1. Computer Science 2
Heaps

2. Learning Objectives Understand the properties of a heap
Understand how to add to a heap
Understand how to remove from a heap

3. Uses for Heaps
Priority Queue. Efficient way of having the first element on the top of the heap.
Heapsort: O(N log N)
Better worst case scenario than using a Tree to sort

4. Properties of a Heap Full Tree
Parents are greater than their children.

5. Heaps: Examples Are heaps BSTs ?? wrong! 24 7 3 30 10 40 80 25 48 2114 17 33 9 18 28 11 22 35 50 20
wrong!
Are heaps BSTs ??

6. Inserting a node Increase the size by 1
Add it to the bottom of the heap
Reheap up.
While the child is greater than its parents
Switch them

7. Heaps: Insertion example
Adding a new node with key = 8 at the “end” of this heap violates the order property.
New node has larger key than its parent, swap nodes to fix this. 16 5 11 3 8
Is the order property restored?
Is 3 < 8? Yes.
Is 8 < 16? Yes.
Is this a coincidence? 16 8 11 3 5

8. Heaps: Insertion example
If the new node had had key = 18, what would have happened? 16 5 11 3 18
Swap with the parent.
Is the order property restored?
Is 3 < 18? Yes.
Is 18 < 16? No. 16 18 11 3 5

9. Heaps: Insertion example
Swap again: order property restored. 18 16 11 3 5
In general, we “bubble up” the new node until the order property is restored.

10. Your Turn Create a heap by adding the following values 20, 10, 30, 15
Add the following to the heap
40, 18, 32

11. Removing from a Heap Take off the top Move the bottom to the top
Decrease the size of the heap
Reheap Down
If the top < greatest child
Switch top and greatest child
Continue until the parent is not < it’s greatest child.

12. Heaps: Removing largest key
The largest key is at the top.
Remove it.
Move “last” node to the top, to maintain shape.
‘Re-heap down’ until the order property is restored. 18 16 11 16 11 3 5 3 5

13. Heaps: Removing largest key5 16 16 11 5 11 3 3
Did we have to reheap down the left? Why?

14. Removal: another exampley p r d f a c k b c p r d f a k b r p c d f a k b r p k d f a c b

15. Your Turn Using the heap created previously
Show what it looks like after removing one item.

16. Implementing it into an array: Adding
How do you find the parents of a child? …
ChildPosition / 2
Adding to a heap
Increase the size by 1
Add it to the bottom of the heap
Reheap up.
While the child is greater than its parent
Switch them
Size 8 r p k d f a c b
Dry run.
Add a z and a q to the heap. 1 2 3 4 5 6 7 8 9 10 R P K D F C B A

17. Implementing it into an array: Removing
How do you find the children? …
parentPosition*2 and parentPosition*2 +1
Size 8
Removing
Take off the top
Move the bottom value to the top
Decrease the size of the heap
Reheap Down
While there are children and added < greatest child
Switch added and greatest child r p k d f a c b 1 2 3 4 5 6 7 8 9 10 R P K D F C B A

18. Heaps of Fun Program Menu Push: Add Show All
Show and remove the top of the heap.
Push:
Implement a heap sort
Switch top and bottom
Rehead down, excluding the last element(s)