Heapify operation in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

What is the primary purpose of the heapify operation?

Choose the best description of what the heapify operation does in a binary heap.

A It removes the root element from the heap.

B It sorts all elements in ascending order.

C It rearranges elements to satisfy the heap property starting from a given node.

D It duplicates the heap structure to create a new heap.

💡 Hint

Think about how a heap maintains its special order after changes.

The heapify operation adjusts the subtree rooted at a given node to maintain the heap property, ensuring the parent node is ordered correctly relative to its children.

📋 Factual

intermediate

What is the time complexity of heapify on a subtree of size n?

Identify the time complexity of the heapify operation when applied to a subtree with n nodes.

A O(n log n)

B O(1)

C O(n)

D O(log n)

💡 Hint

Consider the height of the subtree and how many swaps heapify might perform.

Heapify works by moving down the tree levels, and since the height of a heap is log n, the time complexity is O(log n).

🔍 Analysis

advanced

What is the output array after heapifying the subtree rooted at index 1 in the array [4, 10, 3, 5, 1] (0-based index)?

Given the array representing a binary heap: [4, 10, 3, 5, 1], apply heapify at index 1 and select the resulting array.

Data Structures Theory

array = [4, 10, 3, 5, 1]
heapify at index 1

A [4, 10, 3, 5, 1]

B 4, 10, 3, 5, 1]

C [4, 10, 3, 5, 1

D ]1 ,5 ,3 ,01 ,4[

💡 Hint

Check if the subtree rooted at index 1 already satisfies the heap property.

At index 1, the value is 10 with children 5 and 1. Since 10 is greater than both children, the subtree already satisfies the max-heap property, so the array remains unchanged.

❓ Comparison

advanced

Which option correctly describes the difference between heapify and build-heap operations?

Select the statement that best explains how heapify and build-heap differ.

A Heapify fixes the heap property at one node; build-heap applies heapify from bottom up to the entire array.

B Heapify sorts the entire array; build-heap removes the largest element.

C Heapify duplicates the heap; build-heap deletes the heap.

D Heapify inserts a new element; build-heap extracts the root.

💡 Hint

Think about the scope of each operation.

Heapify adjusts the heap property at a single node, while build-heap runs heapify on all non-leaf nodes from bottom to top to create a valid heap.

❓ Reasoning

expert

If you apply heapify on all nodes from the last non-leaf node up to the root in an array of size n, what is the overall time complexity?

Determine the total time complexity of building a heap by applying heapify from the last non-leaf node up to the root in an array of size n.

A O(n log n)

B O(n)

C O(log n)

D O(n^2)

💡 Hint

Consider that heapify takes less time on nodes lower in the tree and more time near the root.

Although heapify takes O(log n) time per node, most nodes are near the leaves and require less work. Summing all heapify calls results in O(n) total time.