GATE CS 2021 Set 2 — Question 49
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Algorithms → Asymptotic Analysis → Master Theorem
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Question
For constants and , consider the following recurrence defined on the non-negative integers:Which one of the following options is correct about the recurrence ?
Correct answer
(C) If f(n) is O(n^(_b(a) - ε)) for some ε 0, then T(n) is Θ(n^(_b(a))).
Solution
This question tests the understanding of the Master Theorem for solving recurrences of the form .Let . The Master Theorem has three main cases comparing to :
1.Case 1: If for some constant , then .
2.Case 2: If , then .
3.Case 3: If for some constant , and if for some constant and sufficiently large , then .
Analyzing the options:- (A) This is not generally true. For example, if , then . If , this falls into an extended case where , not .
- (B) This is not generally true. The behavior depends on the relationship between and .
- (C) This statement matches Case 1 of the Master Theorem exactly. If is polynomially smaller than , then the solution is dominated by the leaf level of the recursion tree, which is . This is the correct statement.
- (D) This corresponds to Case 2 of the Master Theorem. However, the conclusion is incorrect. If , then , not .
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