Fix interval generation inconsistency between naive and intersection tree implementations

[Copilot]

Problem

When comparing the number of intervals returned by naive_intersectionic_queries.execute_queries and intersection_tree.execute_queries with identical databases and query intervals, the results were different. This inconsistency made it impossible to verify the correctness of the intersection tree optimization.

Root Cause

The bug was in the generate_interval function in intersection_tree.py. While the naive implementations correctly followed the mathematical specification from the notebook that intervals must satisfy end - start > 1, the intersection tree implementation used a weaker constraint:

# Naive implementations (correct)
end = random.randint(start + 2, max_end)  # Ensures end - start > 1

# Intersection tree (incorrect) 
end = random.randint(start + 1, max_end)  # Only ensures end - start >= 1

This caused the implementations to generate completely different intervals when using the same random seed, leading to different databases and consequently different query results.

Solution

• Updated intersection_tree.generate_interval() to use start + 2 instead of start + 1
• Corrected the docstring from end - start >= 1 to end - start > 1 to match the mathematical specification
• Both implementations now generate identical intervals when using the same random seed

Verification

Comprehensive testing confirms that:

• All three implementations (naive_intersectionic_queries, intersection_tree, and naive_pythonic_intersectionic_queries) now generate identical intervals
• Query results are consistent across all implementations
• The mathematical requirement end - start > 1 is satisfied by all generated intervals
• Performance characteristics of the intersection tree remain unchanged

Example verification:

import random
random.seed(42)
naive_results = naive_intersectionic_queries.execute_queries(queries, db)
random.seed(42) 
tree_results = intersection_tree.execute_queries(queries, db)
assert len(naive_results) == len(tree_results)  # Now passes ✅

---

✨ Let Copilot coding agent set things up for you — coding agent works faster and does higher quality work when set up for your repo.