the critical-path algorithm - hampden-sydney...

The Critical-Path AlgorithmLecture 32

Sections 8.3 - 8.4

Robb T. Koether

Hampden-Sydney College

Wed, Nov 19, 2014

Robb T. Koether (Hampden-Sydney College) The Critical-Path Algorithm Wed, Nov 19, 2014 1 / 15

1 Assignment

2 Critical Paths

3 The Backflow Algorithm

4 The Critical-Path Algorithm


Outline

1 Assignment

2 Critical Paths




Assignment

CollectedTo be collected on Monday, November 24.

Chapter 8: Exercises 37, 43, 53.

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


Outline

1 Assignment

2 Critical Paths




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.


Outline

1 Assignment

2 Critical Paths




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 theprocessing time for that vertex plus the largest critical time of thevertices 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 Assignment

2 Critical Paths




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)


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