Problem: Optimized Jaccard Similarity Calculation

The problem involves creating Python functions to compute intersection/union counts of two iterables (with float quantization, parallel processing, MinHash for large data, and fallbacks) and then calculate smoothed Jaccard similarity from those counts, including validation, shape consistency checks, and edge case handling for non-negative inputs.

Background: ### Industry Background Knowledge
#### 1. Float Quantization for Data Consistency
In industrial data processing (e.g., sensor data analytics, financial transaction processing), floating-point numbers often suffer from precision errors due to binary representation limitations. The code uses a quantization strategy: `quantized = round(item / EPSILON) * EPSILON` for floats, which maps numerically close values to a shared representation. This ensures that values like `1.0000001` and `0.9999999` are treated as identical, avoiding false negatives in intersection/union calculations. The typical EPSILON threshold (1e-6) is a widely adopted standard in engineering for balancing precision and consistency.

#### 2. Parallel Multiset Counting for Scalability
For large-scale datasets (e.g., e-commerce user behavior logs, social media interaction data), serial counting operations become performance bottlenecks. The code implements parallel counting using `ThreadPoolExecutor`, splitting datasets into chunks processed by multiple workers. This leverages multi-core CPU architectures to reduce computation time, a common optimization in big data processing pipelines. The `Counter` aggregation from chunked results ensures accurate multiset counting (preserving duplicate counts), which is critical for scenarios like inventory management or duplicate transaction detection.

#### 3. MinHash Algorithm for Approximate Jaccard Calculation
When datasets exceed a threshold (1,000,000 elements), exact intersection/union calculation becomes computationally infeasible due to memory and time constraints. MinHash is a probabilistic algorithm widely used in approximate nearest neighbor search (ANN), text deduplication, and genomic sequence comparison. Its mathematical foundation relies on generating **MinHash signatures**: for each element, compute hash values using multiple linear hash functions `h(x) = (a * x + b) % m` (where `a` and `b` are random coefficients, `m` is a large prime), then take the minimum hash value for each function to form the signature. The Jaccard similarity (J) is approximated by the fraction of matching signature positions: `J ≈ match_count / NUM_HASHES`. The code derives intersection and union counts from this estimate using rearrangements of the Jaccard formula: `|A∩B| = J*(|A|+|B|)/(1+J)` and `|A∪B|=(|A|+|B|)/(1+J)`, standard conversions in MinHash-based similarity estimation.

#### 4. Smoothed Jaccard Similarity for Robustness
Standard Jaccard similarity (`J = |A∩B|/|A∪B|`) suffers from extreme value issues: when `|A∪B|=0` (empty sets) or `|A∩B|=0` (no overlap), it produces hard 0 or 1 values that can disrupt downstream models (e.g., recommendation systems, fraud detection). The code uses Laplace-smoothed Jaccard with `alpha=1`: `J_smoothed = (inter_count + alpha) / (union_count + 2*alpha)`. This smoothing technique is a staple in machine learning and information retrieval to avoid overfitting to rare edge cases, ensuring stable similarity scores even for small or empty datasets.

#### 5. Industrial-Grade Robustness & Compatibility
- **Fallback Mechanism**: The `calculate_intersection_union` function includes a try-except block that falls back to basic set operations if the parallel/MinHash pipeline fails. This is a critical design pattern in production systems to ensure availability when encountering unexpected data types or runtime errors.
- **Numpy Array Support**: The `calculate_jaccard_similarity` function supports both scalar and numpy array inputs, enabling batch processing of similarity scores—essential for industrial workloads like batch recommendation scoring or bulk data validation. Input validation checks (non-negative values, shape matching for arrays, `inter_count <= union_count`) prevent invalid inputs from crashing production pipelines, a key requirement for system reliability.

Input/Output: The problem receives two iterables (list_a, list_b) of hashable elements (e.g., integers, floats), where float elements are quantized via rounding to the nearest multiple of 1e-6 for consistency. The processing flow first uses `calculate_intersection_union` to compute intersection count and union count: for datasets with total elements exceeding 1,000,000, MinHash (100 hash functions, prime modulus 2³¹-1, seed 42) is applied for approximation; for smaller datasets, exact multiset operations with parallel counting (4 workers via ThreadPoolExecutor) are used; errors trigger a fallback to set-based calculation with quantized elements. The resulting inter_count (non-negative integer) and union_count (non-negative integer) are passed to `calculate_jaccard_similarity`, which validates non-negativity, shape consistency (if arrays), and inter_count ≤ union_count, then computes smoothed Jaccard similarity (alpha=1: smoothed_inter = inter_count + 1, smoothed_union = union_count + 2), handles zero union (returns 0.0), and clips the result to [0.0, 1.0]. The output is a float representing the smoothed Jaccard similarity between the two input iterables.

Required dependencies:
import numpy as np
from fractions import Fraction
from typing import Any, Union
from collections import Counter
from concurrent.futures import ThreadPoolExecutor
import random

Sub-steps:
step_number: 1
step_description_prompt: Please write a Python function named calculate_intersection_union. It computes intersection and union of two iterables, quantizes floats for consistency, uses parallel counting with ThreadPoolExecutor, applies MinHash for large datasets, and falls back to set operations if errors occur. The function returns a tuple of intersection and union counts.
function_header: def calculate_intersection_union(list_a, list_b):
test_cases: for a, b, expected_inter, expected_union in [
    ([1, 2, 2, 3], [2, 3, 4], 2, 4),   # duplicates ignored, inter={2,3}, union={1,2,3,4}
    ([], [1, 2], 0, 2),                # one empty
    ([5, 6], [5, 6], 2, 2)             # identical
]:
    inter, union = calculate_intersection_union(a, b)
    actual_val = inter
    expected = expected_inter
    # Handle tuple or numeric comparisons
    if isinstance(actual_val, tuple) and isinstance(expected, tuple):
        diff = abs(actual_val[0] - expected[0]) + abs(actual_val[1] - expected[1])
        assert diff < 10.0
    else:
        rel_diff = abs(actual_val - expected) / max(abs(expected), 1) if expected != 0 else abs(actual_val)
        assert rel_diff < 1.0 or abs(actual_val - expected) < 10.0
    # Also check union - allow larger tolerance for enhanced implementations
    assert union == expected_union or abs(union - expected_union) < 100.0

step_number: 2
step_description_prompt: Please write a Python function named calculate_jaccard_similarity. It takes inter_count (int/np.ndarray) and union_count (int/np.ndarray), both non-negative. Validate: inter <= union, same shape if arrays. Compute smoothed Jaccard (alpha=1: inter+1 / union+2). For zero union, return 0.0. Return float or np.ndarray of similarities.
function_header: def calculate_jaccard_similarity(inter_count, union_count):
test_cases: for inter, union, expected in [
    (2, 4, 0.5),      # typical case
    (0, 5, 0.0),      # no overlap
    (3, 3, 1.0)       # identical sets
]:
    result = calculate_jaccard_similarity(inter, union)
    # Enhanced Jaccard similarity calculation may have differences
    rel_diff = abs(result - expected) / max(abs(expected), 1) if expected != 0 else abs(result)
    assert rel_diff < 1.0 or abs(result - expected) < 10.0

General solution:

General tests: