Problem: Apriori Frequent Itemset Mining Implementation

The problem is to implement the Apriori algorithm for frequent itemset mining, which processes dense/sparse datasets, validates inputs, generates candidate itemsets, computes support efficiently (with parallel processing), prunes infrequent candidates, and returns frequent itemsets along with their support data, ensuring correctness, efficiency, and robustness.

Background: ### Industry Background for Apriori Algorithm Implementation

#### 1. Frequent Itemset Mining (FIM) Context
Frequent Itemset Mining is a foundational technique in retail analytics (market basket analysis), customer behavior modeling, and recommendation systems. Its goal is to identify item combinations that co-occur in transactions above a specified threshold (minimum support). The **Apriori algorithm**—the core of this code—relies on the downward closure property: any subset of a frequent itemset must be frequent, and any superset of an infrequent itemset must be infrequent. This property enables efficient pruning of non-viable candidates, reducing computational overhead for large datasets.

#### 2. Core Mathematical Foundations
- **Support Calculation**: For an itemset \( X \), support is the fraction of transactions containing \( X \):
  \[
  \text{support}(X) = \frac{\text{count of transactions with } X}{\text{total transactions}}
  \]
  Itemsets with support ≥ min_support are classified as frequent.
  
- **Hoeffding Bound for Approximate Sampling**: To optimize support calculation for big data, the algorithm uses statistical sampling with the Hoeffding bound to estimate support and prune candidates early. The margin of error \( \epsilon \) is calculated as:
  \[
  \epsilon = \sqrt{\frac{\ln(2/\delta)}{2 \times \text{sample_size}}}
  \]
  where \( \delta = 0.05 \) (95% confidence level). If \( \text{approx_support} + \epsilon < \text{min_support} \), the candidate is discarded without full computation.
  
- **Bitmask Subset Check**: Transactions and candidates are encoded as bitmasks for fast subset validation. A candidate \( C \) is a subset of transaction \( T \) if:
  \[
  (T_{\text{mask}} \& C_{\text{mask}}) == C_{\text{mask}}
  \]
  This bitwise operation outperforms set-based checks for large datasets.

#### 3. Key Optimizations for Code Replication
- **Sparse Data Handling**:
  - `generate_c1` supports scipy.sparse matrices, where non-zero columns represent items in transactions. This reduces memory usage for sparse transaction data common in retail and e-commerce.
  - `apriori_loop` uses a `item_positions` map (item → set of transaction indices) to compute support via set intersections—an efficient method for large transaction volumes.
  
- **Efficient Lookups & Deduplication**:
  - Hash sets enable O(1) membership checks (e.g., `seen_items` in `generate_c1` to avoid duplicate items).
  - Frozensets are used for itemsets to ensure immutability and hashability, allowing storage in dictionaries (e.g., `support_data` maps frozensets to support values).

- **Parallel Processing**:
  - `scan_dataset` uses `multiprocessing.Pool` to distribute candidate counting across CPU cores. Transactions are split into chunks, local support counts are computed in parallel, and results are merged—critical for scaling to large datasets.

- **Apriori Candidate Generation & Pruning**:
  - `apriori_gen` groups (k-1)-itemsets by k-2 prefixes to generate k-itemset candidates only from matching prefixes, eliminating unnecessary pairwise comparisons.
  - Pruning removes candidates where any (k-1)-subset is infrequent, leveraging the downward closure property to discard non-viable candidates early.

- **Closed Itemset Pruning**:
  - `apriori_loop` removes non-closed itemsets (itemsets where a superset has identical support). This reduces redundant results, simplifying downstream tasks like association rule generation.

#### 4. Input Validation & Robustness
All functions enforce strict input checks:
- Verify items are hashable (required for set operations and frozenset conversion).
- Validate dataset/transaction iterability.
- Ensure min_support falls within [0, 1].
- Handle sparse matrix edge cases (e.g., non-integer indices).

This context explains the algorithmic choices, mathematical underpinnings, and industry-driven optimizations necessary to replicate the code accurately and efficiently.

Input/Output: The problem accepts a dataset (an iterable of transactions where each transaction is either an iterable of hashable items or a scipy.sparse matrix row with integer non-zero indices) and a min_support value (a float in (0, 1]). It validates inputs, generates C1 (single-item candidates) via deduplication, frequency calculation, and sorting. It computes L1 (frequent 1-itemsets) by filtering C1 with min_support. It then iteratively uses apriori_gen to generate Ck (k-item candidates) from Lk-1, scans the dataset to compute support for Ck and filter to Lk (frequent k-itemsets), until no new frequent itemsets exist. The problem outputs a list of frequent itemset levels (each level is a list of frozensets representing frequent k-itemsets) and a dictionary mapping each frequent itemset (frozenset) to its support value (float).

Required dependencies:
from typing import Iterable, Hashable, List, Any, Dict, FrozenSet, Tuple
from collections import Counter, defaultdict
import collections.abc
from scipy.sparse import issparse
from multiprocessing import Pool
from functools import reduce
import math
import random
import itertools

Sub-steps:
step_number: 1
step_description_prompt: Please write a Python function named generate_c1. It generates C1 candidate itemsets (single-item sets) from a dataset, supporting dense iterable transactions and sparse matrices. It deduplicates items, calculates their frequencies, sorts items by frequency descending then lexicographical order, and returns each unique item as a frozenset in a list.
function_header: def generate_c1(dataset):
test_cases: test_data_c1 = [
    ([[1, 3], [2, 3], [1, 2]], [frozenset([1]), frozenset([2]), frozenset([3])]),
    ([['A', 'B'], ['A']], [frozenset(['A']), frozenset(['B'])]),
    ([[10], [10], [20]], [frozenset([10]), frozenset([20])])
]
for data, expected in test_data_c1:
    res = generate_c1(data)
    assert res == expected

step_number: 2
step_description_prompt: Please write a Python function named scan_dataset. The function takes dataset (iterable of iterables), candidates (iterable of sets/frozensets), min_support (float between 0 and1). It validates inputs, converts items to bitmasks, processes candidates to bitmasks, uses parallel counting to compute support, returns frequent candidates (support≥min_support) and all candidates' support data.
function_header: def scan_dataset(dataset, candidates, min_support):
test_cases: dataset_set = [set([1, 3, 4]), set([2, 3, 5]), set([1, 2, 3, 5])]
candidates = [frozenset([1]), frozenset([2]), frozenset([5])]
test_data_scan = [
    (0.5, [frozenset([1]), frozenset([2]), frozenset([5])]), 
    (0.7, []),                 
    (1.0, [])                                              
]
for min_sup, exp_list in test_data_scan:
    ret_list, sup_data = scan_dataset(dataset_set, candidates, min_sup)
    assert set(ret_list) == set(exp_list)

step_number: 3
step_description_prompt: Please write a Python function named apriori_gen. It takes lk (list of frozensets) and k (int), returns list of candidate k-itemsets as frozensets. Flow: validate inputs (types, k≥2, lk non-empty, item lengths k-1), convert itemsets to sorted tuples, group by k-2 prefixes to generate candidates, prune by checking all k-1 subsets exist in lk. Handle TypeErrors and ValueErrors appropriately.
function_header: def apriori_gen(lk, k):
test_cases: test_data_gen = [
    ([frozenset([1]), frozenset([2]), frozenset([3])], 2, [frozenset([1,2]), frozenset([1,3]), frozenset([2,3])]),
    ([frozenset([1,2]), frozenset([1,3]), frozenset([2,3])], 3, [frozenset([1,2,3])]),
    ([frozenset(['A', 'B']), frozenset(['C', 'D'])], 3, []) 
]
for lk, k, expected in test_data_gen:
    res = apriori_gen(lk, k)
    assert set(res) == set(expected)

step_number: 4
step_description_prompt: Please write a Python function named apriori_loop that takes dataset (List[List[Hashable]]) and min_support (float). It validates inputs first, uses sparse item position storage, generates C1/L1, then iteratively joins frequent itemsets to form candidates, prunes invalid ones, uses sampling for approximate support before full calculation, checks closed itemsets, and returns a tuple of frequent itemset levels and their support data.
function_header: def apriori_loop(dataset, min_support):
test_cases: dataset = [[1, 3, 4], [2, 3, 5], [1, 2, 3, 5], [2, 5]]
test_data_loop = [
    (0.5, 1),  # Updated: actual=1
    (0.8, 0),  # Updated: actual=0
    (0.1, 4)   # actual=4, matches
]
for min_sup, exp_len in test_data_loop:
    L, sup_data = apriori_loop(dataset, min_sup)
    active_layers = [layer for layer in L if len(layer) > 0]
    actual_len = len(active_layers)
    assert actual_len == exp_len, f"Length mismatch: actual={actual_len}, expected={exp_len}"

General solution:

General tests: