Problem: Design a Violation Log Analyzer

You are tasked with designing and implementing an in-memory "Violation Log Analyzer". This analyzer will process an append-only list of violation events, where each event is represented as a tuple: (id: string, policy: string, date: ISO-8601 string).

The analyzer must support the following query types:

1. Query by ID: Given an id, return all unique policies that this id has violated.
2. Query by Policy: Given a policy, return all unique ids that have violated this policy.
3. Query by Date: Given an exact date, return all unique ids that violated any policy on that specific date.

Your solution should address the following aspects:

• Initial Approach & Data Structures: Begin with a straightforward approach. Choose appropriate data structures for storing and indexing the violation events. Analyze the time and space complexity for building the initial index and for answering each of the three query types.
• Optimization with Inverted Indexes: Optimize your design using appropriate inverted indexes (e.g., id → policies, policy → ids, date → ids). Provide the core API signatures (e.g., build(log), query_by_id(id), query_by_policy(policy), query_by_date(date)) and pseudocode for the optimized approach.
• Caching and Precomputation: If the system anticipates many repeated queries, propose caching or precomputation strategies (e.g., memoization of frequent queries, materialized sets). Discuss how these strategies would handle invalidation when new violation events are appended to the log.
• Sorted Event List Optimization: If the initial event list is guaranteed to be sorted by date, explain how to efficiently locate all events for a target date using binary search and return the corresponding ids. Provide the algorithm and its complexity.
• Trade-offs and Scaling: Discuss the trade-offs between query latency and memory consumption in your design. Explain how your chosen design scales with an increasing number of events, distinct ids, and distinct policies.