The document discusses greedy methods, matroids, and applying greedy algorithms to matroids. It provides examples of greedy scheduling algorithms like shortest job first. It defines matroids as a pair (S, I) where S is a finite set and I is a family of independent subsets satisfying the exchange property. The document gives an example matroid and shows applying a greedy algorithm to iteratively build independent sets in the matroid by adding elements one by one.

1. Matroids and
Greedy Methods

2. Greedy Methods

3. Introduction
Algorithmic approaches by taking the most profitable decision in the current
state only.
Straight forward and simplest Algorithmic approaches.
Might lead to worst solution.
Examples : Shortest Running Time Next ( Batch System Scheduling ),
Shortest Job First ( Batch System Scheduling ), Shortest Process Next (
Interactive System Scheduling ), etc.

4. Matroids

5. Principles
A matroid is an ordered pair M = ( S, I ). Statifying the following conditions :
S is a Finite Set.
I is a nonempty family of subsets of S ( Independent Subsets of S ).
If |A| ∈ I, |B| ∈ I, and |A| < |B|, then there is some element x ∈ B – A such that A ∪ {
X } ∈ I. ( M statifies Exhchange Property )

6. Greedy Algorithm
on a weighted
matroid

7. Pseudocode*
*Picture taken from Thomas H. Cormen, “Introduction to Algorithms”, 3rd Edition, MIT Press, 2009 , Page 440

8. Examples
Consider S = { a, b, c, d}. Make the Smallest Matroid! with ( S, F ) when { a, b }
and { c, d } included in F.
F = { , {a}, {b}, {c}, {d}, {a,b}, {c,d}  Hereditary
{a,c}, {b,c}, {a,d}, {b,d}  Exchange property

9. Pseudocode*
*Picture taken from https://www.sciencedirect.com/science/article/pii/S0899825614001651 on 15/04/2019

10. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}

11. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A}

12. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A} {B}

13. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A} {B} {C}

14. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A} {B} {C} {D}

15. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A} {B} {C} {D} {E}  SET 1

16. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B}

17. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C}

18. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D}

19. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D} {A, E}

20. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D} {A, E} {B, C}

21. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D}

22. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}

23. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}
{C, D}

24. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}
{C, D} {C, E}  SET 2

25. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B, C}

26. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B, C} {A, B, D}

27. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B, C} {A, B, D} {A, B, E}

28. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B, C} {A, B, D} {A, B, E} {A, C, D}

29. Solution
4 3
2
1
A
C
B
D E
Set Possible = {∅}
= {A, B, C} {A, B, D} {A, B, E} {A, C, D} {A, C, E}
 SET 3

30. Solution
Set Possible = {∅}
= {A} {B} {C} {D} {E}  SET 1
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}
{C, D} {C, E}  SET 2
= {A, B, C} {A, B, D} {A, B, E} {A, C, D} {A, C, E}
 SET 3

31. References
[1] Thomas H. Cormen, “Introduction to Algorithms”,
3rd Edition, MIT Press, 2009. ( Page 437 – 443 )
[2] Andrew S. Tanenbaum, “Modern Operating Systems”,
Pearson, Mar. 20, 2014. ( Page 149 – 167 )