The heaps notes provide a comprehensive exploration of the heap data structure, offering a detailed understanding of its properties and operations. Starting with the fundamentals, the notes explain the concept of heaps, including their binary tree structure and the heap property that maintains the ordering of elements. They cover both max heaps and min heaps, discussing their respective characteristics and use cases. The notes delve into the various operations associated with heaps, such as inserting elements, removing elements, and heapifying arrays. Step-by-step explanations guide readers through the process of maintaining the heap property during these operations, ensuring a clear understanding of how heaps function.

1. Tree
CSC-114 Data Structure and Algorithms

2. Outline
 Binary Tree Variations
 Binary Heap Tree
 Max Heap
 Min Heap
 Insertion
 Deletion
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3. Binary Heap
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 A binary tree which holds two properties:
 Heap Order Property:
 Min-Heap Property: Every node is smaller than or equal to each of its children
 Max-Heap Property: Every node is larger than or equal to each of its children
 Shape Property:
 Tree is complete.
 A tree that is full at all levels except last level, and nodes are filled from left to right
 Which tree is binary heap?
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4. Finding Min
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 Finding minimum or maximum?
 It will always be root node with max or min value depending upon it is min heap or
max heap.
 So constant time required
 What about removal?
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5. Deletion
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 Deleting minimum?
 Root node will be removed
 Which node will be next root node?
 It must be next minimum that is 25
 Then which node will come at place of 25?
 Again, the minimum in sub tree of 25
 Is this heap tree any more?
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6. Deletion
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 Deleting minimum?
 What if we replace root node with last inserted node?
 Now the completeness property is reserved
 But another problem is created, what is that?
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7. Deletion
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 We need to perform another process to maintain heap order
 After replacing root node, check if it is greater than its children, swap with appropriate
child.
 Repeat this process till leaf or parent node becomes smaller than its children
 This process is called Heapify-Down or Down Heap
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8. Heapify-Down
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 Example-2
 Deletion involves following steps:
1. Replace root node with last inserted node, to maintain shape
2. Heapify-Down process, to maintain heap order
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9. Insertion
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 Insertion of a new node
 Always insert from left to right to maintain shape property of left completeness
 Let say starting from root node how to decide that we should go left or right?
 Always remember where you inserted last node
 But heap property is not maintained
 Solution?
 Repeatedly check if parent is larger than node, then swap the node with parent
 Process is called Heapify-Up or Up Heap
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10. Insertion
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 Insertion of a new node
 Always insert from left to right to maintain shape property of left completeness
 How?
 Always remember where you inserted last node
 But heap property is not maintained
 Solution?
 Repeatedly check if parent is larger than node, then swap the node with parent
 Process is called Heapify-Up or Up Heap
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11.  Insert 20
Swap with 35
Swap with 25
Heapify-Up
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 Insert 35
 Insert 25
 Swap with parent if parent is larger
 Insert 30
 Parent is already smaller
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12.  Insert 36
 Insert 41
 Insertion involves two steps:
1. Inserting node at correct position using last node, to maintain left completeness
2. Heapify-Up process, to maintain heap order
Insertion
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13. Implementation
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 Using Linked Memory Allocation
 Maintain two nodes:
 Root
 Last node
 Using Array
 Children of node at location k
 Left -> 2K+1
 Right -> 2K+2
 Parent of a node located at k
 (k-1)/2 (consider integer division)
0 1 2 3 4 5 6
20 25 30 35 36 41 null
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14. Using Array
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 Root is always 1st index
 Last node’s index is = size-1
 No need of functions for left, right, parent
 Just do calculation
 Deletion is always replacing root with last
 Then Heapify-Down
 Insertion is always at end of current nodes
 And then Heapify-Up
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0 1 2 3 4 5 6
20 25 30 35 36 41 null

15. Using Linked Memory Allocation
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 Need to maintain last node?
 In Deletion
1. Root is replaced with Last node
2. Last node is updated
3. And then Heapify-Down
 In Insertion
1. Last node used to find correct location for new node
2. Node is inserted
3. Last node is updated
4. And then Heapify-Up
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0 1 2 3 4 5 6
20 25 30 35 36 41 null

16. Example
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 See the figure, it’s a max heap
 Here last node is 4
 If we delete 48
 Which node will become last node now?
 In array?
 In Linked allocation?
0 1 2 3 4 5 6
48 35 40 10 16 34 4
0 1 2 3 4 5 6
40 35 34 10 16 4 null
After deletion

17. Example
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 Last node is 4
 Insert 50
 Which node will become last node now?
 In array?
 In Linked allocation?
 Perform few more insertions and deletions
 It will give you good understanding of last node
0 1 2 3 4 5 6
40 35 34 10 16 4 null
0 1 2 3 4 5 6
50 35 40 10 16 4 34
After insertion

18. Deletion: Linked Implementation
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 Delete
 Last is right node
 Last= sibling
Last
Last
Heapify-Down
Last
Last

19. Deletion: Linked Implementation
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 Delete
 Last is left node
 go up, until root or node is right
 21 is right
 last= most right of sibling 17
Heapify-Down
Last
Last
Last

20. Deletion: Linked Implementation
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 Perform delete in each figure to see how last is updated
Last
Last
Last
Last

21. Deletion: Linked Implementation
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1. Replace root with last node
2. Update last node
1. Find parent of last node
2. If last is right node
 last=parent.left
 parent.right=null
3. Else If last is left node
 Parent.left=null
 Repeatedly move up in hierarchy to parent nodes of parent until you reach root node or a node that is right node
 If root node
 grand parent is left child
 Last= most right node of root
 Else //its right node
 Last= most right of sibling of this node
3. Perform Heapify-Down process

22. Insertion: Linked Implementation
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 Insert 35
 Insert 25
 Last node is itself a root node
 New node will become left node
 Heapify-up(last)
 Now 35 is last node
 Insert 30
 Last node is a left node
 New node will become right node
 No need of Heapify-Up
 Now 30 is last node
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Heapify-Up
Last
Last Last
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23.  Insert 20
 Last node is right child
 Go up until you reach a node that is either left or root node
 25 is root, go to left until you reach leaf node that is 35
 Insert new node as left node
 Now 35 is last node
Insertion- Linked Implementation
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Heapify-Up
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24.  Insert 36
 Last node is a left node
 New node will become right node
 No need of Heapify-Up
 Now 36 is last node
 Insert 41
 Last node is right child
 Go up until you reach a node that is either left or root node
 25 is left child, go to its sibling that is 30
 Insert new node as left node
 No need of Heapify-Up
 Now 41 is last node
 How much time it will take to find insertion location?
Insertion- Linked Implementation
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25. Insertion: Linked Implementation
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1. If last node is root node
 new node is left child
2. If last node is left node
 new node is right node of parent of last
3. Otherwise, repeatedly go to parent nodes of last until you reach
a node that is either root node or a left child of its parent.
 If node is root node
 then simply go to its left until you reach leaf node
 If node is left child
 then go to its sibling node
 new node is left child of this node // (sibling will be a leaf node)
 last=new node //update last
 Perform Heapify_Up process
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Last

26. Applications
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 Priority Queue
 Heap data structure is mainly as priority queue, often they are used as synonyms
 Highest priority is always on top.
 Time complexity of binary tree depends upon height of tree which is equivalent to logN. Where
N is total number of nodes.
 What can be heap tree’s worst case?
Data
Structure
FindMin Add Remove
Unsorted Array O(n) O(1) O(n)
Sorted Array O(1) O(n) O(1)
Unsorted List O(n) O(1) O(n)
Sorted List O(1) O(n) O(1)
Heap O(1) O(log n) O(log n)

27. Applications
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 Heapsort
 Heap tree can also be used to sort a list of numbers
 How?
 Build the heap tree from given list
 Reconstruct list by repeatedly doing the following:
 Remove min and put in list until tree becomes empty
 Time Complexity? O(nlogn)
 Remove min?
 Total number of nodes?

28. Applications
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 Selection Algorithms
 Finding kth minimum or maximum number from list? Depends upon structure?
 Insertion?
Data Structure Find Insert
Unsorted List O(n) O(1)
Sorted List O(1) O(n)
Heap O(1) O(log n)

29. HomeWork-3
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 Book Problems:
 9.24
 9.25
 9.26