You are an expert Systems Engineer and Concurrency Specialis...

Prompt

You are an expert Systems Engineer and Concurrency Specialist. I need you to implement a highly robust, thread-safe, priority-based Task Scheduler in Python (using only the standard library, no third-party packages). The scheduler must satisfy the following strict requirements: 1. Core Mechanics: - Supports tasks with different priorities (0 is highest, 10 is lowest). - Supports setting a "timeout" for each task. If a task waits in the queue for longer than its timeout before execution begins, it must be discarded (and its rejection must be logged). - Uses a thread pool for execution. 2. Advanced Constraints (The Trap): - Starvation Prevention: To prevent low-priority tasks from starving, implement an "aging" mechanism. Every 5 seconds a task waits in the queue, its priority number should dynamically decrease by 1 (e.g., from 10 to 9, making it more important). - Deadlock Avoidance: Multiple workers will pull tasks from the queue. You must ensure there are no race conditions regarding the dynamic priority updates and the timeout checks. - Graceful Shutdown: Implement a `shutdown()` method that stops accepting new tasks, waits for currently running tasks to finish, and safely discards all queued tasks. 3. Deliverables: - Provide the complete, runnable Python code. - Provide a highly concurrent unit test (using `threading` or `concurrent.futures`) that simulates 50 simultaneous tasks with mixed priorities, some of which will inevitably timeout, to prove your scheduler handles race conditions correctly without deadlocking. - Explain the exact time complexity of `push` and `pop` operations in your implementation and justify your choice of data structure.