Build Heap from Array Heapify in DSA Typescript - 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 array using heapify.

Specifically, how the time grows as the array size increases.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

function heapify(arr: number[], n: number, i: number): void {
  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 unsorted array by calling heapify from the bottom up.

Identify Repeating Operations

• Primary operation: The heapify function called on each non-leaf node.
• How many times: Called once for each node from the middle to the root, about n/2 times.
• Inside heapify: It may recurse down the tree, moving elements to maintain heap property.

How Execution Grows With Input

Heapify runs faster on nodes near the bottom because they have fewer levels to move down.

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

Pattern observation: The total work grows roughly linearly with input size, not quadratically.

Final Time Complexity

Time Complexity: O(n)

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

Common Mistake

[X] Wrong: "Heapify runs in O(n log n) because heapify is called n/2 times and each call takes O(log n)."

[OK] Correct: Most heapify calls run on nodes near the bottom with small heights, so they take less time. The total adds up to O(n), not O(n log n).

Interview Connect

Understanding this helps you explain why heap construction is efficient and shows you can analyze recursive code carefully.

Self-Check

What if we called heapify starting from the root down to the last node instead of bottom-up? 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 find the maximum element in the array

B. To sort the array in ascending order

C. To reverse the array elements

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 parent nodes follow a specific order compared to children (max-heap or min-heap).

2. Step 2: Role of heapify

   Heapify rearranges elements so that the heap property holds for every subtree, starting from the bottom.

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 order to build a valid heap fast [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 TypeScript syntax to start heapifying from the last non-leaf node in an array arr of length n?

easy

A. for(let i = Math.floor(n / 2); i < n; i++) { heapify(arr, n, i); }

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

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

D. for(let i = 0; i < n; i++) { heapify(arr, n, i); }

Solution

1. Step 1: Identify last non-leaf node index

   In a heap array, last non-leaf node is at index Math.floor(n/2) - 1.

2. Step 2: Loop from last non-leaf to root

   We heapify from this index down to 0 to ensure all subtrees satisfy heap property.

3. Final Answer:

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

4. Quick Check:

   Start heapify at last non-leaf node [OK]

Hint: Start heapify at Math.floor(n/2)-1 down to 0 [OK]

Common Mistakes:

• Starting heapify from 0 upwards
• Starting heapify from last element
• Looping beyond last non-leaf node

3. Given the array arr = [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. [5, 4, 3, 1, 9, 2]

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

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

Solution

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

   Heapify at index 2: children 5 (index 5), swap 2 and 5 to get [3,9,5,1,4,2].

2. Step 2: Heapify at index 1 and 0

   Index 1: children 1 and 4, max is 9 at index 1, no swap. Index 0: children 9 and 5, swap 3 and 9. Final array: [9,4,5,1,3,2]

3. Final Answer:

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

4. Quick Check:

   Max-heap root is 9, children smaller [OK]

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

Common Mistakes:

• Not swapping correctly during heapify
• Confusing min-heap with max-heap
• Leaving original array unchanged

4. Consider this TypeScript code snippet to build a max heap:

function heapify(arr: number[], n: number, i: number) {
  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, n, largest);
  }
}

function buildHeap(arr: number[]) {
  let n = arr.length;
  for (let i = 0; i <= Math.floor(n / 2); i++) {
    heapify(arr, n, i);
  }
}

What is the error in this code?

medium

A. The loop in buildHeap should run from Math.floor(n/2)-1 down to 0

B. The heapify function swaps elements incorrectly

C. The left and right child indices are calculated wrongly

D. The buildHeap function should not call heapify

Solution

1. Step 1: Analyze buildHeap loop range

   The loop runs from 0 up to Math.floor(n/2), but heapify must start from last non-leaf node down to root.

2. Step 2: Correct loop direction and range

   Loop should be: for(let i = Math.floor(n/2) - 1; i >= 0; i--) to ensure all subtrees heapify properly.

3. Final Answer:

   The loop in buildHeap should run from Math.floor(n/2)-1 down to 0 -> Option A

4. Quick Check:

   Heapify bottom-up loop direction matters [OK]

Hint: Loop heapify bottom-up from last non-leaf node [OK]

Common Mistakes:

• Looping heapify top-down instead of bottom-up
• Incorrect child index calculations
• Not swapping elements during heapify

5. You have an array arr = [4, 10, 3, 5, 1, 2]. After building a min-heap using heapify, what will be the array representation?

hard

A. [1, 4, 2, 10, 5, 3]

B. [1, 3, 2, 5, 10, 4]

C. [1, 4, 2, 5, 10, 3]

D. [1, 2, 3, 4, 5, 10]

Solution

1. Step 1: Build min-heap bottom-up

   Start heapify from index 2 (Math.floor(6/2)-1): heapify at index 2 swaps 3 and 2 to get [4,10,2,5,1,3].

2. Step 2: Heapify at index 1 and 0

   At index 1: children 5 and 1, swap 10 and 1 -> [4,1,2,5,10,3]. At index 0: children 1 and 2, swap 4 and 1 -> [1,4,2,5,10,3].

3. Final Answer:

   [1, 4, 2, 5, 10, 3] -> Option C

4. Quick Check:

   Min-heap root is smallest element 1 [OK]

Hint: Min-heap root is smallest; heapify swaps bottom-up [OK]

Common Mistakes:

• Confusing min-heap with max-heap results
• Not swapping correctly during heapify
• Assuming sorted array equals heap