Build Heap from Array Heapify in DSA Javascript - Time & Space Complexity

Time Complexity: Build Heap from Array Heapify

O(n)

Understanding Time Complexity

We want to understand how long it takes to build a heap from an unordered array using heapify.

How does the time needed grow as the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function buildHeap(arr) {
  const n = arr.length;
  for (let i = Math.floor(n / 2) - 1; i >= 0; i--) {
    heapify(arr, n, i);
  }
}

function heapify(arr, n, i) {
  let largest = i;
  const left = 2 * i + 1;
  const right = 2 * i + 2;

  if (left < n && arr[left] > arr[largest]) largest = left;
  if (right < n && arr[right] > arr[largest]) largest = right;

  if (largest !== i) {
    [arr[i], arr[largest]] = [arr[largest], arr[i]];
    heapify(arr, n, largest);
  }
}
    

This code builds a max heap from an array by calling heapify on each non-leaf node starting from the bottom.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The heapify function called on each node from the bottom up.
• How many times: The for loop runs about n/2 times, and each heapify can recurse down the tree height.

How Execution Grows With Input

Heapify runs faster near the bottom because those nodes have smaller heights, and slower near the top with bigger heights.

Input Size (n)	Approx. Operations
10	About 20-30 operations
100	About 300-400 operations
1000	About 4000-5000 operations

Pattern observation: The total work grows a bit faster than linear but much slower than quadratic, closer to n log n.

Final Time Complexity

Time Complexity: O(n)

This means building a heap from an array takes time roughly proportional to the number of elements.

Common Mistake

[X] Wrong: "Heapify on each node takes O(log n), so building heap is O(n log n)."

[OK] Correct: Nodes near the bottom have smaller heights, so heapify is faster there. The total work adds up to O(n), not O(n log n).

Interview Connect

Understanding this time complexity shows you know how to analyze recursive and layered operations, a key skill for many coding challenges.

Self-Check

"What if we called heapify starting from the root down to the leaves instead? How would the time complexity change?"

Practice

1. What is the main purpose of the heapify process when building a heap from an array?

easy

A. To reverse the array elements

B. To sort the array in ascending order

C. To find the maximum element in the array

D. To rearrange the array elements to satisfy the heap property

Solution

1. Step 1: Understand heap property

   A heap is a special tree where each parent node is greater (max-heap) or smaller (min-heap) than its children.

2. Step 2: Role of heapify

   Heapify rearranges elements so the heap property holds for every subtree, starting from a node downwards.

3. Final Answer:

   To rearrange the array elements to satisfy the heap property -> Option D

4. Quick Check:

   Heapify = Rearrange for heap property [OK]

Hint: Heapify fixes subtree to maintain heap order [OK]

Common Mistakes:

• Thinking heapify sorts the array
• Confusing heapify with finding max element
• Assuming heapify reverses the array

2. Which of the following is the correct way to start heapifying an array arr of length n in JavaScript to build a max heap?

easy

A. for(let i = 0; i < n; i++) heapify(arr, i, n);

B. for(let i = 1; i < n; i++) heapify(arr, i, n);

C. for(let i = Math.floor(n/2) - 1; i >= 0; i--) heapify(arr, i, n);

D. for(let i = n - 1; i >= 0; i--) heapify(arr, i, n);

Solution

1. Step 1: Identify heapify start index

   Heapify starts from the last non-leaf node, which is at index Math.floor(n/2) - 1.

2. Step 2: Loop backwards to root

   We heapify each node from this index down to 0 to ensure the entire tree satisfies heap property.

3. Final Answer:

   for(let i = Math.floor(n/2) - 1; i >= 0; i--) heapify(arr, i, n); -> Option C

4. Quick Check:

   Start heapify at last non-leaf node [OK]

Hint: Start heapify from last parent node backwards [OK]

Common Mistakes:

• Starting heapify from index 0 forwards
• Heapifying leaf nodes unnecessarily
• Using incorrect loop boundaries

3. Given the array [3, 9, 2, 1, 4, 5], what is the array after building a max heap using heapify?

medium

A. [9, 4, 5, 1, 3, 2]

B. [9, 4, 5, 3, 1, 2]

C. [5, 4, 9, 1, 3, 2]

D. [9, 3, 5, 1, 4, 2]

Solution

1. Step 1: Start heapify from index 2 (Math.floor(6/2)-1)

   Heapify at index 2: children are indices 5 (5) and no right child. 2 < 5, swap to get [3,9,5,1,4,2].

2. Step 2: Heapify at index 1

   Children at indices 3 (1) and 4 (4). 9 > 4 and 1, no swap needed.

3. Step 3: Heapify at index 0

   Children at indices 1 (9) and 2 (5). 3 < 9, swap 3 and 9: [9,3,5,1,4,2]. Heapify at index 1 again: children 3 (1) and 4 (4). 3 < 4, swap: [9,4,5,1,3,2].

4. Final Answer:

   [9, 4, 5, 1, 3, 2] -> Option A

5. Quick Check:

   Max heap root is 9, children smaller [OK]

Hint: Heapify bottom-up swaps to fix heap [OK]

Common Mistakes:

• Not heapifying children after swap
• Swapping with wrong child
• Stopping heapify too early

4. Consider this heapify function snippet in JavaScript:

function heapify(arr, i, n) {
  let largest = i;
  let left = 2 * i + 1;
  let right = 2 * i + 2;
  if (left < n && arr[left] > arr[largest]) largest = left;
  if (right < n && arr[right] > arr[largest]) largest = right;
  if (largest !== i) {
    [arr[i], arr[largest]] = [arr[largest], arr[i]];
    heapify(arr, largest, n);
  }
}

What is the bug if the function is called with heapify(arr, 0, arr.length - 1) instead of heapify(arr, 0, arr.length)?

medium

A. It will ignore the last element of the array during heapify

B. It will cause an infinite recursion

C. It will throw a syntax error

D. It will heapify the entire array correctly

Solution

1. Step 1: Understand parameter n meaning

   Parameter n is the array length limit for heapify to avoid out-of-bound access.

2. Step 2: Effect of passing arr.length - 1

   Passing arr.length - 1 excludes the last element index (n-1) from heapify checks, so last element is ignored.

3. Final Answer:

   It will ignore the last element of the array during heapify -> Option A

4. Quick Check:

   n must be array length, not length-1 [OK]

Hint: Use full array length as heapify limit [OK]

Common Mistakes:

• Confusing length and last index
• Expecting syntax error instead of logic error
• Ignoring off-by-one errors

5. You have an array [4, 10, 3, 5, 1] and want to build a min heap using heapify. Which of the following changes is necessary in the heapify function to build a min heap instead of a max heap?

hard

A. Swap the left and right child indices calculations

B. Change comparisons from '>' to '<' when comparing children with parent

C. Change comparisons from '<' to '>' when comparing children with parent

D. No change needed; heapify works for both min and max heaps

Solution

1. Step 1: Understand max vs min heap comparison

   Max heap keeps parent >= children; min heap keeps parent <= children.

2. Step 2: Adjust comparison operators

   To build min heap, change '>' to '<' so parent is smaller than children.

3. Final Answer:

   Change comparisons from '>' to '<' when comparing children with parent -> Option B

4. Quick Check:

   Min heap uses less-than comparisons [OK]

Hint: Flip comparison signs for min heap [OK]

Common Mistakes:

• Swapping child indices instead of comparisons
• Leaving comparisons unchanged
• Assuming heapify auto-detects heap type