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Compressed Sparse eXtended (CSX) sparse matrix format. This proof-of-concept version is now deprecated. It has been replaced from the SparseX sparse kernel optimization library []


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CSX library v0.2

1. Introduction

This package is a proof-of-concept release of the Compressed Sparse
eXtended format for sparse matrices. This format seeks to minimize the
memory footprint of the column index array of the typical Compressed
Sparse Row (CSR) format by exploiting dense substructures inside the
sparse matrix. Instead of storing a single index for every nonzero
element of the sparse matrix, CSX stores a short description for each
substructure found in the matrix (and selected for encoding). This
technique can save significant amount of main memory storage and
minimize the bandwidth requirements of the Sparse Matrix-Vector
Multiplication (SpMV) kernel. Finally, the CSX format employs runtime
code generation (using the LLVM compiler infrastructure) for emitting
optimized SpMV routines for each encoded pattern.

More information about the CSX format can be found in

K. Kourtis, V. Karakasis, G. Goumas, and N. Koziris, "CSX: An extended
compression format for SpMV on shared memory systems," 16th ACM
SIGPLAN Annual Symposium on Principles and Practice of Parallel
Programming (PPoPP'11) San Antonio, TX, USA, February 12-16, 2011.

V. Karakasis, G. Goumas, K. Nikas, N. Koziris, J. Ruokolainen, and
P. Råback, "Using State-of-the-Art Sparse Matrix Optimizations for
Accelerating the Performance of Multiphysics Simulations" PARA 2012:
Workshop on State-of-the-Art in Scientific and Parallel Computing
Helsinki, Finland, June 10-13, 2012.

2. Summary of changes with the previous version

The CSX format and library has matured considerably since its first
release in February, 2011. A summary of the changes follows:

* Complete re-implementation of the runtime code generation module
  using Clang and LLVM
* Efficient support of NUMA architectures
* Support for symmetric matrices
* Proof-of-concept non-preconditioned CG implementation using CSX
* Revised preprocessing (better sampling, multi-threading, careful
  memory management)
* Code and Makefile restructuring
* Support for multi-threaded compilation
* Efficient implementations, incl. NUMA-aware versions, of other
  sparse matrix storage formats, such as CSR, BCSR, VBL, SSS
* Self-contained library interface ( for use with sparse
  linear algebra s/w, specifically tuned for the Elmer multiphysics
  simulation s/w.
* Several bug fixes

3. Prerequisites and external dependencies

* A fairly recent Linux OS
* LLVM >= 2.9 and Clang
* Boost Regex Library >= 1.48
* numactl library >= 2.0.7
* Cairo library (optionally)

4. Compiling the package

This package is written in C and C++. We have compiled it successfully
with several GCC up to 4.6. Running `make' at the top-level source
directory will be enough to compile the whole package for a typical
installation of LLVM and Boost. If you have installed LLVM 2.9 in a
non-standard location that is not in your path, you can instruct the
compilation process to use your preferred LLVM 2.9 installation by

$ make llvmconf=/path/to/llvm-2.9/bin/llvm-config

If, for any reason, you might want to alter the predefined
preprocessor, compiler flags and/or linker flags, you can use the
standard CPPFLAGS, CFLAGS, CXXFLAGS and LDFLAGS. You need not worry
for the dependencies within the CSX library, since we append the
required flags to the user supplied ones.

Finally, you can speed up the compilation by using multiple tasks with
the `-j' flag of make:

$ make -j8

5. Running CSX

After you have successfully compiled CSX, the final executable, named
`spmv', will be placed inside the `csx' directory. The spmv executable
may be invoked as follows:

[ENV=<value>] ... ./spmv [-b | -s] <matrix_file> [...]

The execution of CSX is controlled by a set of environment variables
and takes as input one or more files describing sparse matrices. For
each matrix, `spmv' will perform an analysis of the patterns found and
print the results, the preprocessing time and the wall-clock time (and
performance in Mflop/s) for the execution of 128 consecutive SpMV

5.1 Input matrix format

The `spmv' executable requires the input matrices to be in a variation
of the Matrix Market Exchange format. Specifically, the first line of
the file must contain the number of rows, columns and non-zeros,
respectively, separated by whitespace. The rest of the lines contain
the nonzero elements of the matrix (one per line) in the form of
`<row> <column> <value>', sorted lexicographically (i.e.,
row-major). To facilitate the conversion to the desired format, the
utility script `scripts/' is supplied.

See also

5.2 Environment

The execution of `spmv' is solely controlled by environment variables,
which are described in detail below.

** Variables controlling multithreaded execution

MT_CONF         Set the CPU affinity for running `spmv'. This is the
                only way to setup a multithreaded execution for
                `spmv'. You should supply the CPU numbers (see
                /sys/devices/) of the desired CPUs as a
                comma-separated list. If MT_CONF is not specified, it
                is set to `0', thus assuming single-threaded

** Variables controlling the construction of CSX

XFORM_CONF      Set the pattern types to search for in the
                matrix. This is a comma-separated list of the pattern
                type IDs as specified in the enumeration
                `SpmIterOrder' in `spm.h'. The mapping between type
                IDs and types is the following:

                           0 -> NONE
                           1 -> HORIZONTAL
                           2 -> VERTICAL
                           3 -> DIAGONAL
                           4 -> REV_DIAGONAL
                           5 -> BLOCK_TYPE_START (not used)
                           6 -> BLOCK_R1 (not used)
                           7 -> BLOCK_R2
                           8 -> BLOCK_R3
                           9 -> BLOCK_R4
                          10 -> BLOCK_R5
                          11 -> BLOCK_R6
                          12 -> BLOCK_R7
                          13 -> BLOCK_R8
                          14 -> BLOCK_COL_START (not used)
                          15 -> BLOCK_C1 (not used)
                          16 -> BLOCK_C2
                          17 -> BLOCK_C3
                          18 -> BLOCK_C4
                          19 -> BLOCK_C5
                          20 -> BLOCK_C6
                          21 -> BLOCK_C7
                          22 -> BLOCK_C8
                          23 -> BLOCK_TYPE_END (not used)
                          24 -> XFORM_MAX
                The default value of XFORM_CONF is `0', i.e.,
                NONE. You can optionally enable the BLOCK_R1 and
                BLOCK_C1 types, i.e., one-dimensional fixed size
                blocks, programmatically through the DRLE_Manager

ENCODE_DELTAS   Set specific deltas to encode for one-dimensional
                patterns or block dimensions for two-dimensional
                patterns. ENCODE_DELTAS is a comma-separated list of
                "delta descriptors". A delta descriptor is a tuple in
                the form of `{d1,d2,...,dn}', where `di' is a specific
                delta to be encoded. For two-dimensional patterns
                types, the `di' signifies the second dimension of the
                block pattern. The ENCODE_DELTAS variable must be
                specified along with XFORM_CONF, in which case there
                is a one-to-one mapping between the pattern types and
                delta descriptors. For example, using
                XFORM_CONF=16,1,3 and ENCODE_DELTAS={4,7},{1},{11},
                first `BLOCK_C2' patterns with a second dimension of 4
                and 7, respectively (i.e., block patterns 4x2 and 7x2)
                will be encoded into the matrix. Second, `HORIZONTAL'
                patterns with delta 1 will be encoded and, finally,
                `DIAGONAL' patterns with delta 11. The number of
                pattern types specified in XFORM_CONF must equal the
                number of delta descriptors specified in

** Variables controlling preprocessing

The preprocessing phase of CSX can be optimized with the use of
statistical sampling. Preprocessing is done in multiple threads using
the thread configuration specified by the MT_CONF variable. The
following variables control the sampling process.

SAMPLES         Specify the number of sampling windows to consider for
                searching per thread. This variable actually enables
                the optimized preprocessing. It can be used in
                conjunction with one of the following variables, which
                specify the sampling window size, either explicitly or
                implicitly. If the specified number of samples exceeds
                the maximum possible number of samples (depending on
                the sampling policy), it will be set to this maximum
                number. In this case, no statistical sampling takes
                place, but the preprocessing is performed simply with
                the use of preprocessing windows.

WINDOW_SIZE     Set the preprocessing window size explicitly. The
                window size is set in terms of non-zero elements by
                default. The size of the window will be "rounded" up
                to the next full row. You can sample the matrix
                row-wise, in which case WINDOW_SIZE is the number or
                rows of each window. However, to enable this policy,
                you should alter the sampling policy during the
                creation of the DRLE_Manager in `'. This kind
                of sampling is not recommended, since it cannot
                provide a good coverage for irregular matrices.

SAMPLING_PORTION Specify the portion of the matrix to sample as a real
                number in [0,1]. If this variable is specified, the
                window size in non-zero elements will be automatically
                computed to meet the following requirement:

                SAMPLING_PORTION*non_zeros = SAMPLES*computed_win_size

5.3 Other options controlling execution

You can specify the two following options when running CSX:

-b      Disable the split-block optimization. The split-blocks
        optimization leads to better compression, since it improves
        the statistics (number of nonzero elements covered by a
        pattern) of smaller blocks making them eligible for

-s      Use the version of CSX for symmetric matrices. In this
        variation, the elements of the main diagonal are stored
        separately and the lower triangular matrix is stored in the
        CSX format.

5.4 Output

The typical output of running the CSX is the following (lines starting
with `#' are not part of the output, but serve as explanatory comments):

## CSX set-up information
Window size: Not set
Number of samples: Not set
Sampling portion: Not set
## Per-thread preprocessing results
## s: heuristic score
## <int>-> : the delta value or the second dimension of the block for
##           blocked substructures
## np: number of patterns encoded by every substructure instantiation
## nnz (<percent>): number of non-zeros encoded by every substructure
##                  instantiation followed by the coverage percentage.
##                  This percentage is an estimation if sampling is
##                  used.
==> Thread: #0
HORIZONTAL  s:12    1-> np:4 nnz: 16 (42.1053%)
VERTICAL    s:15    1-> np:5 nnz: 20 (52.6316%)
DIAGONAL    s:8    1-> np:1 nnz: 6 (15.7895%)    2-> np:1 nnz: 4 (10.5263%)
REV_DIAGONAL    s:3    1-> np:1 nnz: 4 (10.5263%)
BLOCK_R2    s:16    2-> np:3 nnz: 12 (31.5789%)    4-> np:1 nnz: 8 (21.0526%)
BLOCK_R3    s:10    2-> np:2 nnz: 12 (31.5789%)
BLOCK_C2    s:19    3-> np:1 nnz: 6 (15.7895%)    4-> np:2 nnz: 16 (42.1053%)
BLOCK_C3    s:8    3-> np:1 nnz: 9 (23.6842%)
## Substructure type selected for encoding and the detection process
## starts over
Encode to BLOCK_C2
HORIZONTAL  s:3    1-> np:1 nnz: 4 (10.5263%)
VERTICAL    s:3    1-> np:1 nnz: 4 (10.5263%)
DIAGONAL    s:3    2-> np:1 nnz: 4 (10.5263%)
REV_DIAGONAL    s:3    1-> np:1 nnz: 4 (10.5263%)
VERTICAL    s:3    1-> np:1 nnz: 4 (10.5263%)
DIAGONAL    s:3    2-> np:1 nnz: 4 (10.5263%)
REV_DIAGONAL    s:3    1-> np:1 nnz: 4 (10.5263%)
Encode to VERTICAL
DIAGONAL    s:3    2-> np:1 nnz: 4 (10.5263%)
REV_DIAGONAL    s:3    1-> np:1 nnz: 4 (10.5263%)
Encode to DIAGONAL
REV_DIAGONAL    s:3    1-> np:1 nnz: 4 (10.5263%)
## The substructure types were selected for encoded in the selection
## order
## The exact instantiations that were encoded
## delta:<int> : the delta value of the instantiation
## dim:<int>x<int> : the block dimensions of the encoded blocks
## npatterns:<int> : the number of patterns encoded by the instantiation
## nnz:<int> : the number of non-zeros encoded by the instantiation
type:HORIZONTAL delta:1 npatterns:1 nnz:4
type:VERTICAL delta:1 npatterns:1 nnz:4
type:DIAGONAL delta:2 npatterns:1 nnz:4
type:REV_DIAGONAL delta:1 npatterns:1 nnz:4
type:BLOCK_C2 dim:3x2 npatterns:1 nnz:6
type:BLOCK_C2 dim:4x2 npatterns:2 nnz:16
Checking ... Check Passed
## Performance info
## m:<method> : method used (csx or csx-sym)
## f:<file> : the matrix file (basename)
## s:<size> : size in bytes of the CSX format
## pt:<time> : preprocessing time in seconds
## t:<time> : execution time of 128 SpMV operations in seconds
## r:<perf> : performance in Mflop/s
m:csx f:demopatt.mtx.sorted s:327 pt:0.063501 t:0.000086 r:113.363468

6. Running CG

After you have successfully compiled CG, the final executable, named
`cg', will be placed inside the `cg' directory. The cg executable may
be invoked as follows:

[ENV=<value>] ... ./cg [-s | -x | -l <n> | -L <n>] <matrix_file>

The execution of CG is controlled by a set of environment variables
and command line parameters and takes as input a file describing the
sparse matrix. The solution vector of equation Ax = b (s) is
randomized with double precision elements from -1000 to 1000, the
vector b is calculated from the multiplication b = As and the
estimated solution vector (x) is initialized with elements equal to
0.01. The result shows the square distance between the estimated
vector (x) and the real solution vector values (s), the time breakdown
into spmv multiplication phase, spmv reduction time and rest vector
operations, and the total cg time.

6.1 Input matrix format

See 5.1. 

6.2 Environment

See 5.2.

6.3 Other options controlling execution

You can control the execution of the CG method with the following four
command line options:

-x      Activate the CSX format. By default (if the options is not
        set), CSR will be used.

-s      Replace with the respective symmetric formats. Instead of CSR,
        SSS will be deployed and instead of CSX, CSX-Sym would be

-l <n>  Specify the maximum number of iterations of the CG method
        (default is 1024).

-L <n>  Specify how many times to run the specified benchmark (default
        is once).

6.4 Output

The typical output of running CG is the following (lines starting with `#' are
not part of the output, but serve as explanatory comments):

## The CPU configuration.
MT_CONF: 0,1,2,3
## The specified format output (if CSX or CSX-Sym is activated the details about
## stats, encoding etc. is shown.
## Performance info
## m:<matrix> : the matrix file (basename)
## l:<loops> : number of iterations of CG
## rd:<distance> : square distance from actual solution
## st:<time> : SpMxV multiplication operations time
## rt:<time> : SpMxV reduction operations time
## ct:<time> : CG total time
## * the rest vector operations time of CG can be found by subtracting 'st' and
##   'rt' from 'ct' time.
m:26.nd12k.mtx.sorted l:1024 rd:0.004506 st:10.816884 rt:0.145372 ct:11.367220

7. Licence

This package is distributed under the BSD Licence. See `LICENCE.txt' for more

8. Authors

This package is maintained by 

Vasileios Karakasis        <>
Theodoros Gkountouvas      <>
Kornilios Kourtis          <>


Compressed Sparse eXtended (CSX) sparse matrix format. This proof-of-concept version is now deprecated. It has been replaced from the SparseX sparse kernel optimization library []







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