Lecture 155:- Sparse Matrix Representation

Sparse matrix representation is a technique used to efficiently store matrices that have a significant number of zero elements. In many real-world scenarios, matrices contain a large number of zeros, making it inefficient to store them in a conventional dense matrix format, where every element occupies memory, including the zeros. Sparse matrix representation aims to save memory by only storing non-zero elements and their respective positions.

There are several common methods for representing sparse matrices:

1. Array of Lists (Row-wise or Column-wise): In this representation, we use an array of linked lists to store the non-zero elements of the matrix. Each node in the linked list contains the value of the element and its column (for row-wise representation) or its row (for column-wise representation).
2. Array of Triplets (COO - Coordinate List): In the COO representation, we use three arrays to store the non-zero elements' values and their corresponding row and column indices. The first array holds the non-zero values, the second array holds the row indices, and the third array holds the column indices.
3. Compressed Sparse Row (CSR) or Compressed Row Storage (CRS): In this representation, we use three arrays: one for non-zero elements, another for their respective column indices, and a third for indicating where each row's elements start and end in the previous two arrays. This format is efficient for matrix-vector multiplication.
4. Compressed Sparse Column (CSC) or Compressed Column Storage (CCS): Similar to CSR, CSC also uses three arrays, but in this case, one array stores non-zero elements, another stores their row indices, and the third indicates the start and end positions of each column in the first two arrays.

The choice of representation depends on the specific application and the matrix's sparsity pattern. Each representation has its advantages and trade-offs in terms of memory usage, ease of access, and efficiency in performing specific matrix operations.

Sparse matrix representation is widely used in scientific computing, image processing, and various other fields where large matrices with many zero elements are encountered. It helps to conserve memory and speeds up computations involving sparse matrices compared to using dense matrix representations.