Heap Extract Min or Max Bubble Down in DSA Java - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the purpose of the bubble down operation in a heap after extracting the min or max element?

Bubble down restores the heap property by moving the new root element down the tree until it is in the correct position, ensuring the heap remains a valid min-heap or max-heap.

beginner

In a max-heap, after extracting the max element, which child does the bubble down operation compare the root with first?

It compares the root with its larger child to decide if a swap is needed to maintain the max-heap property.

intermediate

What is the time complexity of the extract min or max operation including bubble down in a heap of size n?

The time complexity is O(log n) because bubble down moves down the height of the heap, which is logarithmic in the number of elements.

beginner

Why do we replace the root with the last element in the heap before performing bubble down during extract operation?

Replacing the root with the last element keeps the heap complete (no gaps) and allows bubble down to restore the heap property from the root.

intermediate

Describe the difference between bubble down in a min-heap and a max-heap.

In a min-heap, bubble down swaps the root with the smaller child to maintain the min-heap property. In a max-heap, it swaps with the larger child to maintain the max-heap property.

What element is removed during the extract operation in a min-heap?

A The smallest element (root)

B The largest element (leaf)

C A random element

D The last element

In a min-heap, the root holds the smallest element, which is removed during extract.

After removing the root in a max-heap, which element replaces it before bubble down?

A The smallest child

B A new random element

C The last element in the heap

D The second largest element

The last element replaces the root to keep the heap complete before bubble down.

What is the main goal of the bubble down operation?

A Remove the last element

B Add a new element at the bottom

C Sort the entire heap

D Restore heap property by moving element down

Bubble down moves the root element down to restore the heap property.

Which of these is true about the height of a heap with n elements?

A Height is n

B Height is about log n

C Height is constant

D Height is n squared

Heap height grows logarithmically with the number of elements.

In a min-heap bubble down, if both children are larger than the root, what happens?

A No swap, bubble down ends

B Swap with the left child

C Swap with the right child

D Remove the root

If both children are larger, the heap property holds and bubble down stops.

Explain step-by-step how the bubble down operation works after extracting the root in a max-heap.

Think about how to keep the largest element at the root.

Describe why the extract min or max operation in a heap is efficient compared to searching for min or max in an unsorted array.

Focus on how heap structure helps quick access.

Practice

1. What is the main purpose of the bubble down operation in a heap after extracting the min or max element?

easy

A. To remove all leaf nodes from the heap

B. To restore the heap property by moving the root element down to its correct position

C. To swap the last element with the root without any further changes

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

Solution

1. Step 1: Understand the extract operation

   When we remove the min or max element (usually the root), the heap may lose its order property.

2. Step 2: Purpose of bubble down

   Bubble down moves the new root element down the tree to restore the heap property by swapping it with the smaller (min-heap) or larger (max-heap) child.

3. Final Answer:

   To restore the heap property by moving the root element down to its correct position -> Option B

4. Quick Check:

   Bubble down fixes heap order after extraction [OK]

Hint: Bubble down fixes heap order after removing root [OK]

Common Mistakes:

• Thinking bubble down inserts new elements
• Assuming bubble down removes leaf nodes
• Believing bubble down only swaps root and last element

2. Which of the following Java method signatures correctly represents a bubble down operation for a min-heap stored in an array?

easy

A. void bubbleDown(int[] heap, int index, int size)

B. int bubbleDown(int[] heap, int size)

C. void bubbleDown(int index, int[] heap)

D. boolean bubbleDown(int[] heap, int index)

Solution

1. Step 1: Identify parameters needed for bubble down

   We need the heap array, the current index to bubble down from, and the size of the heap to avoid out-of-bounds.

2. Step 2: Match method signature

   void bubbleDown(int[] heap, int index, int size) has all parameters in correct order and returns void as bubble down modifies the heap in place.

3. Final Answer:

   void bubbleDown(int[] heap, int index, int size) -> Option A

4. Quick Check:

   Correct parameters and void return for bubble down [OK]

Hint: Bubble down needs array, index, and size parameters [OK]

Common Mistakes:

• Missing size parameter causing index errors
• Wrong return type expecting a value
• Incorrect parameter order

3. Given the min-heap array [2, 5, 3, 10, 7], after extracting the min element and performing bubble down, what will be the resulting heap array?

medium

A. [3, 5, 7, 10]

B. [5, 7, 3, 10]

C. [3, 7, 5, 10]

D. [5, 3, 7, 10]

Solution

1. Step 1: Remove min and replace root

   Remove 2 (min), replace root with last element 7: [7, 5, 3, 10]

2. Step 2: Bubble down from root

   Compare 7 with children 5 and 3; smallest is 3. Swap 7 and 3: [3, 5, 7, 10]. Now 7 has children 10 only, 7 < 10, so stop.

3. Final Answer:

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

4. Quick Check:

   Bubble down restores min-heap order [OK]

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

Common Mistakes:

• Not replacing root with last element before bubble down
• Swapping with wrong child (larger instead of smaller)
• Stopping bubble down too early

4. Consider this bubble down code snippet for a max-heap. Identify the bug:

void bubbleDown(int[] heap, int index, int size) {
  int largest = index;
  int left = 2 * index + 1;
  int right = 2 * index + 2;
  if (left < size && heap[left] > heap[largest]) {
    largest = left;
  }
  if (right < size && heap[right] > heap[largest]) {
    largest = right;
  }
  if (largest != index) {
    int temp = heap[index];
    heap[index] = heap[largest];
    heap[largest] = temp;
    bubbleDown(heap, largest, size);
  }
}

medium

A. The comparison operators should be < instead of > for max-heap

B. The left and right child indices calculation is incorrect

C. The recursive call should be bubbleDown(heap, index, size) instead of bubbleDown(heap, largest, size)

D. There is no bug; the code correctly performs bubble down

Solution

1. Step 1: Check child index calculations

   Left = 2*index+1 and right = 2*index+2 are correct for binary heap stored in array.

2. Step 2: Verify comparisons and recursion

   Comparisons use > for max-heap to find largest child. Recursive call uses largest index to continue bubbling down correctly.

3. Final Answer:

   There is no bug; the code correctly performs bubble down -> Option D

4. Quick Check:

   Child indices and recursion are correct [OK]

Hint: Check indices and recursion carefully; they are often correct [OK]

Common Mistakes:

• Confusing child index formulas
• Wrong comparison operators for max-heap
• Incorrect recursive call index

5. You have a max-heap represented by the array [20, 18, 15, 10, 12, 9, 8]. After extracting the max element and performing bubble down, what is the resulting heap array?

hard

A. [18, 15, 12, 10, 9, 8]

B. [18, 12, 15, 10, 9, 8]

C. [18, 12, 15, 10, 8, 9]

D. [18, 10, 15, 12, 9, 8]

Solution

1. Step 1: Extract max and replace root

   Remove 20, replace root with last element 8: [8, 18, 15, 10, 12, 9]

2. Step 2: Bubble down from root

   Compare 8 with children 18 and 15; largest is 18. Swap 8 and 18: [18, 8, 15, 10, 12, 9]. Now bubble down at index 1.

3. Step 3: Continue bubble down

   At index 1, children are 10 and 12; largest is 12. Swap 8 and 12: [18, 12, 15, 10, 8, 9]. At index 4, children do not exist, stop.

4. Final Answer:

   [18, 12, 15, 10, 8, 9] -> Option C

5. Quick Check:

   Bubble down swaps with largest child until heap restored [OK]

Hint: Replace root with last, swap down with largest child repeatedly [OK]

Common Mistakes:

• Not continuing bubble down after first swap
• Swapping with smaller child by mistake
• Forgetting to update heap size after extraction