Heapify operation in Data Structures Theory - Step-by-Step Execution

Concept Flow - Heapify operation

Start at index i

↓

Find left and right child indices

↓

Compare parent with children

↓

Is parent smallest/largest?

Yes→Done

No↓

Swap parent with smallest/largest child

↓

Repeat heapify at swapped child's index

↓

End

Heapify fixes the heap property starting from a node by comparing it with its children and swapping if needed, then continuing down the tree.

Execution Sample

Data Structures Theory

Array: [4, 10, 3, 5, 1]
Heapify at index 0 (value 4)
Left child index 1 (value 10)
Right child index 2 (value 3)
Swap 4 with 3
Heapify at index 2

This example shows heapify starting at root index 0, swapping with the smaller child to maintain min-heap property.

Analysis Table

Step	Current Index	Parent Value	Left Child Index	Left Child Value	Right Child Index	Right Child Value	Swap Occurred	Array State
1	0	4	1	10	2	3	Yes (4 <-> 3)	[3, 10, 4, 5, 1]
2	2	4	5	N/A	6	N/A	No	[3, 10, 4, 5, 1]
Exit	-	-	-	-	-	-	No swap needed, heapify complete	[3, 10, 4, 5, 1]

💡 Heapify stops when the parent is smaller than both children or no children exist.

State Tracker

Variable	Start	After Step 1	After Step 2	Final
Array	[4, 10, 3, 5, 1]	[3, 10, 4, 5, 1]	[3, 10, 4, 5, 1]	[3, 10, 4, 5, 1]
Current Index	0	2	2	2
Parent Value	4	4	4	4
Left Child Index	1	5	5	5
Right Child Index	2	6	6	6

Key Insights - 3 Insights

Why do we swap the parent with the smallest/largest child during heapify?

Swapping ensures the heap property is restored at the current node. For example, in step 1 of the execution_table, 4 is swapped with 3 because 3 is smaller, maintaining the min-heap property.

What happens if the current node has no children during heapify?

Heapify stops because there are no children to compare with. In step 2, indices 5 and 6 are out of array bounds, so no swap occurs and heapify ends.

Why do we continue heapify at the child's index after swapping?

Because swapping may break the heap property further down the tree. Continuing heapify at the swapped child's index fixes any violations deeper in the heap, as shown in step 2.

Visual Quiz - 3 Questions

Look at the execution_table at step 1, which values are swapped?

A 4 and 3

B 4 and 10

C 10 and 3

D No swap

💡 Hint

Check the 'Swap Occurred' and 'Array State' columns in step 1.

At which step does heapify stop because there are no valid children?

A Step 1

B Exit

C Step 2

D Heapify never stops

💡 Hint

Look at 'Left Child Index' and 'Right Child Index' values in step 2.

If the initial array was already a min-heap, what would the 'Swap Occurred' column show at step 1?

A Yes (swap happened)

B No

C Maybe

D Error

💡 Hint

Refer to the exit condition in the execution_table where no swap means heap property is satisfied.

Concept Snapshot

Heapify operation fixes the heap property starting at a node.
Compare parent with children.
Swap with smallest (min-heap) or largest (max-heap) child if needed.
Repeat heapify at swapped child's index.
Stops when heap property is restored or no children exist.

Full Transcript

Heapify is a process used in heaps to maintain the heap property. Starting at a given index, it compares the node's value with its children. If the heap property is violated, it swaps the node with the appropriate child (smallest for min-heap, largest for max-heap). Then it continues heapifying at the child's position. This continues until the node is in the correct position or it has no children. The example shows heapify on array [4, 10, 3, 5, 1] starting at index 0. The value 4 is swapped with 3 at index 2, then heapify continues at index 2 but stops because there are no children. This ensures the heap property is restored efficiently.