The Critical-Path Algorithm - Hampden-Sydney...

The Critical-Path AlgorithmLecture 34

Sections 8.3 - 8.4

Robb T. Koether

Hampden-Sydney College

Mon, Apr 20, 2015

Robb T. Koether (Hampden-Sydney College) The Critical-Path Algorithm Mon, Apr 20, 2015 1 / 20

1 Critical Paths

2 The Backflow Algorithm

3 The Critical-Path Algorithm

4 Example

5 Assignment


Outline

1 Critical Paths



4 Example

5 Assignment


Definitions

Definition (Critical Path for a Vertex)The critical path for a vertex is the path from that vertex to END withthe longest processing time.

Definition (Critical Path for a Project)The critical path for a project is the critical path from START to END.

Definition (Critical Time)The critical time for a vertex or project is the processing time of itscritical path.


Example

Start(0)B(5)

D(2)

A(6)

C(7)

End(0)

E(5) H(4)

G(3)

F(3)

The critical time of a project is the shortest possible time requiredto complete the project.

It is also the longest path (in terms of time) from START to END.


Example

Start(0)B(5)

D(2)

A(6)

C(7)

End(0)

E(5) H(4)

G(3)

F(3)

The critical time of a project is the shortest possible time requiredto complete the project.It is also the longest path (in terms of time) from START to END.


Outline

1 Critical Paths



4 Example

5 Assignment


The Backflow Algorithm

Definition (The Backflow Algorithm)The backflow algorithm finds the critical path by the following method.

1 Beginning with END and working back to START, find the criticaltime for each vertex.

The critical time for a vertex is the processing time for that vertexplus the largest critical time of the vertices incident from that vertex.

2 The critical path for the project is the path from START to ENDwhose edges connect each vertex to its successor with thegreatest critical time.



Example (The Backflow Algorithm)

Start(0)B(5)

D(2)

A(6)

C(7)

End(0)

E(5) H(4)

G(3)

F(3)




Start(0)B(5)

D(5)

A(6)

C(7)

End(0)

E(9) H(4)

G(3)

F(3)




Start(0)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)




Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


Outline

1 Critical Paths



4 Example

5 Assignment


The Critical-Path Algorithm

Definition (The Critical-Path Algorithm)The critical-path algorithm creates a schedule by the following method.

1 Use the backflow algorithm to find the critical time of every task inthe project.

2 Create a priority list with the tasks listed in order of decreasingcritical time.

3 Use the priority list to create a schedule.


Example

Example (The Critical-Path Algorithm)

Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)

The priority list is C, B, A, E , D, H, F , G.


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)

Priority list: C, B, A, E , D, H, F , G

0 5 10 15 20

Processor 1Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

Processor 1Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5)Processor 1Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5) A(6)Processor 1Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5) A(6)E(5)Processor 1

Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5) A(6)E(5)

D(2)Processor 1Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5) A(6)E(5)

D(2)H(4)Processor 1

Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5) A(6)E(5)

D(2)H(4)

F(3)Processor 1Processor 2


Example


Start(16)B(14)

D(5)

A(11)

C(16)

End(0)

E(9) H(4)

G(3)

F(3)


C(7)

0 5 10 15 20

B(5) A(6)E(5)

D(2)H(4)

F(3)Processor 1Processor 2 G(3)


Outline

1 Critical Paths



4 Example

5 Assignment


Example (Exercise 55)

Precedencent ProcessingTask Tasks Time

A 10B 5C B 4D C, G 1E D 8F 3G H 7H F 1J H 6



Start(0)B(5) D(1)

A(10)

C(4)End(0)

E(8)

H(1)

G(7)

F(3) J(6)

Schedule the tasks using 2 processors.



Precedencent Processing CriticalTask Tasks Time Time

A 10

10

B 5

18

C B 4

13

D C, G 1

9

E D 8

8

F 3

20

G H 7

16

H F 1

17

J H 6

6




A 10 10B 5

18

C B 4

13

D C, G 1

9

E D 8

8

F 3

20

G H 7

16

H F 1

17

J H 6

6




A 10 10B 5

18

C B 4

13

D C, G 1

9

E D 8 8F 3

20

G H 7

16

H F 1

17

J H 6

6




A 10 10B 5

18

C B 4

13

D C, G 1

9

E D 8 8F 3

20

G H 7

16

H F 1

17

J H 6 6




A 10 10B 5

18

C B 4

13

D C, G 1 9E D 8 8F 3

20

G H 7

16

H F 1

17

J H 6 6




A 10 10B 5

18

C B 4 13D C, G 1 9E D 8 8F 3

20

G H 7

16

H F 1

17

J H 6 6




A 10 10B 5

18

C B 4 13D C, G 1 9E D 8 8F 3

20

G H 7 16H F 1

17

J H 6 6




A 10 10B 5

18

C B 4 13D C, G 1 9E D 8 8F 3

20

G H 7 16H F 1 17J H 6 6




A 10 10B 5 18C B 4 13D C, G 1 9E D 8 8F 3

20

G H 7 16H F 1 17J H 6 6




A 10 10B 5 18C B 4 13D C, G 1 9E D 8 8F 3 20G H 7 16H F 1 17J H 6 6



Start(20)B(18) D(9)

A(10)

C(13)End(0)

E(8)

H(17)

G(16)

F(20) J(6)



Outline

1 Critical Paths



4 Example

5 Assignment


Assignment

AssignmentChapter 8: Exercises 49, 52, 53, 56.


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