REPRESENTING HIGHER DIMENSIONAL ARRAYS INTO A GENERALIZED TWO-DIMENSIONAL ARRAY

Authors

K.M. Azharul Hasan

Md Abu Hanif Shaikh

Dept. of Computer Science and Engineering

Khulna University of Engineering & Technology,

Khulna, Bangladesh

PRESENTATION LAYOUTMotivation toward

G2A

Why G2A

Previous Work

WHY G2A ?

Modeling and analyzing scientific phenomena strongly requires handling large scale data of higher dimension efficiently and effectively

The cost of index computation becomes high for higher dimensional array

The cache miss rate increases for higher dimensional arrays as more cache lines need to be accessed

Visualization and operation on higher dimensional array is tough for Traditional Multidimensional Array (TMA)

PREVIOUS WORK

Extendible Karnaugh Map Representation (EKMR) by Lin et al. But it works well till four dimensions.

Matricization of n-way tensor by Brett W. Bader and Tamara G. Kolda has a higher storage rate though operation on stored data good hear.

G2A

N-dimensional array will be fitted in a 2-D space by placing odd dimensions in row direction and even dimensions along column direction

Forward Mapping: Equivalent G2A indices from TMA(n)

G2A (CONTINUED….)

Backward Mapping: Finding TMA from G2A

Initialize x1’ := 0, x2’ : = 0

Repeat i := 1 to n Repeat j:= i +2 to n xi := xi × lj j := j + 2 x’2 - i%2 := x’2 - i%2 + xi

G2A: 3-D TO 2-D

G2A A[l1'][ l2'] for

TMA(3) A[2][3][4] where l1'=l1×l3=8

and l2'=l2=3

Element A[1][1]2] of TMA(3) is equi-

valent G2A is A[x1'][ x2'] where x1'= 1 × 4 + 2 = 6 and x2'= x2 =1

G2A is A[x1'][ x2'] is known then it’s equivalent TMA(4) becomes A[x1][x2][x3] where x3=x1'%l3=6%4=2, x1=x1'/l3 =1 and x2 =x2'=1

G2A: 4-D TO 2-D

TMA(4) A[l1][l2][l3][l4] of size

[2, 3, 3, 2], the correspondingG2A A[l1'][ l2‘] where l1'=l1×l3=2×3=6 and l2'=l2×l4=3×2=6

Element A[1][1][2][0] of TMA(4) is equivalent G2A is A[x1'][ x2'] where x1'= 1 × 3 + 2 = 5 and x2'= 1 × 2 + 0 = 2

G2A is A[x1'][ x2'] is known then it’s equivalent TMA(3) becomes A[x1][x2][x3] [x4] where x3=x1'%l3=5%3=2, x1=x1'/l3 =1 and x4 =x2'%l4=2%2=0, x2 =x2'/l4=2/2=1

G2A: 6-D TO 2-D

Similarly for G2A of size[12,12] equivale-nt to TMA(6) of size [2, 2, 2, 3, 3, 2]x'1= x1×l3×l5+x3×l5+x5,

x'2=x2×l4×l6+x4× l6+ x6

For backward mappingx6 = 7 % l6 = 7 % 2 = 1,

x4=(7/l6)%l4=(7/2)%3=0

x2 =(7/l6)/l4=(7/2)/3=1

…………

MATRIX-MATRIX ADDITION/SUBTRACTIONAlgorithm 1:matrix-

matrix_addition_TMA_nbegin for x1 = 0 to (l1-1) do for x2 = 0 to (l2-1) do

………………….………………….

for xn =0 to (ln-1) do C[x1][x2]…[xn] = A[x1][x2]…[xn] + B[x1][x2]…[xn];End.

Algorithm 2:matrix-matrix_addition_G2A begin for = 0 to (-1) do for = 0 to (-1) do

C'[x'1][x'2] = A'[x'1][x'2] + B'[x'1][x'2]; End.

MATRIX-MATRIX MULTIPLICATIONAlgorithm 3:

matrix-matrix_multiplication_TMA_n

begin for x1 = 0 to (l1-1) do for x2 = 0 to (l2-1) do

for x3 =0 to (l3-1) do …………………… …………………… for xn-1 =0 to (l-1) do for xn =0 to (l-1) do

for i =0 to (l-1) doC[x1][x2]…[xn-1][xn]= C[x1][x2]…

[xn-1][xn] + A[x1][x2]…[xn-1][i]×B[x1][x2]…[i][ xn];

End.

Algorithm 4:matrix-matrix_multiplication_G2A

begin for = 0 to (-1) do

begin m= - for = 0 to (-1) do

beginn = -

for i =0 to (l-1) do C'[x'1][x'2] = C'[x'1][x'2] + A'[x'1][n+i] × B'[m+i][x‘2];

end End.

MATRIX OPERATION (BLOCK BY BLOCK)

EXPERIMENTAL RESULT (ADDITION)

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Execution time is less for our proposed G2A scheme than TMA. Because the algorithm for TMA has many loops than G2A based algorithm. Hence TMA based algorithm has higher cache miss rate than that of G2A based algorithm.

EXPERIMENTAL RESULT (MULTIPLICATION)

Execution time is less for our proposed G2A scheme than TMA. Because the algorithm for TMA has many loops than G2A based algorithm. Hence TMA based algorithm has higher cache miss rate than that of G2A based algorithm.

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FUTURE SCOPE

Efficient Storage Scheme for higher dimensional sparse array

Better Memory Management with G2A than TMA

Operations on Stored data with G2A Parallelization on operation on G2A operation

Thank you