algorithm Analysis

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Questions and Answers

Which algorithm has the worst time complexity for computing 1+2+...+n?

  • Algorithm C (correct)
  • Algorithm B
  • Algorithm A
  • The time complexity is the same for all algorithms

In Algorithm B, what is the purpose of the inner loop (for j = n to i)?

  • To subtract 1 from sum
  • To perform division operations
  • To add 1 to sum (correct)
  • To multiply i by n and add it to sum

Which algorithm provides a closed-form expression for the sum of 1 to n?

  • Algorithm B
  • Algorithm A
  • None of the algorithms
  • Algorithm C (correct)

What is the purpose of the loop in Algorithm A that runs from 1 to n?

<p>To add 1 to sum each iteration (B)</p> Signup and view all the answers

Which algorithm requires the least number of total operations to compute 1+2+...+n?

<p>Algorithm A (C)</p> Signup and view all the answers

If n is the input size, what is the complexity of Algorithm B in terms of T(n)?

<p>$O(n^2)$ (A)</p> Signup and view all the answers

What is the definition of f(n) = Ω(g(n))?

<p>There exist positive constants c and n0 such that f(n) ≥ c.g(n) for all n ≥ n0 (D)</p> Signup and view all the answers

In Example 2, what is the value of c and n0 used to prove that T(n) = 3n + 2 is an element of Ω(n)?

<p>c = 3, n0 = 1 (D)</p> Signup and view all the answers

What is the relationship between the Θ(g(n)) and O(g(n)) notations?

<p>If f(n) = Θ(g(n)), then f(n) = O(g(n)) and f(n) = Ω(g(n)) (C)</p> Signup and view all the answers

In Example 4, what is the value of c and n0 used to prove that T(n) = 10n^2 + 4n + 2 is an element of O(n^2)?

<p>c = 11, n0 = 5 (C)</p> Signup and view all the answers

What is the purpose of the Ω(g(n)) notation?

<p>To provide a loose lower bound for f(n) in terms of g(n) (D)</p> Signup and view all the answers

In Example 3, what are the values of c1, c2, and n0 used to prove that T(n) = 3n + 2 is an element of Θ(n)?

<p>c1 = 3, c2 = 4, n0 = 2 (D)</p> Signup and view all the answers

How do we prove that $T(n) = \Omega(n^2)$?

<p>We need to find constants $c &gt; 0$ and $n_0 \geq 0$ such that $T(n) \geq c \ n^2$ for all $n \geq n_0$. (D)</p> Signup and view all the answers

For the function $T(n) = 10n^2 + 4n + 2$, what values of $c$ and $n_0$ can be used to prove $T(n) = \Omega(n^2)$?

<p>$c = 1, n_0 = 1$ (B)</p> Signup and view all the answers

What is the Big-Oh time complexity of multiplying two arrays of size $n$?

<p>$O(n^2)$ (D)</p> Signup and view all the answers

What is the best case time complexity of a function $f(n)$, denoted as $\Omega(g(n))$?

<p>The case where $f(n)$ is always greater than or equal to $g(n)$. (A)</p> Signup and view all the answers

For the function $f(n) = 3n^2 + 20$, what values of $c$ and $n_0$ can be used to prove $f(n) = O(n^2)$?

<p>$c = 4, n_0 = 5$ (B)</p> Signup and view all the answers

What is the time complexity of the function $T(n) = 2n + 3$?

<p>$O(n)$ (B)</p> Signup and view all the answers

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