Heap extraction (bubble down) in Data Structures Theory - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to identify the root element in a max-heap.

Data Structures Theory

root = heap[[1]]

Drag options to blanks, or click blank then click option'

A0

B1

C-1

Dlen(heap)

Attempts:

3 left

💡 Hint

Common Mistakes

Using 1 as the root index, which is incorrect for zero-based arrays.

Using -1 or len(heap) which are invalid indices for the root.

The root element of a max-heap is always at index 0 in the array representation.

2fill in blank

medium

Complete the code to find the left child index of a node at index i in a heap.

Data Structures Theory

left_child = 2 * i [1] 1

Drag options to blanks, or click blank then click option'

A-

B+

C*

D//

Attempts:

3 left

💡 Hint

Common Mistakes

Using subtraction instead of addition.

Using division or multiplication incorrectly.

In a heap, the left child of node at index i is at 2*i + 1.

3fill in blank

hard

Fix the error in the condition to check if the right child exists in the heap array.

Data Structures Theory

if right_child_index [1] len(heap):

Drag options to blanks, or click blank then click option'

A<

B<=

C>

D>=

Attempts:

3 left

💡 Hint

Common Mistakes

Using <= which allows out-of-range index.

Using > or >= which are incorrect for existence check.

The right child exists only if its index is less than the length of the heap array.

4fill in blank

hard

Fill both blanks to correctly swap the parent with the larger child during bubble down.

Data Structures Theory

if heap[[1]] < heap[[2]]:
    heap[[1]], heap[[2]] = heap[[2]], heap[[1]]

Drag options to blanks, or click blank then click option'

Aparent_index

Bleft_child_index

Cright_child_index

Droot_index

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping with the left child when the right child is larger.

Using root_index instead of parent_index.

During bubble down, the parent is swapped with the larger child, here the right child index is used.

5fill in blank

hard

Fill all three blanks to complete the bubble down step choosing the larger child correctly.

Data Structures Theory

larger_child = [1] if heap[[2]] > heap[[3]] else [3]

Drag options to blanks, or click blank then click option'

Aleft_child_index

Bright_child_index

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up left and right child indices in the comparison.

Using the same index for both comparison sides.

This line selects the larger child index by comparing left and right child values.

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