Build Heap from Array Heapify in DSA Python - 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.

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

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def heapify(arr, n, i):
    largest = i
    left = 2 * i + 1
    right = 2 * i + 2

    if left < n and arr[left] > arr[largest]:
        largest = left
    if right < n and arr[right] > arr[largest]:
        largest = right

    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        heapify(arr, n, largest)


def build_heap(arr):
    n = len(arr)
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)

This code builds a max heap from an array by calling heapify on all non-leaf nodes from bottom up.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The heapify function called on each non-leaf node.
• How many times: Called about n/2 times (for each non-leaf node).
• Inside heapify: It may recurse down the tree, but the depth depends on the node's height.

How Execution Grows With Input

Heapify work depends on node height; nodes near bottom do less work, nodes near top do more.

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 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 is called n/2 times and each call can go down log n levels, so total time is O(n log n)."

[OK] Correct: Most heapify calls are on nodes near the bottom with small height, so they do less work. The total sums up to O(n), not O(n log n).

Interview Connect

Knowing that building a heap is O(n) helps you explain efficient data structure setup in interviews confidently.

Self-Check

"What if we used a simple insert operation repeatedly to build the heap instead of heapify? 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 rearrange the array elements to satisfy the heap property

B. To sort the array in ascending order

C. To reverse the array elements

D. To remove duplicate elements from the array

Solution

1. Step 1: Understand heap property

   The heap property requires each parent node to be greater (max-heap) or smaller (min-heap) than its children.

2. Step 2: Role of heapify

   Heapify rearranges elements to ensure this property holds for every subtree starting from a node.

3. Final Answer:

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

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 reversing array
• Assuming heapify removes duplicates

2. Which of the following is the correct Python syntax to call the heapify function from the heapq module on a list named arr?

easy

A. arr.heapify()

B. heapify.heapq(arr)

C. heapq.heapify(arr)

D. heapq.heapify = arr

Solution

1. Step 1: Identify module and function

   The heapq module provides the heapify function to convert a list into a heap.

2. Step 2: Correct function call syntax

   The correct syntax is heapq.heapify(arr) after importing heapq.

3. Final Answer:

   heapq.heapify(arr) -> Option C

4. Quick Check:

   heapq.heapify(list) is correct call [OK]

Hint: Call heapify as heapq.heapify(list) after import [OK]

Common Mistakes:

• Using wrong order like heapify.heapq(arr)
• Calling heapify as a method on list
• Assigning heapify function instead of calling

3. What will be the output of the following Python code?

import heapq
arr = [3, 1, 6, 5, 2, 4]
heapq.heapify(arr)
print(arr)

medium

A. [6, 5, 4, 3, 2, 1]

B. [1, 2, 4, 5, 3, 6]

C. [1, 3, 2, 5, 6, 4]

D. [3, 1, 6, 5, 2, 4]

Solution

1. Step 1: Understand heapq.heapify behavior

   heapq.heapify converts list into a min-heap, smallest element at root (index 0).

2. Step 2: Apply heapify on given list

   After heapify, the list becomes [1, 2, 4, 5, 3, 6], which satisfies min-heap property.

3. Final Answer:

   [1, 2, 4, 5, 3, 6] -> Option B

4. Quick Check:

   heapify(min-heap) = [1, 2, 4, 5, 3, 6] [OK]

Hint: heapq.heapify creates min-heap, smallest first [OK]

Common Mistakes:

• Expecting sorted list instead of heap order
• Confusing max-heap with min-heap output
• Assuming original list order remains

4. The following code attempts to build a heap from an array but does not produce the correct heap. What is the error?

def build_heap(arr):
    n = len(arr)
    for i in range(n):
        heapify(arr, i, n)

def heapify(arr, i, n):
    largest = i
    left = 2 * i + 1
    right = 2 * i + 2
    if left < n and arr[left] > arr[largest]:
        largest = left
    if right < n and arr[right] > arr[largest]:
        largest = right
    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        heapify(arr, largest, n)

arr = [4, 10, 3, 5, 1]
build_heap(arr)
print(arr)

medium

A. The loop should run from n//2 - 1 down to 0, not from 0 to n

B. The heapify function should use min-heap logic instead of max-heap

C. The swap inside heapify is missing

D. The array length n is calculated incorrectly

Solution

1. Step 1: Analyze build_heap loop range

   Heap building requires heapify calls from last non-leaf node down to root, i.e., from n//2 - 1 down to 0.

2. Step 2: Identify incorrect loop direction

   The code loops from 0 to n, which is incorrect and misses proper heapify order.

3. Final Answer:

   The loop should run from n//2 - 1 down to 0, not from 0 to n -> Option A

4. Quick Check:

   Build heap loop must go bottom-up [OK]

Hint: Build heap bottom-up from last parent to root [OK]

Common Mistakes:

• Looping top-down instead of bottom-up
• Confusing min-heap and max-heap logic
• Forgetting to swap elements in heapify

5. Given an array arr = [9, 4, 7, 1, -2, 6, 5], you want to build a max-heap using the heapify method. Which sequence of indices should you call heapify on to build the heap correctly?

hard

A. From index 0 to len(arr)-1 in ascending order

B. Only on index 0

C. From index len(arr)-1 down to 0 in descending order

D. From index len(arr)//2 - 1 down to 0 in descending order

Solution

1. Step 1: Identify last non-leaf node

   For array length 7, last non-leaf node is at index 7//2 - 1 = 2.

2. Step 2: Correct heapify call order

   Heapify must be called from index 2 down to 0 to ensure all subtrees satisfy max-heap property.

3. Final Answer:

   From index len(arr)//2 - 1 down to 0 in descending order -> Option D

4. Quick Check:

   Heapify bottom-up from last parent to root [OK]

Hint: Start heapify from last parent index down to root [OK]

Common Mistakes:

• Calling heapify in ascending order
• Calling heapify only on root
• Calling heapify on all indices including leaves