import numpy as np
from sklearn.decomposition import PCA

def analyze_matrices(A, B):
    """
    Analyze the given matrices to determine their properties such as
    sparsity, symmetry, and other characteristics.
    
    Parameters:
    A - First input matrix.
    B - Second input matrix.
    
    Returns:
    characteristics - A dictionary with various properties of the matrices.
    """
    characteristics = {}
    
    # Calculate the size of the matrices (number of elements)
    characteristics["size_A"] = A.size
    characteristics["size_B"] = B.size
    
    # Calculate sparsity of the matrices
    sparsity_A = np.count_nonzero(A) / A.size
    sparsity_B = np.count_nonzero(B) / B.size
    characteristics["sparsity_A"] = sparsity_A
    characteristics["sparsity_B"] = sparsity_B
    
    # Check for symmetry
    characteristics["symmetric_A"] = np.allclose(A, A.T)
    characteristics["symmetric_B"] = np.allclose(B, B.T)
    
    # Check if matrices are square
    characteristics["square"] = A.shape[0] == A.shape[1] and B.shape[0] == B.shape[1]
    
    # Add more properties and heuristics as needed
    # …

    return characteristics

def select_algorithm(characteristics):
    """
    Select the most efficient matrix multiplication algorithm based
    on the analyzed characteristics of the input matrices.
    
    Parameters:
    characteristics - A dictionary containing properties of the matrices.
    
    Returns:
    algorithm - A function that performs matrix multiplication.
    """
    # Example heuristic rules based on matrix properties:

    # If matrices are sparse, use a sparse-matrix specialized method
    if characteristics["sparsity_A"] < 0.1 and characteristics["sparsity_B"] < 0.1:
        algorithm = multiply_sparse_matrices
    # If matrices are both symmetric and square, other optimizations may be possible
    elif characteristics["symmetric_A"] and characteristics["symmetric_B"] and characteristics["square"]:
        algorithm = strassen_matrix_mult  # Placeholder for a potential symmetry-exploiting method
    # For dense and square matrices, consider Strassen’s or Coppersmith-Winograd algorithms
    elif characteristics["square"]:
        algorithm = strassen_matrix_mult  # Should consider matrix size and other factors
    # As a default, use NumPy’s highly optimized matrix multiplication
    else:
        algorithm = np.dot

    return algorithm

# Placeholder for the example sparse matrix multiplication algorithm
def multiply_sparse_matrices(A, B):
    # In practice, you’d use an optimized sparse matrix library like SciPy
    from scipy.sparse import csr_matrix
    A_sparse = csr_matrix(A)
    B_sparse = csr_matrix(B)
    return A_sparse.dot(B_sparse).toarray()

# Placeholder for Strassen’s matrix multiplication algorithm.
# Here you would put the actual implementation or the call to an optimized library that implements Strassen’s algorithm.
def strassen_matrix_mult(A, B):
    # A placeholder function that should implement Strassen’s algorithm or forward the call to an efficient implementation
    # For our example purposes, we’ll call NumPy’s dot product as a placeholder
    return np.dot(A, B)

def should_reduce_dim(characteristics):
    """
    Determine if dimensionality reduction would be beneficial prior to matrix multiplication.
    
    Parameters:
    characteristics - A dictionary containing properties of the matrices.

    Returns:
    True if dimensionality reduction should be applied, False otherwise.
    """
    # Example criteria for dimensionality reduction
    # (in reality, these thresholds should be determined through empirical analysis
    # and could be much more complex, possibly involving machine learning models)

    # Define a size threshold for dimensionality reduction.
    # This is just a mock-up example and assumes that for larger matrices,
    # there could be a benefit for such reduction if they are not sparse.
    SIZE_THRESHOLD = 1000  # Assume matrices larger than 1000 elements could benefit from reduction
    
    # Define a minimum sparsity threshold to avoid reduction for already sparse matrices,
    # as reducing dimensions might not yield additional benefits.
    SPARSITY_THRESHOLD = 0.1  # Arbitrary example threshold

    # Ensure matrices are not too sparse to benefit from dimensionality reduction
    not_too_sparse = characteristics["sparsity_A"] > SPARSITY_THRESHOLD and characteristics["sparsity_B"] > SPARSITY_THRESHOLD

    # Ensure matrices are large enough to consider reduction
    large_enough = characteristics["size_A"] >= SIZE_THRESHOLD or characteristics["size_B"] >= SIZE_THRESHOLD

    # Example conditional logic for simplification
    return large_enough and not_too_sparse

# Note: this is a very simplified possible function. In practice, the decision to apply dimensionality reduction would include
# a cost-benefit analysis considering the overhead of the reduction process itself, the importance of the result’s accuracy,
# and the specific domain of application. In many cases, such methods are employed in the fields of data compression and
# signal processing, where approximate results can be acceptable, rather than in strict linear algebra computation.

def dimensionality_reduction(A, B, num_components):
    """
    Apply PCA dimensionality reduction to both matrices A and B.

    Parameters:
    A - First input matrix.
    B - Second input matrix.
    num_components - Number of components to keep.

    Returns:
    A_reduced - The reduced representation of matrix A.
    B_reduced - The reduced representation of matrix B.
    reverse_mapper - A function that maps the reduced representation back to the original space.
    """
    pca_A = PCA(n_components=num_components)
    pca_B = PCA(n_components=num_components)

    A_reduced = pca_A.fit_transform(A)
    B_reduced = pca_B.fit_transform(B)

    def reverse_mapper(A_reduced, B_reduced):
        A_original = pca_A.inverse_transform(A_reduced)
        B_original = pca_B.inverse_transform(B_reduced)
        return A_original, B_original

    return A_reduced, B_reduced, reverse_mapper

# Adaptive Fast Matrix Multiplication :

def AFMM(A, B):
    # Step 1: Analyze matrices A and B for sparsity, structure, etc.
    characteristics = analyze_matrices(A, B)
    
    # Step 2: Select the fastest approach based on the analysis
    algorithm = select_algorithm(characteristics)
    
    # Step 3: Optionally, reduce dimensions of A and B if appropriate
    if should_reduce_dim(characteristics):
        A_reduced, B_reduced, reverse_mapper = dimensionality_reduction(A, B)
        product_reduced = algorithm(A_reduced, B_reduced)
        product = reverse_mapper(product_reduced)
    else:
        product = algorithm(A, B)
    
    # Step 4: Return the result
    return product

# Example cases :

# Printing result :
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
print(AFMM(A, B))