Data labeling scheduler with prefix-balanced models

A data labeling platform pairs tasks, AI models and human annotators. IDs are 1-indexed strings: tasks "t1"…"t{totalTask}", models "m1"…"m{totalModel}", humans "h1"…"h{totalHuman}".

An assignment is a triple [task, model, human]: that human labels that task with help from that model. A task may appear in many assignments, with different humans and models.

Part 1: any valid schedule

Implement schedule(totalTask, totalModel, totalHuman, k) -> list[list[str]]. It returns a list of assignments such that:

1. Every human appears in at least k assignments.
2. No human is assigned the same task twice.
3. Every ID is within its range.

Return any valid schedule. If no valid schedule exists, return [].

Constraints: 1 <= totalTask, totalModel, totalHuman <= 100, 0 <= k.

Example: totalTask = 4, totalModel = 1, totalHuman = 3, k = 2. One valid answer:

[["t1","m1","h1"], ["t2","m1","h2"], ["t3","m1","h3"],
 ["t2","m1","h1"], ["t3","m1","h2"], ["t4","m1","h3"]]

k = 5 with totalTask = 4 has no answer, so it returns [].

Part 2: prefix-balanced schedule

Now the order of the list matters. Look at every prefix, meaning the first x assignments for each x >= 1. In every prefix:

• Per task: for each task, the counts of each model it was used with differ by at most 1. Models used 0 times count as 0.
• Per human: for each human, the counts of each model they used differ by at most 1, counted the same way.

Implement balanced_schedule(totalTask, totalModel, totalHuman, k). It must meet all of the Part 1 rules plus this one, or return [] if that is impossible.

Example: totalTask = 3, totalModel = 2, totalHuman = 2, k = 2. One valid answer:

[["t1","m1","h1"], ["t1","m2","h2"], ["t2","m2","h1"], ["t2","m1","h2"]]

After the first assignment, t1 has m1:1, m2:0, which is fine. After all four, every task and every human has used each model once.

Part 3 (if time allows): streaming

Humans join one at a time. When a human joins, emit their assignments immediately, and never reorder what you already emitted. The whole stream must stay valid and prefix-balanced after every emission. State the time and the memory your scheduler keeps.