GATE CS 2023 Set 1 — Question 54
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Algorithms → Asymptotic Analysis → Asymptotic Notations
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Question
Consider functions Function_1 and Function_2 expressed in pseudocode as follows:Function 1
Function 2
Let and denote the number of times the statement "" is executed in Function_1 and Function_2, respectively.Which of the following statements is/are TRUE?
while n > 1 do
for i = 1 to n do
x = x + 1;
end for
n = floor(n/2);
end while
for i = 1 to 100 * n do
x = x + 1;
end for
Correct answer
(A) f₁(n) ∈ Θ(f₂(n)); (D) f₁(n) ∈ O(n)
Solution
Let us analyze the time complexity of both functions by counting the number of times the statement is executed.For Function 1:
The outer loop runs while , and in each iteration, is halved ().
The inner loop runs times in the current iteration.
The total number of executions is the sum of the geometric series:This is a geometric series with sum .
Thus, .For Function 2:
The loop runs from to .
The statement is executed exactly times.
Thus, .Evaluating the options:
(A) : Since both and are , this is TRUE.
(B) : Little-o notation denotes strictly smaller growth. Since both grow linearly, this is FALSE.
(C) : Little-omega notation denotes strictly larger growth. Since both grow linearly, this is FALSE.
(D) : Since , it implies . This is TRUE.Therefore, the correct options are (A) and (D).
The outer loop runs while , and in each iteration, is halved ().
The inner loop runs times in the current iteration.
The total number of executions is the sum of the geometric series:This is a geometric series with sum .
Thus, .For Function 2:
The loop runs from to .
The statement is executed exactly times.
Thus, .Evaluating the options:
(A) : Since both and are , this is TRUE.
(B) : Little-o notation denotes strictly smaller growth. Since both grow linearly, this is FALSE.
(C) : Little-omega notation denotes strictly larger growth. Since both grow linearly, this is FALSE.
(D) : Since , it implies . This is TRUE.Therefore, the correct options are (A) and (D).
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