Task Scheduler Assignment

You are tasked with implementing a task scheduler responsible for assigning N tasks to W workers.

Task Details

Each task i (IDs 0 to N-1) arrives at a specific arrival[i] time (non-decreasing) and has the following attributes:

• duration[i]: A positive integer representing the base duration of the task.
• type[i]: A string or integer label categorizing the task type.
• account[i]: A string or integer label identifying the account associated with the task.

Worker Details

Each worker w (IDs 0 to W-1) has the following attributes:

• skills[w]: A set of task types that the worker is capable of processing.
• speed[w][t]: A positive integer representing the processing speed of worker w for a task of type t. If t is not in skills[w], the worker cannot process that task type.

When worker w runs task i of type t, the actual runtime is calculated as: runTime(w, i) = duration[i] × speed[w][t]

A worker can process at most one task at a time. If a worker starts a task at time s, it becomes free again at time s + runTime.

Assignment Rules

Tasks must be processed in increasing task index order (i = 0 to N-1), which corresponds to their arrival order. For each task, a worker must be chosen based on the following priority rules:

1. Account Stickiness:

   • Maintain a mapping owner[account] -> workerId for accounts that have been assigned tasks previously.
   • If owner[account[i]] = w exists and worker w supports type[i], then task i must be assigned to worker w.
   • If the mapped worker w does not support type[i], ignore this rule for the current task and proceed to Rule 2.

2. Fastest Eligible Worker:

   • Among all workers that support type[i], assign the task to the worker that will finish it earliest, assuming it starts as soon as possible.
   • The earliest start time for worker w for task i is start = max(arrival[i], nextFreeTime[w]).
   • The finish time for worker w for task i is finish = start + runTime(w, i).
   • Choose the worker with the minimum (finish, workerId) lexicographically. This means prioritizing the earliest finish time, and then tie-breaking by choosing the worker with the smaller workerId.

3. Update Ownership:

   • After assigning task i to worker w, update the owner mapping: owner[account[i]] = w.

Output

Return an array assigned where assigned[i] is the workerId to which task i was assigned.

Constraints and Efficiency Expectations

• W, N ≤ 2e5
• The total number of (worker, type) skill entries can be large.
• Your solution should be efficient, avoiding a full scan of all workers for each task where possible.