Asymptotic Notations in Algorithm Analysis

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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?

To add 1 to sum each iteration Signup and view all the answers

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

Algorithm A Signup and view all the answers

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

$O(n^2)$ Signup and view all the answers

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

There exist positive constants c and n0 such that f(n) ≥ c.g(n) for all n ≥ n0 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)?

c = 3, n0 = 1 Signup and view all the answers

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

If f(n) = Θ(g(n)), then f(n) = O(g(n)) and f(n) = Ω(g(n)) 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)?

c = 11, n0 = 5 Signup and view all the answers

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

To provide a loose lower bound for f(n) in terms of g(n) 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)?

c1 = 3, c2 = 4, n0 = 2 Signup and view all the answers

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

We need to find constants $c > 0$ and $n_0 \geq 0$ such that $T(n) \geq c \ n^2$ for all $n \geq n_0$. 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 $T(n) = \Omega(n^2)$?

$c = 1, n_0 = 1$ Signup and view all the answers

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

$O(n^2)$ Signup and view all the answers

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

The case where $f(n)$ is always greater than or equal to $g(n)$. 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 $f(n) = O(n^2)$?

$c = 4, n_0 = 5$ Signup and view all the answers

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

$O(n)$ Signup and view all the answers