Algorithm_Matroids and greedy methods

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.

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 )

Greedy Algorithm
on a weighted
matroid

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

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

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

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

Solution
4 3
2
1
A
C
B
D E
= {A}

Solution
4 3
2
1
A
C
B
D E
= {A} {B}

Solution
4 3
2
1
A
C
B
D E
= {A} {B} {C}

Solution
4 3
2
1
A
C
B
D E
= {A} {B} {C} {D}

Solution
4 3
2
1
A
C
B
D E
= {A} {B} {C} {D} {E}  SET 1

Solution
4 3
2
1
A
C
B
D E
= {A, B}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D} {A, E}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D} {A, E} {B, C}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}
{C, D}

Solution
4 3
2
1
A
C
B
D E
= {A, B} {A, C} {A, D} {A, E} {B, C} {B, D} {B, E}
{C, D} {C, E}  SET 2

Solution
4 3
2
1
A
C
B
D E
= {A, B, C}

Solution
4 3
2
1
A
C
B
D E
= {A, B, C} {A, B, D}

Solution
4 3
2
1
A
C
B
D E
= {A, B, C} {A, B, D} {A, B, E}

Solution
4 3
2
1
A
C
B
D E
= {A, B, C} {A, B, D} {A, B, E} {A, C, D}

Solution
4 3
2
1
A
C
B
D E
= {A, B, C} {A, B, D} {A, B, E} {A, C, D} {A, C, E}
 SET 3

Solution
= {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

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 )

Algorithm_Matroids and greedy methods

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