2.6 Directed Acyclic Graphs (DAGs) & Scheduling

Give the correct bound! Here we're concerned with the general problem of parallel task scheduling, but we stick to the language of our running example of class scheduling.

1. Number of semesters in a shortest possible schedule, with no limit on classes per term \(=\)

   Exercise 1

     ceil(total classes needed / max chain size) max chain size # of vertices / max anti-chain size max anti-chain size max chain size 



2. Maximum classes per term needed to achieve shortest schedule \(\leq\)

   Exercise 2

     ceil(total classes needed / max chain size) max chain size # of vertices / max anti-chain size max anti-chain size max anti-chain size 



3. Maximum classes per term needed to achieve shortest schedule \(\geq\)

   Exercise 3

     ceil(total classes needed / max chain size) max chain size # of vertices / max anti-chain size max anti-chain size ceil(total classes needed / max chain size) 



4. Max chain size \(\geq\)

   Exercise 4

     ceil(total classes needed / max chain size) # of vertices / max anti-chain size max anti-chain size # of vertices / max anti-chain size 
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