GATE CS 2020 Set 1 — Question 12
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Algorithms → Asymptotic Analysis → Master Theorem
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Question
For parameters and , both of which are , , and .
Then is
Then is
Correct answer
(A) Θ(ₐ _b n)
Solution
Given the recurrence relation:with the base case .Let us solve this by substitution. Assume can be written in terms of and . Let .
Taking on both sides:
Taking on both sides:
Now substitute into the recurrence:
Let . Then:
This is a simple arithmetic progression where each step adds 1. The base case is when the argument of is , which corresponds to , i.e., .
Given , we have .So, .
Since , we have:
Thus, .
Taking on both sides:
Taking on both sides:
Now substitute into the recurrence:
Let . Then:
This is a simple arithmetic progression where each step adds 1. The base case is when the argument of is , which corresponds to , i.e., .
Given , we have .So, .
Since , we have:
Thus, .
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