Scheduling where tasks take varying amounts of time:

Let $G=(V,E)$ be a directed acyclic graph where nodes represent tasks that need to be completed and edges represent the order in which tasks need to be completed, i$.$e$.,$ edge $(u,v)$ indicates that task $u$ must be completed before task $v$ begins. Each node $v$ has a weight $T(v)$ indicating how long it takes to complete task $v.$

Example: There are $4$ tasks, $A,B,C,$ and $D.$ A depends on both $B$ and $D,B$ depends on $D,$ and $C$ precedes $D.$ Task A takes two time units, $B$ takes three, $C$ takes four, and $D$ takes one. The shortest time to complete all tasks is $6(C$ can be completed concurrently with A and $B).$

Describe an efficient algorithm that determines the shortest possible time it takes to complete all tasks in network G$.$

Guidelines: In this problem you should solve the task by transferring it into "classical" network task. Describe the network $G=(V,E)$ that "encodes" the original task, which $($classical$)$ network algorithm is used, and how the solution to the original task is found subsequently. Use the following framework:

i$)$ What do the nodes in the network represent and how many are there?

ii$)$ What do the edges represent and how many are there?

iii$)$ What property of the network corresponds to the original task?

iv$)$ Which network algorithm should be executed?

v$)$ What is the time complexity in terms of the size of the original task? Here, consider the time it takes to construct the network, execute the network algorithm, and $($if necessary$)$ transfer the solution to the original task.