Heap extraction (bubble down) in Data Structures Theory - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main purpose of the bubble down process in heap extraction?

The bubble down process restores the heap property after removing the root by moving the new root element down the tree until it is in the correct position.

beginner

In a max-heap, after extracting the root, which element replaces the root before bubbling down?

The last element in the heap replaces the root before the bubble down process begins.

intermediate

Describe the condition that determines when to stop the bubble down process.

Bubble down stops when the current node is greater than or equal to its children (in a max-heap) or when it has no children to compare with.

intermediate

Why is bubble down important for maintaining heap structure?

Bubble down ensures the heap property is maintained by repositioning the root element after extraction, keeping the heap organized for efficient operations.

intermediate

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

The time complexity of bubble down is O(log n), where n is the number of elements in the heap, because it moves down at most the height of the heap.

What element replaces the root in a heap during extraction before bubble down?

A The smallest element in the heap

B The last element in the heap

C A new random element

D The second element in the heap

The last element in the heap replaces the root to maintain the complete tree structure before bubble down.

When does the bubble down process stop in a max-heap?

A When the current node is smaller than its children

B When the heap is empty

C When the current node is larger than or equal to its children

D After one swap only

Bubble down stops when the current node is larger than or equal to its children, maintaining the max-heap property.

What is the main goal of the bubble down operation?

A To restore heap property after extraction

B To insert a new element

C To sort the heap

D To delete the last element

Bubble down restores the heap property after the root is removed during extraction.

What is the time complexity of bubble down in a heap with n elements?

A O(n log n)

B O(n)

C O(1)

D O(log n)

Bubble down takes O(log n) time because it moves down the height of the heap, which is logarithmic in the number of elements.

Which heap property does bubble down help maintain?

A Heap order property (max or min)

B Balanced tree height

C Complete binary tree structure

D Sorted order of elements

Bubble down maintains the heap order property by repositioning elements after extraction.

Explain the steps involved in the heap extraction process using bubble down.

Think about what happens after removing the top element in a heap.

Why is bubble down necessary after extracting the root from a heap?

Consider what happens to the heap structure after root removal.

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