GATE | GATE-CS-2003 | Question 20

Consider the following three claims

1. (n + k)m = Θ(nm), where k and m are constants
2. 2n + 1 = O(2n)
3. 22n + 1 = O(2n) 

Which of these claims are correct ?

(A) 1 and 2
(B) 1 and 3
(C) 2 and 3
(D) 1, 2, and 3


Answer: (A)

Explanation: (n + k)^m and Θ(n^m) are asymptotically same as theta notation can always be written by taking the leading order term in a polynomial expression.

2^{n + 1} and O(2ⁿ) are also asymptotically same as 2^{n + 1} can be written as 2 * 2ⁿ and constant multiplication/addition doesn’t matter in theta notation.

2^{2n + 1} and O(2ⁿ) are not same as constant is in power.

See Asymptotic Notations for more details.

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