GATE CS 2015 Set 2 — Question 22
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Programming & Data Structures → Heaps → Heapify & Build-Heap
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Question
Consider a complete binary tree where the left and the right subtrees of the root are max-heaps. The lower bound for the number of operations to convert the tree to a heap is
Correct answer
(A) Ω(log n)
Solution
In a complete binary tree where both the left and right subtrees are already max-heaps, the only node that might violate the max-heap property is the root. To convert this tree into a max-heap, we perform the max-heapify operation starting at the root. The max-heapify operation compares the root with its children and swaps it with the larger child if necessary, continuing down the tree until the property is restored. In a complete binary tree with nodes, the height is . The number of comparisons and swaps in the worst case is proportional to the height of the tree. Therefore, the complexity is , and the lower bound is .
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