rocSPARSE storage formats

A sparse matrix is a matrix in which most of the items are zero, although there is no strict criteria for the number of non-zero items. The sparsity of a matrix refers to the number of non-zero elements divided by the total number of elements. For example, if an 8 x 8 matrix with 64 elements has 16 non-zero elements, it has a sparsity value of 0.25.

The main reason for storing and processing sparse matrices differently is to take advantage of lower memory requirements and potentially faster processing times. It is inefficient or impractical to store every element of a sparse matrix in memory (as a dense matrix) because most of the elements are zero. Instead, sparse matrices are compressed in storage using multiple vectors that map the individual non-zero values to their position in the original matrix. However, more complex algorithms are required to store the values in, and retrieve them from, these compound data structures.

rocSPARSE offers several storage formats for sparse matrices, each with specialized algorithms for matrix storage, retrieval, and manipulation. For additional information about the storage formats and their associated algorithms, see the Sparse matrix vector multiplication blog post.

Storage formats

This section describes the supported matrix storage formats.

Note

The different storage formats support indexing from a base of 0 or 1 as described in Storage schemes and indexing base.

COO storage format

The Coordinate (COO) storage format represents an \(m \times n\) matrix by:

m	Number of rows (integer).
n	Number of columns (integer).
nnz	Number of non-zero elements (integer).
coo_val	Array of nnz elements containing the data (floating point).
coo_row_ind	Array of nnz elements containing the row indices (integer).
coo_col_ind	Array of nnz elements containing the column indices (integer).

The COO matrix is sorted by row indices and column indices per row. Furthermore, each pair of indices should appear only once. The following \(3 \times 5\) matrix and corresponding COO structures, with \(m = 3\), \(n = 5\), and \(\text{nnz} = 8\), use zero-based indexing:

\[\begin{split}A = \begin{pmatrix} 1.0 & 2.0 & 0.0 & 3.0 & 0.0 \\ 0.0 & 4.0 & 5.0 & 0.0 & 0.0 \\ 6.0 & 0.0 & 0.0 & 7.0 & 8.0 \\ \end{pmatrix}\end{split}\]

where

\[\begin{split}\begin{array}{ll} \text{coo_val}[8] & = \{1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0\} \\ \text{coo_row_ind}[8] & = \{0, 0, 0, 1, 1, 2, 2, 2\} \\ \text{coo_col_ind}[8] & = \{0, 1, 3, 1, 2, 0, 3, 4\} \end{array}\end{split}\]

COO (AoS) storage format

The Coordinate (COO) Array of Structure (AoS) storage format represents an \(m \times n\) matrix by:

m	Number of rows (integer).
n	Number of columns (integer).
nnz	Number of non-zero elements (integer).
coo_val	Array of nnz elements containing the data (floating point).
coo_ind	Array of 2 * nnz elements containing alternating row and column indices (integer).

The COO (AoS) matrix is sorted by row indices and column indices per row. Each pair of indices should appear only once. The following \(3 \times 5\) matrix and corresponding COO (AoS) structures, with \(m = 3\), \(n = 5\), and \(\text{nnz} = 8\), use zero-based indexing:

\[\begin{split}A = \begin{pmatrix} 1.0 & 2.0 & 0.0 & 3.0 & 0.0 \\ 0.0 & 4.0 & 5.0 & 0.0 & 0.0 \\ 6.0 & 0.0 & 0.0 & 7.0 & 8.0 \\ \end{pmatrix}\end{split}\]

where

\[\begin{split}\begin{array}{ll} \text{coo_val}[8] & = \{1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0\} \\ \text{coo_ind}[16] & = \{0, 0, 0, 1, 0, 3, 1, 1, 1, 2, 2, 0, 2, 3, 2, 4\} \\ \end{array}\end{split}\]

CSR storage format

The Compressed Sparse Row (CSR) storage format represents an \(m \times n\) matrix by:

m	Number of rows (integer).
n	Number of columns (integer).
nnz	Number of non-zero elements (integer).
csr_val	Array of nnz elements containing the data (floating point).
csr_row_ptr	Array of m+1 elements that point to the start of every row (integer).
csr_col_ind	Array of nnz elements containing the column indices (integer).

The CSR matrix is sorted by column indices within each row. Each pair of indices should appear only once. The following \(3 \times 5\) matrix and corresponding CSR structures, with \(m = 3\), \(n = 5\), and \(\text{nnz} = 8\), use one-based indexing:

\[\begin{split}A = \begin{pmatrix} 1.0 & 2.0 & 0.0 & 3.0 & 0.0 \\ 0.0 & 4.0 & 5.0 & 0.0 & 0.0 \\ 6.0 & 0.0 & 0.0 & 7.0 & 8.0 \\ \end{pmatrix}\end{split}\]

where

\[\begin{split}\begin{array}{ll} \text{csr_val}[8] & = \{1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0\} \\ \text{csr_row_ptr}[4] & = \{1, 4, 6, 9\} \\ \text{csr_col_ind}[8] & = \{1, 2, 4, 2, 3, 1, 4, 5\} \end{array}\end{split}\]

CSC storage format

The Compressed Sparse Column (CSC) storage format represents an \(m \times n\) matrix by:

m	Number of rows (integer).
n	Number of columns (integer).
nnz	Number of non-zero elements (integer).
csc_val	Array of nnz elements containing the data (floating point).
csc_col_ptr	Array of n+1 elements that point to the start of every column (integer).
csc_row_ind	Array of nnz elements containing the row indices (integer).

The CSC matrix is sorted by row indices within each column. Each pair of indices should appear only once. The following \(3 \times 5\) matrix and corresponding CSC structures, with \(m = 3\), \(n = 5\), and \(\text{nnz} = 8\), use one-based indexing:

\[\begin{split}A = \begin{pmatrix} 1.0 & 2.0 & 0.0 & 3.0 & 0.0 \\ 0.0 & 4.0 & 5.0 & 0.0 & 0.0 \\ 6.0 & 0.0 & 0.0 & 7.0 & 8.0 \\ \end{pmatrix}\end{split}\]

where

\[\begin{split}\begin{array}{ll} \text{csc_val}[8] & = \{1.0, 6.0, 2.0, 4.0, 5.0, 3.0, 7.0, 8.0\} \\ \text{csc_col_ptr}[6] & = \{1, 3, 5, 6, 8, 9\} \\ \text{csc_row_ind}[8] & = \{1, 3, 1, 2, 2, 1, 3, 3\} \end{array}\end{split}\]

BSR storage format

The Block Compressed Sparse Row (BSR) storage format represents an \((mb \cdot \text{bsr_dim}) \times (nb \cdot \text{bsr_dim})\) matrix by:

mb	Number of block rows (integer).
nb	Number of block columns (integer).
nnzb	Number of non-zero blocks (integer).
bsr_val	Array of nnzb * bsr_dim * bsr_dim elements containing the data (floating point). Blocks can be stored in column-major or row-major format.
bsr_row_ptr	Array of mb+1 elements that point to the start of every block row (integer).
bsr_col_ind	Array of nnzb elements containing the block column indices (integer).
bsr_dim	Dimension of each block (integer).

The BSR matrix is sorted by column indices within each row. This matrix is defined as having a number of rows equivalent to \(\text{block_dim} \times \text{number_of_row_blocks}\). The following \(4 \times 3\) matrix and corresponding BSR structures, with \(\text{bsr_dim} = 2\), \(mb = 2\), \(nb = 2\), and \(\text{nnzb} = 4\), use zero-based indexing and column-major storage:

\[\begin{split}A = \begin{pmatrix} 1.0 & 0.0 & 2.0 \\ 3.0 & 0.0 & 4.0 \\ 5.0 & 6.0 & 0.0 \\ 7.0 & 0.0 & 8.0 \\ \end{pmatrix}\end{split}\]

with the blocks \(A_{ij}\)

\[\begin{split}A_{00} = \begin{pmatrix} 1.0 & 0.0 \\ 3.0 & 0.0 \\ \end{pmatrix}, A_{01} = \begin{pmatrix} 2.0 & 0.0 \\ 4.0 & 0.0 \\ \end{pmatrix}, A_{10} = \begin{pmatrix} 5.0 & 6.0 \\ 7.0 & 0.0 \\ \end{pmatrix}, A_{11} = \begin{pmatrix} 0.0 & 0.0 \\ 8.0 & 0.0 \\ \end{pmatrix}\end{split}\]

such that

\[\begin{split}A = \begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \\ \end{pmatrix}\end{split}\]

with arrays represented as

\[\begin{split}\begin{array}{ll} \text{bsr_val}[16] & = \{1.0, 3.0, 0.0, 0.0, 2.0, 4.0, 0.0, 0.0, 5.0, 7.0, 6.0, 0.0, 0.0, 8.0, 0.0, 0.0\} \\ \text{bsr_row_ptr}[3] & = \{0, 2, 4\} \\ \text{bsr_col_ind}[4] & = \{0, 1, 0, 1\} \end{array}\end{split}\]

GEBSR storage format

The General Block Compressed Sparse Row (GEBSR) storage format represents an \((mb \cdot \text{bsr_row_dim}) \times (nb \cdot \text{bsr_col_dim})\) matrix by:

mb	Number of block rows (integer).
nb	Number of block columns (integer).
nnzb	Number of non-zero blocks (integer).
bsr_val	Array of nnzb * bsr_row_dim * bsr_col_dim elements containing the data (floating point). Blocks can be stored in column-major or row-major format.
bsr_row_ptr	Array of mb+1 elements that point to the start of every block row (integer).
bsr_col_ind	Array of nnzb elements containing the block column indices (integer).
bsr_row_dim	Row dimension of each block (integer).
bsr_col_dim	Column dimension of each block (integer).

The GEBSR matrix is sorted by column indices within each row. If \(m\) is not evenly divisible by the row block dimension or \(n\) is not evenly divisible by the column block dimension, then zeros are padded to the matrix, such that \(mb = (m + \text{bsr_row_dim} - 1) / \text{bsr_row_dim}\) and \(nb = (n + \text{bsr_col_dim} - 1) / \text{bsr_col_dim}\). The following \(4 \times 5\) matrix and corresponding GEBSR structures, with \(\text{bsr_row_dim} = 2\), \(\text{bsr_col_dim} = 3\), \(mb = 2\), \(nb = 2\), and \(\text{nnzb} = 4\), use zero-based indexing and column-major storage:

\[\begin{split}A = \begin{pmatrix} 1.0 & 0.0 & 0.0 & 2.0 & 0.0 \\ 3.0 & 0.0 & 4.0 & 0.0 & 0.0 \\ 5.0 & 6.0 & 0.0 & 7.0 & 0.0 \\ 0.0 & 0.0 & 8.0 & 0.0 & 9.0 \\ \end{pmatrix}\end{split}\]

with the blocks \(A_{ij}\)

\[\begin{split}A_{00} = \begin{pmatrix} 1.0 & 0.0 & 0.0 \\ 3.0 & 0.0 & 4.0 \\ \end{pmatrix}, A_{01} = \begin{pmatrix} 2.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 \\ \end{pmatrix}, A_{10} = \begin{pmatrix} 5.0 & 6.0 & 0.0 \\ 0.0 & 0.0 & 8.0 \\ \end{pmatrix}, A_{11} = \begin{pmatrix} 7.0 & 0.0 & 0.0 \\ 0.0 & 9.0 & 0.0 \\ \end{pmatrix}\end{split}\]

such that

\[\begin{split}A = \begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \\ \end{pmatrix}\end{split}\]

with arrays represented as

\[\begin{split}\begin{array}{ll} \text{bsr_val}[24] & = \{1.0, 3.0, 0.0, 0.0, 0.0, 4.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0, 0.0, 6.0, 0.0, 0.0, 8.0, 7.0, 0.0, 0.0, 9.0, 0.0, 0.0\} \\ \text{bsr_row_ptr}[3] & = \{0, 2, 4\} \\ \text{bsr_col_ind}[4] & = \{0, 1, 0, 1\} \end{array}\end{split}\]

ELL storage format

The Ellpack-Itpack (ELL) storage format represents an \(m \times n\) matrix by:

m	Number of rows (integer).
n	Number of columns (integer).
ell_width	Maximum number of non-zero elements per row (integer).
ell_val	Array of m * ell_width elements containing the data (floating point).
ell_col_ind	Array of m * ell_width elements containing the column indices (integer).

The ELL matrix is assumed to be stored in column-major format. Rows with less than ell_width non-zero elements are padded with zeros (ell_val) and \(-1\) (ell_col_ind). The following \(3 \times 5\) matrix and corresponding ELL structures, with \(m = 3\), \(n = 5\), and \(\text{ell_width} = 3\), use zero-based indexing:

\[\begin{split}A = \begin{pmatrix} 1.0 & 2.0 & 0.0 & 3.0 & 0.0 \\ 0.0 & 4.0 & 5.0 & 0.0 & 0.0 \\ 6.0 & 0.0 & 0.0 & 7.0 & 8.0 \\ \end{pmatrix}\end{split}\]

where

\[\begin{split}\begin{array}{ll} \text{ell_val}[9] & = \{1.0, 4.0, 6.0, 2.0, 5.0, 7.0, 3.0, 0.0, 8.0\} \\ \text{ell_col_ind}[9] & = \{0, 1, 0, 1, 2, 3, 3, -1, 4\} \end{array}\end{split}\]

Blocked ELL storage format

The Blocked Ellpack (ELL) storage format represents an \((mb \cdot \text{block_dim}) \times (nb \cdot \text{block_dim})\) matrix by:

mb	Number of block rows (integer).
nb	Number of block columns (integer).
ell_width	Maximum number of non-zero block elements per row (integer).
ell_val	Array of mb * ell_width * block_dim * block_dim elements containing the data (floating point).
ell_col_ind	Array of mb * ell_width elements containing the column indices (integer).
block_dim	Dimension of each block (integer).

The Blocked ELL is similar to the ELL format except that column entries now indicate the location of two dimensional blocks of size block_dim * block_dim instead of single matrix entries. The block values can be stored in either row or column ordering. Rows with less than ell_width non-zero blocks are padded with zero blocks (ell_val) and \(-1\) (ell_col_ind). The following \(6 \times 6\) matrix and corresponding Blocked ELL structures, with \(mb = 3\), \(nb = 3\), \(block_dim = 2\), and \(\text{ell_width} = 2\), use zero-based indexing and row ordering for the blocks:

\[\begin{split}A = \begin{pmatrix} 1.0 & 2.0 & 0.0 & 0.0 & 3.0 & 1.0 \\ 2.0 & 4.0 & 0.0 & 0.0 & 4.0 & 3.0 \\ 0.0 & 0.0 & 6.0 & 4.0 & 7.0 & 8.0 \\ 0.0 & 0.0 & 4.0 & 5.0 & 3.0 & 2.0 \\ 1.0 & 2.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 2.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ \end{pmatrix}\end{split}\]

with the blocks \(A_{ij}\)

\[\begin{split}A_{00} = \begin{pmatrix} 1.0 & 2.0 \\ 2.0 & 4.0 \\ \end{pmatrix}, A_{02} = \begin{pmatrix} 3.0 & 1.0 \\ 4.0 & 3.0 \\ \end{pmatrix}, A_{11} = \begin{pmatrix} 6.0 & 4.0 \\ 4.0 & 5.0 \\ \end{pmatrix}, A_{12} = \begin{pmatrix} 7.0 & 8.0 \\ 3.0 & 2.0 \\ \end{pmatrix}, A_{21} = \begin{pmatrix} 1.0 & 2.0 \\ 2.0 & 1.0 \\ \end{pmatrix}\end{split}\]

such that

\[\begin{split}A = \begin{pmatrix} A_{00} & 0 & A_{02} \\ 0 & A_{11} & A_{12} \\ A_{21} & 0 & 0 \\ \end{pmatrix}\end{split}\]

where

\[\begin{split}\begin{array}{ll} \text{ell_val}[20] & = \{1.0, 2.0, 2.0, 4.0, 6.0, 4.0, 4.0, 5.0, 1.0, 2.0, 2.0, 1.0, 3.0, 1.0, 4.0, 3.0, 7.0, 8.0, 3.0, 2.0, 0.0, 0.0, 0.0, 0.0\} \\ \text{ell_col_ind}[6] & = \{0, 1, 0, 2, 2, -1\} \end{array}\end{split}\]

HYB storage format

The Hybrid (HYB) storage format represents an \(m \times n\) matrix by:

m	Number of rows (integer).
n	Number of columns (integer).
nnz	Number of non-zero elements of the COO part (integer).
ell_width	Maximum number of non-zero elements per row of the ELL part (integer).
ell_val	Array of m * ell_width elements containing the data for the ELL part (floating point).
ell_col_ind	Array of m * ell_width elements containing the column indices for the ELL part (integer).
coo_val	Array of nnz elements containing the data for the COO part (floating point).
coo_row_ind	Array of nnz elements containing the row indices for the COO part (integer).
coo_col_ind	Array of nnz elements containing the column indices for the COO part (integer).

The HYB format is a combination of the ELL and COO sparse matrix formats. Typically, the regular part of the matrix is stored in ELL storage format, and the irregular part of the matrix is stored in COO storage format. Three different partitioning schemes can be applied when converting a CSR matrix to a matrix in HYB storage format. For further details on the partitioning schemes, see rocsparse_hyb_partition.

Storage schemes and indexing base

rocSPARSE supports zero-based and one-based indexing. The index base is selected by the rocsparse_index_base type, which is either passed as a standalone parameter or as part of the rocsparse_mat_descr type.

Dense vectors are represented with a 1D array, stored linearly in memory. Sparse vectors are represented by a 1D data array that holds all non-zero elements and a 1D indexing array that holds the positions of the corresponding non-zero elements, both stored linearly in memory.