Compute total time to finish all courses

## Problem You are given `n` courses labeled `1..n`. Each course `i` takes `time[i]` units of time to complete. You are also given prerequisite relations `prereqs`, where each pair `[a, b]` means **course `a` must be completed before course `b` can be taken**. Courses are taken in **batches (rounds/semesters)** under the following rule: - At the **start** of a batch, you may start **any number of courses** whose prerequisites are already satisfied. - The batch **finishes only when all courses started in that batch finish**. - The duration of a batch is therefore `max(time[c])` over all courses `c` taken in that batch. - You can only start the next batch after the current batch finishes. ### Task Return the **minimum total time** required to finish all courses. ### Cycle condition If the prerequisite graph contains a cycle (so it is impossible to finish all courses), return `-1`. ## Example Suppose prerequisites imply two batches: first take courses `{1, 3}`, then take `{2, 4}` (e.g., edges `1 -> 2` and `3 -> 4`). - `time = [1, 10, 10, 1]` for courses `1..4` - Batch 1 duration: `max(1, 10) = 10` - Batch 2 duration: `max(10, 1) = 10` - Total = `20` ## Follow-up (variant scheduling rule) How would your approach change if courses can start **immediately when their prerequisites are satisfied**, without waiting for the rest of the current batch to finish (i.e., no batch barrier; courses can overlap asynchronously)? In that variant, return the minimum total completion time or `-1` if a cycle exists. ## Constraints (typical) - `1 <= n <= 2 * 10^5` - `0 <= prereqs.length <= 2 * 10^5` - `1 <= time[i] <= 10^9`