Design a Message Queue with Multi-Channel Messages and Channel-Based Subscriptions

Context

You are asked to design a message-queue data structure where:

• Each message may belong to multiple communication channels (tags).
• Consumers subscribe to channels and should receive messages if they share at least one channel with the message.
• The system must support adding new channels dynamically, efficient detection of shared channels via union/intersection, and analysis of operation complexities.
• Additionally, given a set of desired producer–consumer communication pairs, determine how many channels are needed and how to assign them to cover these patterns.

Requirements

1. Data model where messages can have multiple channels and consumers subscribe to channels.
2. Efficient operations:
   • Enqueue message
   • Dequeue/consume message
   • Subscribe/Unsubscribe consumer to channels
   • Query shared channels (e.g., does message M intersect consumer C’s subscriptions?)
3. Representation of channel identifiers enabling fast union/intersection.
4. Support adding channels at runtime.
5. Analyze time and space complexity for all operations.
6. Given a set of producer–consumer communication patterns, determine the minimum (or a good) number of channels required and how to assign them.