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(nm) are asymptotically same as theta notation can always be written by taking the leading order term in a polynomial expression.
2n + 1 and O(2n) are also asymptotically same as 2n + 1 can be written as 2 * 2n and constant multiplication/addition doesn’t matter in theta notation.
22n + 1 and O(2n) are not same as constant is in power.
See Asymptotic Notations for more details.