Describes basic understanding of priority queues, their applications, methods, implementation with sorted/unsorted list, sorting applications with insertion sort and selection sort with their running times.

1. Priority Queues
(Algorithms and Data Structures)
By
Priyanka Rana
By – Priyanka Rana

2. Priority Queues
Which assets (stock, company, real-estate) the company should
purchase for investment.

3. Priority Queues
Many factors determine the future value of an asset.
Let’s call this “Utility”.
Utility(asset) = priority or importance that Utility() return the
minimum value.
Use Utility() as key or order of assets in the list.

4. Definition Priority Queue
Is an Abstract Data Type.
Storing collection of prioritized elements.
Supports
arbitrary element insertion.
removal of elements in order of priority.

5. Priority Queue ADT
Each entry is a pair (key, value).
Key: used to identify or weigh that element.
Two distinct entries in a priority queue can have the same key.
Mathematical concept of total order relation, must satisfy:
Reflexive property: k ≤ k.
Antisymmetric property: if k1 ≤ k2 and k2 ≤ k1 , then k1 = k2.
Transitive property: if k1 ≤ k2 and k2 ≤ k3, then k1 ≤ k3.

6. Applications
Standby flyers.
Auctions.
Stock market.

7. Main methods of Priority Queue ADT
size():
Return the number of entries in the queue.
isEmpty():
Test whether the queue is empty.
min():
Return (do not remove) an entry with smallest key.
insert(k, x):
Insert an entry with key k and value x.
removeMin():
Remove and return the entry with smallest key.

8. Sorting with a Priority Queue
Sorting a collection S of n comparable elements
with a priority queue Q, called PriorityQueueSort.
Hint: Using insert() and removeMin()

9. Exercise
Operation Output
Insert(5,A) nil
Insert(4,B) nil
Insert(7,I) nil
removeMin() (4,B)
insert(3,J) Nil
removeMin() (3,J)
insert(6,L) nil
removeMin() (5,A)

10. Implementing a Priority Queue
• With an unsorted list :
• insert() – adds entry at last -> takes O(1)
• removeMin() – requires complete scan -> takes O(n)
• “fast insertion and slow removal”
4 5 2 3 1

11.  With a Sorted List:
 Insert() – requires complete scan -> takes O(n)
 removeMin() – always access the first element -> takes O(1)
 fast removal and slow insertions.
1 2 3 4 5

12. Method Unsorted List Sorted List
size, isEmpty O(1) O(1)
insert O(1) O(n)
Min, removeMin O(n) O(1)

13. Sorting using priority Queues
• Selection – Sort
• Insertion - Sort

14. Selection Sort – 7 4 8 2 5 3 9
7 4 8 2 5 3 9
2 4 8 7 5 3 9
2 3 8 7 5 4 9
2 3 4 7 5 8 9
2 3 4 5 7 8 9
2 3 4 5 7 8 9
2 3 4 5 7 8 9

15. Selection- Sorting using priority Queues
Sequence S Priority Queue P
Input (7,4,8,2,5,3,9) ()
Phase 1 (a)
(b)
.
.
.
(g)
(4,8,2,5,3,9)
(8,2,5,3,9)
.
.
.
()
(7)
(7,4)
.
.
.
(7,4,8,2,5,3,9)
Phase 2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(2)
(2,3)
(2,3,4)
(2,3,4,5)
(2,3,4,5,7)
(2,3,4,5,7,8)
(2,3,4,5,7,8,9)
(7,4,8,2,5,3,9)
(7,4,8,5,9)
(7,8,5,9)
(7,8,9)
(8,9)
(9)
()

16. Running Time
• First removeMin takes O(n) time
• Second removeMin takes O(n-1) time) and so on
• Till last n operation takes O(1) time
= O(n2)O(n

17. Insertion Sort – 7 4 8 2 5 3 9
7
4 7
4 7 8
2 4 7 8
2 3 4 5 7 8 9
2 3 4 5 7 8
2 4 5 7 8

18. Insertion -Sorting using priority Queues
Sequence S Priority Queue P
Input (7,4,8,2,5,3,9) ()
Phase 1 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(4,8,2,5,3,9)
(8,2,5,3,9)
(2,5,3,9)
(5,3,9)
(3,9)
(9)
()
(7)
(4,7)
(4,7,8)
(2,4,7,8)
(2,4,5,7,8)
(2,3,4,5,7,8)
(2,3,4,5,7,8,9)
Phase 2 (a)
(b)
●
.
(2)
(2,3)
● .
● .
● .
(2,3,4,5,7,8,9)
(3,4,5,7,8,9)
(4,5,7,8,9)
.
.
.
()

19. Running Time
• First element -no comparison
• Second element – 1 comparison
• Third element – 2 comparisons and so on.
= O(n2)O(n

20. THANK YOU