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)¶
One of the most common sparse formats is the CSR (sometimes called 3-array CSR or CSR3) format that is
represented by scalar sizes (num_rows
, num_cols
), as well as three data arrays: rowptr
, colinds
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 rowptr
array as described below in the
description of the rowptr
so we will not include it here.
CSR Matrix Format Elements |
Description |
---|---|
|
Number of rows in the sparse matrix. |
|
Number of columns in the sparse matrix. |
|
Parameter that is used to specify whether the matrix has zero or one-based indexing. |
|
An array that contains the non-zero elements of the sparse matrix stored row by row. |
|
An integer array of column indices for non-zero elements stored in the |
|
An integer array of size equal to |
Examples of CSR format¶
The following 3 examples show how the sparse CSR format can be used:
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.
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3 |
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3 |
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0 |
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0 |
2 |
4 |
5 |
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0 |
2 |
1 |
2 |
0 |
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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.
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5 |
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1 |
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1 |
3 |
6 |
6 |
8 |
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1 |
3 |
2 |
3 |
5 |
1 |
4 |
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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 colinds
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.
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0 |
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0 |
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5 |
8 |
10 |
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0 |
2 |
4 |
1 |
2 |
1 |
2 |
0 |
3 |
0 |
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1.0 |
2.0 |
1.0 |
-1.0 |
4.0 |
2.0 |
3.0 |
1.0 |
1.0 |
3.0 |