1. Heaps A heap is a binary tree that satisfies the following properties:
Structure property: It is a complete binary tree
Heap-order property: Each node satisfies the heap condition:
The key of every node must be smaller than (or equal to) the keys of its children, i.e.
n.info  n.left.info and
n.info  n.right.info, for all nodes n
Note: this is called a min-heap – max-heaps are possible too
COSC 2P03 Week 6

2. removeMin Algorithm – min-heap
temp = root;
root = last node;
current = root;
while current has at least 1 child {
smaller = smallest child;
if (current < smaller) break;
else swap(current, smaller); }
return temp;
What is the complexity?
COSC 2P03 Week 6

3. insert Algorithm – min-heap
put new node at bottom right of tree and call it current;
// trickle up
while current < its parent
swap (current, parent);
What is the complexity?
COSC 2P03 Week 6

4. Array implementation of Heap
array[0..capacity]: the heap array
array[1..currentsize]: elements currently in heap
currentsize: number of nodes in heap
removeMin (see fig for full details):
The smallest element is the root, which is array[1]
The “last” element (rightmost on lowest level) is array[currentsize]
In trickle-down, we need to find the children of a node:
Left child of array[t] is array[2t]
Right child of array[t] is array[2t+1]
COSC 2P03 Week 6

5. Array implementation of Heap – removeMin Example1 2 3 4 5 6 7 35 10 50 60 40 20 1 2 3 4 5 6 20 35 10 50 60 40 1 2 3 4 5 6 10 35 20 50 60 40
COSC 2P03 Week 6

6. Array implementation of Heap (insert – see fig. 6.8 for full details)
Insert in array[currentsize+1]
During trickle-up, we need to find the parent of a node:
Parent of array[t] is array[t/2]
Example: insert 15 1 2 3 4 5 6 7 10 35 20 50 60 40 15 1 2 3 4 5 6 7 10 35 15 50 60 40 20
COSC 2P03 Week 6

7. Transforming a complete binary tree into a heap
In a heap, all subtrees are also heaps.
Idea: transform all subtrees into heaps, starting from smallest. Note that leaves are already heaps.
array[currentsize/2]is the parent of the last node in the heap.
buildHeap algorithm:
for (i = currentsize/2; i >= 1; i--)
apply trickle-down to node [i];
COSC 2P03 Week 6

8. buildHeap example i=5 59 36 58 21 41 97 31 16 26 53 i=4 59 36 58 21 41
End 16 21 31 26 41 97 58 36 59 53
COSC 2P03 Week 6

9. Heapsort – min heap The results are stored in B[1..n]. buildHeap(A);
for(i = 1; i <= n; i++)
B[i] = removeMin(A);
(etc)
COSC 2P03 Week 6

10. Heapsort – max heap
Avoid the use of a second array by storing sorted elements at the end of the array
Use a max-heap:
At each step, swap root with last element, then apply trickle down to new root
Example – after conversion to max heap:
After 1st swap:
After 2nd swap:
COSC 2P03 Week 6