Optimal Resource Allocation for Parallel Processing

You are tasked with optimizing the allocation of a limited set of resources to a collection of independent tasks. Each task requires a specific amount of processing power, and you have a fixed total processing power available across multiple parallel processors. The goal is to find the minimum number of processors needed to complete all tasks within a given time limit, assuming each processor has a uniform processing speed.

Problem Description

Given a list of tasks, each with a required processing power, and a maximum processing power available per processor, determine the minimum number of processors required to execute all tasks. Tasks can be executed in parallel on different processors. A single processor can handle multiple tasks, but its total processing power cannot exceed the given maximum. The objective is to minimize the number of processors used while ensuring all tasks are assigned.

• Tasks: A list of integers, where each integer represents the processing power required by a task.
• Max Processor Power: A single integer representing the maximum processing power a single processor can provide.
• Output: The minimum number of processors required to assign all tasks.

Edge Cases:

• An empty list of tasks should require zero processors.
• If a single task requires more processing power than a single processor can provide, it's impossible to complete. In this case, you should indicate impossibility.
• If all tasks require zero processing power, zero processors are needed.

Examples

Example 1:

Input:
Tasks: [5, 2, 7, 4, 1, 8]
Max Processor Power: 10

Output: 3

Explanation:
We can group tasks as follows:
Processor 1: [8, 2] (Total: 10)
Processor 2: [7, 1] (Total: 8)
Processor 3: [5, 4] (Total: 9)
This uses 3 processors. It's not possible to use fewer than 3 processors. For instance, if we try to use 2, we can't fit all tasks without exceeding the max processor power on at least one processor.

Example 2:

Input:
Tasks: [10, 10, 10]
Max Processor Power: 10

Output: 3

Explanation:
Each task requires the maximum power of a single processor.
Processor 1: [10]
Processor 2: [10]
Processor 3: [10]

Example 3:

Input:
Tasks: [6, 6, 6, 6]
Max Processor Power: 12

Output: 2

Explanation:
Processor 1: [6, 6] (Total: 12)
Processor 2: [6, 6] (Total: 12)

Example 4:

Input:
Tasks: [15]
Max Processor Power: 10

Output: -1

Explanation:
A single task requires 15 units of processing power, which exceeds the maximum capacity of a single processor (10). Therefore, it's impossible to complete this task.

Constraints

• The number of tasks will be between 0 and 1000.
• Each task's processing power will be an integer between 0 and 1000.
• The Max Processor Power will be an integer between 1 and 1000.
• Your solution should aim for an efficient time complexity, ideally better than brute-force enumeration of all task assignments.

Notes

This problem is a variation of the Bin Packing problem. Consider sorting the tasks. How might that help? Think about greedy approaches. If a task cannot fit into any existing partially filled processor without exceeding its capacity, a new processor must be used. Be mindful of the order in which you try to fit tasks into processors.