GATE CS 2026 Set 2 — Question 38
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Question
Consider an array of integers of size . The indices of run from 1 to . An algorithm is to be designed to check whether satisfies the condition given below.
such that .Which one of the following gives the worst case time complexity of the fastest algorithm that can be designed for the problem?
such that .Which one of the following gives the worst case time complexity of the fastest algorithm that can be designed for the problem?
Correct answer
(A) Θ(n)
Solution
Let for . The condition given is:
such that , we have .This condition is equivalent to saying that the sequence of differences is strictly increasing. That is, .To check if a sequence of size is strictly increasing, we can iterate through the sequence once and check if for all . This requires computing the differences (which takes time) and then verifying the sorted order (which takes time).The total time complexity is . Since we must read the input array of size , we cannot do better than . Thus, the worst-case time complexity of the fastest algorithm is .
such that , we have .This condition is equivalent to saying that the sequence of differences is strictly increasing. That is, .To check if a sequence of size is strictly increasing, we can iterate through the sequence once and check if for all . This requires computing the differences (which takes time) and then verifying the sorted order (which takes time).The total time complexity is . Since we must read the input array of size , we cannot do better than . Thus, the worst-case time complexity of the fastest algorithm is .
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