P$8.$ The figure on the side depicts a task graph. We want to execute this task graph on a machine with multiple computing nodes. Each task takes the same amount of time. Assume that the communication time is zero $($wrong assumption to simplify the problem, but you can assume that communication time is built into the task time$).$ The system manager decides to increase computation reliability by deploying a fault detection scheme using duplication. There is a budget to acquire up to six computing nodes. Compute the ideal number of nodes and the total time steps needed in each of the fnllnwing racoc Ploace rnncidor efficiency and latency in computation in mind.

$($a$)$ Develop a schedule without deploying any redundancy, i$.$e$.,$ duplication.

$($b$)$ Develop a schedule with duplication and a constraint that all results must be certified before the final output is available. Each task must be replicated on a different processor to certify.

$($c$)$ Develop a schedule with duplication and a constraint that no task output can be used before the task is certified.

$($d$)$ Describe the outline of the algorithm you used to develop the schedule in each case.