1. Chapter 21
Binary Heap

2. Objective To learn Basic properties of the binary heap
Insert and deletemin operations
Linear time heap-construction algorithm
Heapsort

3. Topics Review and analyze operations on Heap data structure
Review definitions
Tree, binary tree
depth
height
full binary tree
complete binary tree
Heap property

4. Binary trees A tree is an acyclic, connected, undirected graph.
Only one path exists between a pair of nodes
A leaf of a tree is a node with no children.
Binary tree - a tree where each node has 0,1 or 2 children

5. Depth and height of a tree
Depth of a node is:
Depth of the root of a tree is 0.
The depth of its parent +1
Depth of a tree is maximum depth of its leaves.
Height of a node is:
Height of a leaf of a tree is 0.
The maximum height of its children +1
Height of a tree is the height of the root.

6. Depth and height of a binary tree2 1 1 1 2
The depth of the tree = height of the tree = 2

7. A complete binary tree
A full binary tree (also called complete) is a binary tree such that
All internal nodes have 2 children
All leaves have depth d
A complete binary tree (also called essentially complete) is a binary tree such that
All internal nodes have 2 children except the last internal node which may have only 1 child.
All leaves have depth d or d -1
Nodes with depth d are as far to the left as possible.

8. Full binary tree
Complete binary tree
2h n  2h+1 -1
n = 2h+1 -1

9. The height of a complete binary tree
The number of nodes n of a complete binary tree satisfies: h  n  (2h+1-1)
Taking the log base 2 we get:
h  lg n and lg(n+1)  h+1 or
lg(n+1)-1  h  lg n
Since h is integer h = lg(n + 1)-1=  lg n 

10. The height of a complete binary tree
2h  n  (2h+1-1)
Complete binary
tree with 1 node
at depth h
Full binary tree
all leaves same depth

11. Heap Definition
A heap is a complete binary tree that satisfies the heap property.
Minimum Heap Property:
The value stored at each node is less than or equal to the values stored at its children.
OR Maximum Heap Property: greater

12. Implementation of Heap
For simplicity we assume the complete binary tree is an array, and the root is stored at index 1.
For any element in array position i, its left child is at position 2i, and it’s right child is at 2i+1, its parent is at i/2 .

13. Last node in a heap
The last node of a heap is the rightmost internal node of on the last level 2 5 6 9 7
last node

14. Heap viewed as Binary tree implemented as an array.3 2 8 7 18 14 9 29 6 1 4 5 10
last 1 3 2 8 7 29 6 4 5 9 18 14 10

15. insert(v ) Item inserted as last item in the heap2
Item inserted as last item in the heap
Heap property may be violated
Do percolate to restore heap property 5 6 z 9 7
insertion node 2 5 6 z 9 7 1

16. Percolate up
After the insertion of a new key k, the heap-order property may be violated
Algorithm percolate up restores the heap-order property by swapping k along an upward path from the insertion node
percolate up terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6

17. deleteMin() Save root object O(1)10 10 2 3 30 20 4
Save root object O(1)
Remove last element and store in root O(1)
siftDown(1) 80 1
last 80 2 3 30 20 1 20 2 3 30 80

18. 4 3 8 17 12 14 19 6 13 1 siftDown(1) New value at root.5 7 9 10
siftDown(1)
New value at root.
Right Child is smaller Exchange root and right child
Satisfy the Heap property.

19. 4 9 8 17 12 14 19 6 13 1 3 2 Parent Left Child is smaller5 7 10
Parent Left Child is smaller
Exchange parent and left child

20. The worst case run time to do siftDown(index) is 4 6 8 17 12 14 19 9 13 1 2 3 5 7 10
The worst case run time to do siftDown(index) is
O(h(index)) where h(index) is the height of node index
When index=root=1, O(lg n)

21. Build heap---linear time
1. for i ¬ (last /2) downto 1
2. do siftDown( i )
Why we can start siftDown at last/2 ?
because we :
need to siftDown only parents
the rest of the nodes are leaves and
leaves satisfy the heap property

22. siftDown makes it a min heap5 8 12 9 7 10 21 1 2 3 4 6 14 8 12 9 7 6 14 4 10 21 1 2 3 5
siftDown makes it a min heap

23. siftDown makes this into heap8 12 9 4 6 14 7 10 21 1 2 3 5
this is a heap
siftDown makes this into heap

24. 8 12 6 4 9 14 7 10 21 1 2 3 5
i = 3
siftDown makes heap
These are heaps

25. 1 10 12 21 3 6 7 5 8 4 9 14 2
i = 2 4 6 7 9 14 8 2 5 10 4 6 8 9 14 7 2 5 10
siftDown

26. 4 10 6 7 9 14 8 12 21 1 2 3 5
i = 1 10 6 7 9 14 8 12 21 1 2 3 4 5 5

27. Running time n/2 nodes For each node, at most O(log(n))
The running time is O(nlgn)

28. Heap-Sort
Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time
The resulting algorithm is called heap-sort
Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort

29. 97 53 59 58 31 26 41 16 97 53 59 26 41 58 1 2 3 4 5 31 16 21 36 36 21 6 7 8 9 10 59 53 58 36 31 26 41 16 21 97 59 53 58 26 41 36 1 2 3 4 5 31 16 21 97 6 7 8 9 10

30. HeapSort //build heap for i ¬ (last /2) downto 1
do siftDown( a, i, last) //shiftDown from i to last
for j ¬ last-1 downto 1
swap(a[0], a[j]) //deleteMax
do siftDown(a, 0, j) // shiftDown from 0 to j

31. Common errors (from book, page 786)
For heapsort data begins in position 0, so the children of the I are in positions 2i+1 and 2i+2.
STL priority queue is a max heap, not a min heap.
More errors are defined in the book.

32. In class exercises
21.1
Describe the structure and ordering properties of the binary heap
21.2
In a binary heap, for an item in position I where are the parent, left child, and right child located?
21.3
Show the result of inserting 10, 12, 1, 14, 6, 5, 8, 15, 3, 9, 7, 4, 11, 13, and 2, one at a time, in an initially empty heap. Then show the result of using the linear-time buildHep algorithm instead