Sparse Storage Formats

oneMKL - Data Parallel C++ Developer Reference

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Date 6/24/2024

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Sparse Storage Formats

There are a variety of matrix storage formats available for representing a sparse matrix. The Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS library provides support for the following sparse matrix formats:

Sparse Matrix Formats Supported in oneMKL Sparse BLAS
Compressed Sparse Row (CSR)
Coordinate (COO)

Compressed Sparse Row (CSR)

One of the most common sparse formats is the CSR (sometimes called 3-array CSR or CSR3) format that is represented by scalar sizes (nrows, ncols), as well as three data arrays: row_ptr, col_ind and values, and the index_base parameter. Some versions of this format also explicitly store the number of non-zero elements (nnz) but this can be extracted from the row_ptr array as described below in the description of the row_ptr so we will not include it here.

CSR Matrix Format Elements	Description
`nrows`	Number of rows in the sparse matrix.
`ncols`	Number of columns in the sparse matrix.
`index`	Parameter that is used to specify whether the matrix has zero or one-based indexing.
`values`	An array that contains the non-zero elements of the sparse matrix stored row by row.
`col_ind`	An integer array of column indices for non-zero elements stored in the `values` array, such that `col_ind[i]` is the column number (using zero- or one-based indexing) of the element of the sparse matrix stored in `values[i]`.
`row_ptr`	An integer array of size equal to `nrows + 1`. Element `j` of this integer array gives the position of the element in the `values` array that is first non-zero element in a row `j` of `A`. Note that this position is equal to `row_ptr[j] - index`. The last element of the `row_ptr` array (`row_ptr[nrows]`) stores the sum of the number of non-zero elements (nnz) and `index`. That is, `nnz = row_ptr[nrows] - index`.

Examples of CSR format

The following 3 examples show how the sparse CSR format can be used:

CSR Case 1: sorted square matrix with zero-based indexing
CSR Case 2: sorted rectangular matrix with one-based indexing and an empty row
CSR Case 3: unsorted rectangular matrix with zero-based indexing

CSR Case 1: sorted square matrix with zero-based indexing

Assuming zero-based indexing and a real square matrix.

nrows	3
ncols	3
index	0
row_ptr	0	2	4	5
col_ind	0	2	1	2	0
values	1.0	2.0	-1.0	4.0	3.0

CSR Case 2: sorted rectangular matrix with one-based indexing and an empty row

Assuming one-based indexing and real rectangular matrix with an empty row.

nrows	4
ncols	5
index	1
row_ptr	1	3	6	6	8
col_ind	1	3	2	3	5	1	4
values	1.0	2.0	-1.0	4.0	1.0	3.0	1.0

CSR Case 3: unsorted rectangular matrix with zero-based indexing

Unsorted CSR example: Assuming zero-based indexing and a real rectangular matrix, we note that the CSR format does not require column indices to be sorted within a given row, but values and col_ind arrays must be consistent with each other. Having the sorted property is not necessary, but can lead to better performance in actual runs due to better algorithms and data locality being enabled. See sparse::set_matrix_property() and sparse::sort_matrix for more details on how one could set the sorted property when it is applicable or reorder the matrix to achieve it when desired.

nrows	4
ncols	5
index	0
row_ptr	0	2	5	9	10
col_ind	0	2	4	1	2	1	2	0	3	0
values	1.0	2.0	1.0	-1.0	4.0	2.0	3.0	1.0	4.0	3.0

Coordinate (COO) format

The simplest sparse matrix format is the COO format, represented by scalar sizes (nrows, ncols, nnz), as well as three data arrays: row_ind, col_ind, and values, along with an index parameter. Each tuple, (row_ind[i], col_ind[i], values[i]), represents a non-zero value in the sparse matrix.

COO Matrix Format Elements	Description
`nrows`	Number of rows in the sparse matrix.
`ncols`	Number of columns in the sparse matrix.
`nnz`	Number of non-zero in the sparse matrix.
`index`	Parameter that is used to specify whether the matrix has zero or one-based indexing.
`row_ind`	An integer array of row indices of non-zero elements in the `values` array, such that `row_ind[i]` is the row number (using zero- or one-based indexing) of the element of the non-zero value in `values[i]`.
`col_ind`	An integer array of column indices of non-zero elements in the `values` array, such that `col_ind[i]` is the column number (using zero- or one-based indexing) of the element of the non-zero value in `values[i]`.
`values`	An array that contains the non-zero elements of the sparse matrix, preferably but not necessarily stored in a sorted order.

Examples of COO format

The following examples show how the sparse COO format can be used:

COO Case 1: sorted rectangular matrix with one-based indexing
COO Case 2: unsorted rectangular matrix with zero-based indexing

COO Case 1: sorted rectangular matrix with one-based indexing

An example of a rectangular COO matrix in one-based indexing sorted by rows, and then by columns within rows, is given below.

nrows	4
ncols	5
nnz	7
index	1
row_ind	1	1	2	2	2	4	4
col_ind	1	3	2	3	5	1	4
values	1.0	2.0	-1.0	4.0	1.0	3.0	1.0

COO Case 2: unsorted rectangular matrix with zero-based indexing

The COO format does not require sorted arrays so long as (row_ind[i], col_ind[i], val[i]) are grouped together in their unsorted order for all i, representing each non-zero element in the sparse matrix. Having the sorted property is not necessary, but can lead to better performance in actual runs due to better algorithms and data locality being enabled.

An example of an unsorted COO matrix in zero-based indexing and a real rectangular matrix are given below.

nrows	4
ncols	5
index	0
row_ind	2	1	0	1	2	3	0	1	2	2
col_ind	1	4	2	1	0	0	0	2	3	2
values	2.0	1.0	2.0	-1.0	1.0	3.0	1.0	4.0	4.0	3.0

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oneMKL - Data Parallel C++ Developer Reference

Sparse Storage Formats

Compressed Sparse Row (CSR)

Coordinate (COO) format