There are $$$N$$$ jobs, each job $$$i$$$ has a single prerequisite job $$$P_i$$$ that must be done before, except for a global root job which has no prerequisite. Each job takes $$$T_i$$$ time to be finished, and if a job is finished at time $$$t$$$ it contributes with a penalty of $$$t * U_i$$$, where $$$U_i$$$ is the i-th job's penalty coefficient. What is the minimum penalty for finishing all jobs?

Constraints: $$$N <= 2 * 10^5$$$, everything is integer and non-negative

Notes:

• only a single task can be performed at a time (there is no concurrent work / task parallelism)
• the tasks DAG looks like a rooted tree

Any ideas on how to solve this? I was thinking of performing some kind of backtracking over all possible topological orderings of the DAG + some kind of extremely heavy pruning, but I haven't figured out yet a good pruning strategy to avoid exponential time. I also thought of using DP, but then I need to use bitmasks to keep track of unvisited nodes, which leads to exponential time.