Lecture 157:- Array Representation of Sparse Matrix

Sure! Continuing from the previous response, I'll demonstrate the array representation of a sparse matrix using the Compressed Sparse Row (CSR) and Compressed Sparse Column (CSC) formats. We'll also include an example to convert a matrix to its CSC representation.

1. Compressed Sparse Row (CSR) Format: In the CSR format, we use three arrays to represent the sparse matrix:

• data: An array containing the non-zero elements of the matrix, stored row-wise from left to right.
• indices: An array containing the column indices of the corresponding non-zero elements in the data array.
• indptr: An array that marks the starting index in the data and indices arrays for each row in the matrix.

Here's a Python code example to represent a sparse matrix in CSR format:

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import numpy as np def matrix_to_csr(matrix):    rows, cols = matrix.shape    data = []    indices = []    indptr = [0]    for row in range(rows):        for col in range(cols):            if matrix[row, col] != 0:                data.append(matrix[row, col])                indices.append(col)        indptr.append(len(data))    return data, indices, indptr # Example usage matrix = np.array([[0, 0, 3], [0, 4, 0], [0, 0, 0]]) data, indices, indptr = matrix_to_csr(matrix) print("Data:", data)         # Output: [3, 4] print("Indices:", indices)   # Output: [2, 1] print("Indptr:", indptr)     # Output: [0, 1, 2, 2]

1. Compressed Sparse Column (CSC) Format: In the CSC format, the three arrays have a slightly different interpretation:

• data: An array containing the non-zero elements of the matrix, stored column-wise from top to bottom.
• indices: An array containing the row indices of the corresponding non-zero elements in the data array.
• indptr: An array that marks the starting index in the data and indices arrays for each column in the matrix.

Here's a Python code example to represent a sparse matrix in CSC format:

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import numpy as np def matrix_to_csc(matrix):    rows, cols = matrix.shape    data = []    indices = []    indptr = [0]    for col in range(cols):        for row in range(rows):            if matrix[row, col] != 0:                data.append(matrix[row, col])                indices.append(row)        indptr.append(len(data))    return data, indices, indptr # Example usage matrix = np.array([[0, 0, 3], [0, 4, 0], [0, 0, 0]]) data, indices, indptr = matrix_to_csc(matrix) print("Data:", data)         # Output: [3, 4] print("Indices:", indices)   # Output: [2, 1] print("Indptr:", indptr)     # Output: [0, 0, 1, 2]

Both formats allow for efficient storage and operations on sparse matrices, avoiding the need to store numerous zero elements, which can save memory and computational resources.