1. "Developing an Efficient Sparse Matrix Framework Targeting SSI Applications" Diego Rivera and David Kaeli The Center for Subsurface Sensing and Imaging Systems (CenSSIS) Electrical and Computer Engineering Department, Northeastern University, Boston, MA {drivera,
Introduction and Goals
a barrier encountered in many CenSSIS applications is the computational time required to generate a solution
this is most critical in sparse algorithms: because of indirect indexing, the number of non-contiguous memory system accesses increases
it can increase the number of memory stalls and overall application execution time
modifying to the right data layout can significantly improve the performance of solving the system
but predicting and improving locality is complicated due to irregular patterns of references and the non-zeros structure of matrix is unknown a priori
in addition, the selection of a preconditoner as an efficient solution for a particular application is now more difficult
there are a large amount of options and often there is little knowledge of the characteristics of those methods by the people who will use them
we are working on the development and implementation of a framework that will allow us to:
predict the best function to predict locality
for distributed shared memory machines, predict execution times of computations and communications
modify data layout to increase locality
choose Iterative method and preconditioner, including their dynamic parameters
Jagged Diagonal format
Second cache hierarchy simulation
Second set of matrices
replacement policy LRU
level 1 (data and instructions)
cache size 8 KB
line size 64 B
associativity 4-way
level 2
cache size 256 KB
line size 128 B
associativity 8-way
a11
a12
a14
a22
a23
a25
a31
a33
a34
a42
a44
a45
a46
a55
a56
a65
a66
a42
a44
a45
a46
a11
a12
a14
a22
a23
a25
a31
a33
a34
a55
a56
a65
a66
Name: s3dkq4m2
N = 90449
Nnz =
WB = 615
Name: Fidap011
N = 16614
Nnz =
WB = 9680
value list = a42 | a11 | a22 | a31 | a55 | a65 || a44 | a12 | a23 | a33 | a56 | a66 || a45 | a14 | a25 | a34 || a46
column indexes = 2 | 1 | 2 | 1 | 5 | 5 || 4 | 2 | 3 | 3 | 6 | 6 || 5 | 4 | 5 | 4 || 6
start positions = 1 | 7 | 13 | 17 | 18
perm vector = 4 | 2 | 3 | 1 | 5 | 6
for i ← 1 to N
Temp[perm vector[i]] := Y [i]
Y [i] := Temp[i]
Distance vs. Cache misses
disp := 0
for i ← 1 to num jdiag
for j ← 1 to (start position[i+1] - start position[i] - 1)
Y [j] := Y [j] + value list[disp] × X[column indexes[disp]]
disp := disp + 1
end for
Distances x y x x x x x y
aelemes(x,y) = 2
D1(x,y) = max( nelems (x) , nelems(y) ) - aelemes(x,y)
D2(x,y) = nelems (x) + nelems(y) - 2 × aelemes(x,y)
Then:
Note:
JAD indirect addressing on X using column indexes, then distance by row is correct!. But since JAD storages by column, the column indexes should be compared by column.
Clustering - Optimizing x x
Choosing parameters and function to predict locality, also iterative method and preconditioner, including their dynamic parameters.
New
matrix
Feature extraction
Data
Mining
If column indexes = 2 | 1 | 2 | 1 | 5 | 5 || 4 | 2 | 3 | 3 | 6 | 6 || 5 | 4 | 5 | 4 || 6
then:
Note:
CSC indirect addressing on Y using row indexes, then distance by column is correct
JAD indirect addressing on X using column indexes, then distance by row is correct!
aelemes(x,y) = 1
D3 = ( 6 – 3 ) + ( 6 – 2 ) + ( 4 – 2 ) + ( 1 – 0 )
How many different indexes by Jagged diagonal are there?
Modifying locality and choosing Iterative method and preconditioner, including their dynamic parameters.
System
solved
Models
and rules, samples.
Conclusions
Experiments and results
an indirect measure of locality is the reuse distance, which can be calculated for example counting the number of matches between sets of columns/row; but the way that it is measured depends on the storage scheme selected
minimizing a distance function lends to a locality maximization
different distance functions can be definite for a specific format, but their accuracy for describing the locality for a specific matrix could depend on non-zeros structure of the matrix, because in order to measure locality, these functions have to be defined by that structure
First set of matrices
Studying locality
First cache hierarchy simulation
Compressed Column Storage format (CCS)
one level
cache size 16 KB
line size 64 B
associativity 2-way
replacement policy LRU
a11
a12
a14
for i ← 1 to N
for j ← start positions [i] to start positions [i+1] -1
Y [row indexes [j]] := Y [row indexes [j]] + value list[j] × X[i]
end for
a22
a23
a25
a31
a33
a34
Name: Fidap008
N = 3096
Nnz = 90766
WB = 256
Name: Bcsstk16
N = 4884
Nnz =
WB = 282
a42
a44
a45
a46
Generating permutations
a55
a56
Where: N, number of row/columns
Nnz, number of non-zero elements
a65
a66
Plans and future work:
Defining a framework that allows us to:
predict the best function to predict locality
for distributed shared memory machines, predict execution times of computations and communications
modify data layout to increase locality
choose Iterative method and preconditioner, including their dynamic parameters
4000 Permut.
2000 Permut.
200 Permut.
20 Permut.
value list = a11 | a31 | a12 | a22 | a42 | a23 | a33 | a14 | a34 | a44 | a25 | a45 | a55 | a65 | a46 | a56 | a66
row indexes = 1 | 3 | 1 | 2 | 4 | 2 | 3 | 1 | 3 | 4 | 2 | 4 | 5 | 6 | 4 | 5 | 6
start positions = 1 | 3 | 6 | 8 | 11 | 15 | 18
arrays Y, row indexes and value list have Nnz references and arrays X and start positions have 2×N
therefore total number of references 3×Nnz+4×N
X and start positions have good spatial locality : compulsory misses, but poor contribution because usually N << Nnz
arrays value list and row indexes have no temporal locality, but have good spatial locality : compulsory misses  mayor contribution (2×Nnz)
Y [row indexes [j]] := Y [row indexes [j]] + value list[j] × X[i]
complex behavior
conflict and capacity misses
Remarks:
main source of cache misses are misses due to access arrays value list and row indexes, but it can be solved with prefetching; assuming continuous storage
misses due to access array Y are hard to estimate
focusing on data and not instructions
a closer grouping of Nnz in a particular row  higher temporal locality on Y
a closer grouping of Nnz in consecutive rows  higher spatial locality on Y
Name: Bcsstk18
N = 11948
Nnz =
WB = 2488
Name: E30R5000
N = 9661
Nnz =
WB = 686
Points to consider:
the associativity and size of each element of the memory hierarch
the sparse format selected
classic techniques of optimization such as blocking and tiling
Value Added to CenSSIS
This research addresses an important problem in accelerating the execution of applications in the area of Subsurface Sensing and Imaging System. This has a high value for the CenSSIS community.
References
D. Heras, V. Blanco, J. Cabaleiro, and F. Rivera. Modeling and improving locality for the sparse matrix-vector product on cache memories . Future Generation Computer Systems, 18(1):55-67, 2001
Shuting Xu, Jun Zhang, A Data Mining Approach to Matrix Preconditioning Problem, Laboratory for High Performance Scientific Computing and Computer Simulation, Department of Computer Science, University of Kentucky, March, 2005
Distance vs. Cache misses
Acknowledgement
This work is affiliated with CenSSIS, the Center for Subsurface Sensing and Imaging Systems. Support is provided by National Science Foundation under Grant No. NFS ACR: