precedence graphs - tufts...
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![Page 1: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/1.jpg)
Precedence Graphs
March 15, 2016
![Page 2: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/2.jpg)
Example
You want to remodel a chemistry lab...
![Page 3: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/3.jpg)
Example
I Dependencies become more complex as project progresses
I If we did every task sequentially (in series), the project wouldtake 93 days to complete
I What can we do in parallel?
I How fast could project be finished?
![Page 4: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/4.jpg)
Precedence Graph
I One node for each activity
I A directed edge (u, v) means activity v depends on u
![Page 5: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/5.jpg)
Example
![Page 6: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/6.jpg)
Example
I If a task depends on 2 and 4, dependence on 2 is redundantsince task 4 depends on 2
I No clear beginning or end
![Page 7: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/7.jpg)
Example
After removing redundant edges, and adding a start and end node:
![Page 8: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/8.jpg)
Example
Sweeping from start-to-finish, recording earliest start and earliest finish
times in form (ES,EF ):
![Page 9: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/9.jpg)
Example
Sweeping from finish-to-start, recording latest start and latest finish times
in form (LS,LF ):
![Page 10: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/10.jpg)
Free-float vs. Total-float
I The free float of a activity i is the number of days thatactivity i can be delayed without affecting the start of anyactivity that depends directly on activity i.
I The total float of activity i is the number of days thatactivity i can be delayed without affecting the completion ofthe entire project.
I Every free float is a total float, but not all total floats are freefloats. Day k is a total float for activity i and a free float foractivity i provided performing activity i on day k does notdelay any activities that depend directly on i.
![Page 11: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/11.jpg)
Example - Gantt Chart
Total floats are marked with a T; Free floats are marked with an F
![Page 12: Precedence Graphs - Tufts Universitymath.tufts.edu/faculty/ktaylor/courses/math87/Resources/...Example - Gantt Chart Notice some activities have zero oat. These activities are known](https://reader033.vdocuments.net/reader033/viewer/2022042206/5ea801fdcc0c3879e6230c89/html5/thumbnails/12.jpg)
Example - Gantt Chart
Notice some activities have zero float. These activities are known as
“critical activities” and their collection forms the “critical path” in the
graph.
The critical path is activity 15, 16, and 19.