1. Heaps
Section 6.4, Pg. 309 (Section 9.1)

2. Motivation: Prim’s Algorithm
Starting node T
How many times is this edge visited?
3 times
Problem: You keep comparing the same edges over and over.
Solution: Use priority queues containing the edges

3. Priority Queues Typical example:
printing in a Unix/Linux environment. Printing jobs have different priorities.
These priorities override the FIFO policy of the queues
Operations supported in a priority queue:
Insert a new element
Extract/Delete of the element with the highest priority
The priority is a number

4. Implementing Priority Queues: min-HEAPS
We assume that each element has a key, K, and other information, I. K is the priority of I.
Heap
greater
A min-Heap is a binary tree T such that:
The priority of the element in any node is less or equal than the priority of the elements in its children nodes
T is complete
Note: The algorithms for Heaps are the same as for min-Heaps. Just invert the comparisons.

5. (non) Examples 10 > 9! Tree is not complete 5 5 5 8 8 8 10 9 9 1612 9 56 12 16 10 12

6. Insert a New Element
Idea: insert the new element as the last leaf and re-adjust the tree 5 8 9 16 12 18 20 22 44 10 13 56
Insert 7

7. Insert a New Element (II)
Step 1: add it as the last leaf 5 8 9 16 12 18 20 22 44 10 13 56 7

8. Insert a New Element (III)
Steps 2, 3 ,… : swap until key is in the right place 5 8 9 12 56 16 7 22 44 13 10 18 20

9. Insert a New Element (IV)
Steps 2, 3 ,… : swap until it finds its place 5 8 7 12 56 16 9 22 44 13 10 18 20
Complexity is O(log2 n)

10. Extract/Delete the Element with the Lowest Priority
Idea: the root has always the lowest priority. Then delete it and replace it with the last child. Re-adjust the tree. 5 8 9 16 12 18 20 22 44 10 13 56
Extract/Step 1: returns 5

11. Extract/Delete the Element with the Lowest Priority (II)
Step 2: Put the last key as the new root. 13 8 9 12 56 16 10 22 44 18 20

12. Extract/Delete the Element with the Lowest Priority (III)
Steps 3, 4, … : Swap until key is in the correct place. Always swap with node that has the lowest priority. 8 13 9 12 56 16 10 22 44 18 20

13. Extract/Delete the Element with the Lowest Priority (IV)
Steps 3, 4, … : Swap until key is in the correct place. Always swap with node that has the lowest priority. 8 12 9 13 56 16 10 22 44 18 20
Complexity is O(log2 n)

14. Complete Trees can be Represented in Arrays5 8 9 16 12 18 20 22 44 10 13 56 …
Corresponding array:

15. Operations in Array Representation of Complete Trees
Assume that the first index in the array is 0 and that the number of keys is n
root(i):
i = 0
LeftChild(i):
2i + 1
rightChild(i):
2i + 2
isLeaf(i):
Homework
Parent(i):
Homework

16. Using Priority Queues in Prim’s Algorithm
The Fringe (neighbors of T) are maintained in a priority list (min-Heap), which ensures O((|E|+|V|)log2|E|)
Maintaining the Fringe is the crucial aspect of Djisktra’s Algorithm for computing shortest path between two points (next class)