Heap Insert Operation Bubble Up in DSA Typescript - Time & Space Complexity

Time Complexity: Heap Insert Operation Bubble Up

O(log n)

Understanding Time Complexity

When we add a new item to a heap, we need to keep the heap rules intact. This involves moving the new item up the heap if it's bigger or smaller than its parent.

We want to know how the time to do this moving up changes as the heap grows.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function bubbleUp(heap: number[], index: number): void {
  while (index > 0) {
    const parentIndex = Math.floor((index - 1) / 2);
    if (heap[index] <= heap[parentIndex]) break;
    [heap[index], heap[parentIndex]] = [heap[parentIndex], heap[index]];
    index = parentIndex;
  }
}
    

This code moves the newly added element up the heap until the heap property is restored.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that compares and swaps the new element with its parent.
• How many times: At most, this loop runs once per level of the heap, moving from the inserted node up to the root.

How Execution Grows With Input

Each time we add a new item, it might move up several levels. The heap is a tree with levels growing slowly as the heap grows.

Input Size (n)	Approx. Operations
10	About 4 moves (levels)
100	About 7 moves
1000	About 10 moves

Pattern observation: The number of moves grows slowly, roughly with the height of the heap, which increases as the logarithm of the number of items.

Final Time Complexity

Time Complexity: O(log n)

This means the time to insert and bubble up grows slowly as the heap gets bigger, only increasing with the height of the heap.

Common Mistake

[X] Wrong: "The bubble up always takes as long as the number of items in the heap (O(n))."

[OK] Correct: The heap is a balanced tree, so the bubble up only moves up the height of the tree, which grows much slower than the total number of items.

Interview Connect

Understanding how bubble up works and its time cost helps you explain heap insertions clearly and confidently, a skill often valued in interviews.

Self-Check

What if we changed the heap to a linked structure instead of an array? How would the time complexity of bubble up change?

Practice

1. What is the main purpose of the bubble up operation during a heap insert?

easy

A. To swap all elements randomly in the heap

B. To remove the root element from the heap

C. To sort all elements in the heap in ascending order

D. To restore the heap property by moving the new element up if it is smaller (min-heap) or larger (max-heap) than its parent

Solution

1. Step 1: Understand the heap property

   In a min-heap, each parent node must be smaller than its children. In a max-heap, each parent must be larger.

2. Step 2: Purpose of bubble up

   When inserting a new element, it is placed at the end. Bubble up moves it up the tree to restore the heap property if violated.

3. Final Answer:

   To restore the heap property by moving the new element up if it is smaller (min-heap) or larger (max-heap) than its parent -> Option D

4. Quick Check:

   Bubble up fixes heap order after insert [OK]

Hint: Bubble up fixes heap order by moving new element up [OK]

Common Mistakes:

• Confusing bubble up with removing elements
• Thinking bubble up sorts the entire heap
• Assuming bubble up swaps elements randomly

2. Which of the following TypeScript code snippets correctly performs the bubble up step in a min-heap after inserting a new element at the end?

easy

A. while (index > 0 && heap[index] < heap[parentIndex]) { swap(heap, index, parentIndex); index = parentIndex; }

B. while (index > 0 && heap[index] > heap[parentIndex]) { swap(heap, index, parentIndex); index = parentIndex; }

C. for (let i = 0; i < heap.length; i++) { if (heap[i] < heap[i+1]) swap(heap, i, i+1); }

D. if (heap[index] > heap[parentIndex]) { swap(heap, index, parentIndex); }

Solution

1. Step 1: Identify min-heap bubble up condition

   In a min-heap, bubble up swaps while the child is smaller than the parent.

2. Step 2: Check code correctness

   while (index > 0 && heap[index] < heap[parentIndex]) { swap(heap, index, parentIndex); index = parentIndex; } uses a while loop with condition heap[index] < heap[parentIndex], correctly swapping and updating index.

3. Final Answer:

   while (index > 0 && heap[index] < heap[parentIndex]) { swap(heap, index, parentIndex); index = parentIndex; } -> Option A

4. Quick Check:

   Min-heap bubble up swaps smaller child up [OK]

Hint: Bubble up swaps while child < parent in min-heap [OK]

Common Mistakes:

• Using > instead of < for min-heap
• Using for loop instead of while
• Not updating index after swap

3. Given a max-heap represented as an array: [50, 30, 40, 10, 20], what will be the heap array after inserting 45 and performing bubble up?

medium

A. [50, 45, 40, 10, 20, 30]

B. [50, 30, 45, 10, 20, 40]

C. [50, 30, 40, 10, 20, 45]

D. [45, 30, 50, 10, 20, 40]

Solution

1. Step 1: Insert 45 at the end

   Array becomes [50, 30, 40, 10, 20, 45]

2. Step 2: Bubble up 45 in max-heap

   Parent index of 45 (index 5) is 2 (value 40). Since 45 > 40, swap. New array: [50, 30, 45, 10, 20, 40]. Now index 2's parent is 0 (value 50). 45 < 50, stop bubbling up.

3. Final Answer:

   [50, 30, 45, 10, 20, 40] -> Option B

4. Quick Check:

   Bubble up swaps 45 with 40 only [OK]

Hint: Insert at end, swap up while child > parent in max-heap [OK]

Common Mistakes:

• Not swapping 45 with 40
• Swapping 45 with 30 instead of 40
• Stopping bubble up too early

4. Consider this bubble up code snippet for a min-heap insert:

while (index > 0 && heap[index] > heap[parentIndex]) {
  swap(heap, index, parentIndex);
  index = parentIndex;
}

What is the error in this code?

medium

A. The parentIndex is not updated inside the loop

B. The swap function is called incorrectly

C. The comparison operator should be '<' to maintain min-heap property

D. The loop should run while index < 0

Solution

1. Step 1: Check comparison for min-heap bubble up

   In a min-heap, bubble up swaps when child is smaller than parent, so condition should be heap[index] < heap[parentIndex].

2. Step 2: Identify error in code

   The code uses '>' which is incorrect for min-heap bubble up, causing wrong swaps.

3. Final Answer:

   The comparison operator should be '<' to maintain min-heap property -> Option C

4. Quick Check:

   Min-heap bubble up uses < comparison [OK]

Hint: Min-heap bubble up swaps when child < parent [OK]

Common Mistakes:

• Using > instead of < in condition
• Not updating parentIndex after swap
• Loop condition index > 0 is correct, not < 0

5. You have a min-heap represented as [5, 10, 15, 20, 25, 30]. You insert 12 and perform bubble up. After that, you insert 3 and perform bubble up again. What is the final heap array?

hard

A. [3, 5, 12, 10, 25, 30, 15, 20]

B. [3, 5, 15, 10, 25, 30, 12, 20]

C. [3, 5, 12, 20, 25, 30, 15, 10]

D. [3, 5, 15, 20, 25, 30, 12, 10]

Solution

1. Step 1: Insert 12 and bubble up

   Insert 12 at index 6: [5,10,15,20,25,30,12]. Parent of index 6 is 2 (value 15). Since 12 < 15, swap: [5,10,12,20,25,30,15]. Parent of index 2 is 0 (value 5). 12 > 5, stop.

2. Step 2: Insert 3 and bubble up

   Insert 3 at index 7: [5,10,12,20,25,30,15,3]. Parent of index 7 is 3 (value 20). 3 < 20, swap: [5,10,12,3,25,30,15,20]. Parent of index 3 is 1 (value 10). 3 < 10, swap: [5,3,12,10,25,30,15,20]. Parent of index 1 is 0 (value 5). 3 < 5, swap: [3,5,12,10,25,30,15,20]. Now at root, stop.

3. Final Answer:

   [3, 5, 12, 10, 25, 30, 15, 20] -> Option A

4. Quick Check:

   Bubble up swaps until heap property restored [OK]

Hint: Bubble up each insert until parent smaller in min-heap [OK]

Common Mistakes:

• Not swapping 3 all the way to root
• Incorrect parent index calculation
• Stopping bubble up too early