Total Run Time of Ordered Tasks with a Per-Type Cooldown
Company: Waymo
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: easy
Interview Round: Technical Screen
A single worker runs a list of tasks in exactly the order given. Each task is labeled by its type, an uppercase letter, so the list is a string such as `"ABBCC"`. Two tasks of the same type need a cooldown between them: after a task of some type finishes, the next task of that type cannot start until `cooldown` time units have passed, and the worker sits idle while it waits. Tasks may not be reordered. Return the total time needed to finish all the tasks.
### Function Signature
```python
def total_run_time(tasks: str, cooldown: int) -> int:
```
### Rules
- Time starts at `0`. The worker runs one task at a time, and every task takes exactly `1` time unit (the task duration is not part of the input, so this version fixes it at one unit).
- Tasks run in the given order: a task cannot start before the previous task in the string has finished.
- If a task of type `X` finished at time `f`, the next task of type `X` may start no earlier than `f + cooldown`. Tasks of other types are not affected by it.
- Every task starts as early as these rules allow. Return the time at which the last task finishes, or `0` if `tasks` is empty.
### Constraints
- `0 <= len(tasks) <= 10^5`
- `tasks` consists of uppercase English letters `'A'` to `'Z'`.
- `0 <= cooldown <= 10^4`
- The answer is at most `1,000,090,000`, so it fits in a 32-bit signed integer.
### Examples
**Example 1**
```text
Input: tasks = "ABBCC", cooldown = 3
Output: 11
```
`A` runs in `[0, 1)` and the first `B` in `[1, 2)`. The second `B` cannot start before `2 + 3 = 5`, so the worker idles and runs it in `[5, 6)`. The first `C` runs in `[6, 7)`; the second `C` must wait until `7 + 3 = 10` and runs in `[10, 11)`.
**Example 2**
```text
Input: tasks = "ABCA", cooldown = 2
Output: 4
```
The first `A` finishes at `1`, so the second `A` may start at `1 + 2 = 3`, which is exactly when `B` and `C` have finished. There is no idle time.
**Example 3**
```text
Input: tasks = "ABAB", cooldown = 2
Output: 5
```
`A` runs in `[0, 1)` and `B` in `[1, 2)`. The second `A` must wait until `1 + 2 = 3` and runs in `[3, 4)`; the second `B` may start at `2 + 2 = 4` and runs in `[4, 5)`.
Overview: Tasks labeled by type must run in a fixed order, each taking one time unit, and two tasks of the same type need a cooldown gap between them. Compute the total time until the last task finishes. It tests careful simulation, per-type bookkeeping and edge cases such as an empty input or a zero cooldown.