GATE DA 2026 Set 1 — Question 15
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Programming, Data Structures & Algorithms → Divide & Conquer → Recurrence of Merge & Quick Sort
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Question
Consider that the quick sort algorithm is used to sort an array of distinct randomly ordered elements. In every call, the pivot is chosen as the first element of the current subarray.Let denote the expected time to sort the array. Assume that the time to partition is linear in the size of the current subarray. Which of the following recurrence relations correctly represents in this scenario?
Correct answer
(D) T(n) = (1)/(n) Σₖ₌₀ⁿ⁻¹ [T(k) + T(n-k-1)] + O(n)
Solution
When the input array is randomly ordered and the first element is chosen as the pivot, the pivot is equally likely to be the -th smallest element for any rank from to (0-indexed). If the pivot is the -th smallest element, the array is partitioned into a left subarray of size and a right subarray of size . The time taken is the cost of partitioning () plus the time to sort the two subarrays ().The expected time is the average over all possible pivot positions:This matches option (D).
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