Heap extraction (bubble down) in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

What is the primary purpose of the bubble down operation in a heap?

In the context of heap data structures, what does the bubble down operation achieve?

A It restores the heap property after removing the root element by moving the new root downwards.

B It inserts a new element at the bottom and moves it upwards to maintain the heap property.

C It swaps the root with the last element without changing the heap structure.

D It sorts all elements in the heap in ascending order.

💡 Hint

Think about what happens after the root element is removed from a heap.

Bubble down is used to restore the heap property after the root is removed by moving the new root down to its correct position.

📋 Factual

intermediate

Which condition triggers the bubble down operation to continue in a max-heap?

During bubble down in a max-heap, under which condition does the operation continue moving the element down?

A When the current node is equal to its parent.

B When the current node is larger than both children.

C When the current node has no children.

D When the current node is smaller than at least one of its children.

💡 Hint

Consider the heap property for a max-heap.

In a max-heap, bubble down continues if the current node is smaller than any of its children to maintain the max-heap property.

🔍 Analysis

advanced

What is the time complexity of the bubble down operation in a heap of size n?

Analyze the time complexity of the bubble down operation when extracting the root from a heap with n elements.

A O(n log n) because it sorts the entire heap.

B O(n) because it may need to check every element in the heap.

C O(log n) because the element moves down at most the height of the heap.

D O(1) because only one swap is needed.

💡 Hint

Consider the height of a binary heap and how many levels the element can move down.

The height of a binary heap is log n, so bubble down takes O(log n) time in the worst case.

🚀 Application

advanced

Given a max-heap array [50, 30, 40, 10, 20, 35], what is the array after extracting the root and performing bubble down?

Start with the max-heap array: [50, 30, 40, 10, 20, 35]. Extract the root (50) and perform bubble down. What is the resulting array?

A [40, 30, 35, 10, 20]

B [40, 30, 20, 10, 35]

C [35, 30, 40, 10, 20]

D [30, 20, 40, 10, 35]

💡 Hint

Replace root with last element, remove last, then bubble down by swapping with the larger child.

After removing 50, replace root with 35. Bubble down swaps 35 with 40 (larger child), resulting in [40, 30, 35, 10, 20].

❓ Reasoning

expert

Why does bubble down always restore the heap property after root extraction in a binary heap?

Explain why the bubble down operation guarantees the heap property is restored after extracting the root element from a binary heap.

A Because it rebuilds the entire heap from scratch after extraction.

B Because it moves the new root down by swapping with the larger (or smaller) child until the heap property is satisfied at every level.

C Because it only swaps the root with the last element without further adjustments.

D Because it removes all leaf nodes and reinserts them in order.

💡 Hint

Think about how the heap property is defined and how bubble down fixes violations.

Bubble down fixes violations by swapping the new root with the larger (max-heap) or smaller (min-heap) child repeatedly until the heap property holds at all levels.

Practice

1. What is the main purpose of the bubble down operation during heap extraction?

easy

A. To restore the heap property after removing the root element

B. To insert a new element at the bottom of the heap

C. To find the maximum element in the heap

D. To sort all elements in the heap

Solution

1. Step 1: Understand heap extraction

   Heap extraction removes the root element, which may break the heap property.

2. Step 2: Role of bubble down

   Bubble down swaps the root with its smaller child repeatedly to restore the heap order.

3. Final Answer:

   To restore the heap property after removing the root element -> Option A

4. Quick Check:

   Bubble down fixes heap after root removal = D [OK]

Hint: Bubble down fixes heap after root removal [OK]

Common Mistakes:

• Confusing bubble down with insertion
• Thinking bubble down sorts the entire heap
• Believing bubble down finds max element

2. Which of the following correctly describes the condition to continue bubbling down in a min-heap?

easy

A. While the current node is larger than both children

B. While the current node is larger than at least one child

C. While the current node is smaller than both children

D. While the current node is equal to its children

Solution

1. Step 1: Understand min-heap property

   In a min-heap, parent nodes must be smaller than their children.

2. Step 2: Bubble down condition

   Bubble down continues while the current node is larger than at least one child, to swap with the smaller child.

3. Final Answer:

   While the current node is larger than at least one child -> Option B

4. Quick Check:

   Bubble down if parent > any child = C [OK]

Hint: Bubble down if parent bigger than any child [OK]

Common Mistakes:

• Stopping bubble down too early
• Swapping with larger child instead of smaller
• Confusing min-heap with max-heap conditions

3. Given the min-heap array [2, 5, 3, 10, 7], what is the array after extracting the root and performing bubble down?

medium

A. [5, 7, 3, 10]

B. [3, 7, 5, 10]

C. [3, 5, 7, 10]

D. [5, 3, 7, 10]

Solution

1. Step 1: Remove root and replace with last element

   Remove 2, replace root with 7: [7, 5, 3, 10]

2. Step 2: Bubble down to restore heap

   Compare 7 with children 5 and 3; swap with smaller child 3: [3, 5, 7, 10]

3. Final Answer:

   [3, 5, 7, 10] -> Option C

4. Quick Check:

   Bubble down swaps root with smaller child = A [OK]

Hint: Replace root with last, then swap down smaller child [OK]

Common Mistakes:

• Not replacing root with last element
• Swapping with larger child
• Forgetting to bubble down fully

4. Identify the error in this bubble down pseudocode for a min-heap extraction:

function bubbleDown(heap, index):
  left = 2 * index + 1
  right = 2 * index + 2
  smallest = index
  if left < heap.size and heap[left] > heap[smallest]:
    smallest = left
  if right < heap.size and heap[right] > heap[smallest]:
    smallest = right
  if smallest != index:
    swap(heap[index], heap[smallest])
    bubbleDown(heap, smallest)

medium

A. The recursive call should be outside the if condition

B. The indices for left and right children are incorrect

C. The swap should happen only if smallest equals index

D. The comparisons should use '<' instead of '>' to find the smallest child

Solution

1. Step 1: Check comparison logic

   In a min-heap, bubble down swaps with the smaller child, so comparisons must find the smallest value.

2. Step 2: Identify incorrect operator

   The code uses '>' which finds the largest child; it should use '<' to find the smallest child.

3. Final Answer:

   The comparisons should use '<' instead of '>' to find the smallest child -> Option D

4. Quick Check:

   Use < to find smallest child in min-heap bubble down = A [OK]

Hint: Use '<' to find smaller child in min-heap bubble down [OK]

Common Mistakes:

• Using '>' instead of '<' in comparisons
• Mixing up left/right child indices
• Calling bubbleDown outside swap condition

5. You have a max-heap represented as [50, 30, 40, 10, 20, 35]. After extracting the root and performing bubble down, what is the resulting array?

hard

A. [40, 30, 35, 10, 20]

B. [40, 30, 20, 10, 35]

C. [40, 35, 30, 10, 20]

D. [35, 30, 40, 10, 20]

Solution

1. Step 1: Remove root and replace with last element

   Remove 50, replace root with 35: [35, 30, 40, 10, 20]

2. Step 2: Bubble down to restore max-heap

   Compare 35 with children 30 and 40; swap with larger child 40: [40, 30, 35, 10, 20]

3. Step 3: Continue bubbling down

   Now 35 at index 2 has children indices 5 and 6 which don't exist, so stop.

4. Final Answer:

   [40, 30, 35, 10, 20] -> Option A

5. Quick Check:

   Bubble down swaps with larger child in max-heap = C [OK]

Hint: Swap root with last, bubble down with larger child in max-heap [OK]

Common Mistakes:

• Swapping with smaller child in max-heap
• Not replacing root with last element
• Stopping bubble down too early