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 LAYOUT
Motivation
toward G2A
G2A
Matrix
Operation
Algorithm
Experimental
Data
Why G2A
3-D to 2-D
Addition/Subtraction
Addition/Subtraction
Previous Work
4-D to 2-D
Multiplication
Multiplication
6-D to 2-D
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 equivalent 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 corresponding
G2A 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] equivalent 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 mapping
x6 = 7 % l6 = 7 % 2 = 1,
x4=(7/l6)%l4=(7/2)%3=0
x2 =(7/l6)/l4=(7/2)/3=1
…………

MATRIX-MATRIX ADDITION/SUBTRACTION
Algorithm 1:
matrixmatrix_addition_TMA_n
begin
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 MULTIPLICATION
Algorithm 3:
Algorithm 4:
matrixmatrix-matrix_multiplication_G2A
matrix_multiplication_TMA_n
begin
begin
for = 0 to (-1) do
for x1 = 0 to (l1-1) do
begin
for x2 = 0 to (l2-1) do
m= for x3 =0 to (l3-1) do
for = 0 to (-1) do
……………………
begin
……………………
n= for xn-1 =0 to (l-1) do
for i =0 to (l-1) do
for xn =0 to (l-1) do
C'[x'1][x'2] = C'[x'1][x'2] +
for i =0 to (l-1) do
A'[x'1][n+i] × B'[m+i][x‘2];
C[x1][x2]…[xn-1][xn]=
end
C[x1][x2]…[xn-1][xn] +
End.
A[x1][x2]…[xn-1][i]×B[x1][x2]…[i][
xn];
End.
MATRIX OPERATION (BLOCK BY BLOCK)
EXPERIMENTAL RESULT (ADDITION)
2500
4000
TMA(4) Addition
G2A Addition
3000
1500
Execution Time (ms)
Execution Time (ms)
2000
1000
500
TMA(8) Addition
G2A Addition
2000
1000
0
0
0
20
40
60
80
Length of Dimensions
100
120
3
4
5
6
7
8
9
Length of Dimensions
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.
10
11
12
EXPERIMENTAL RESULT
(MULTIPLICATION)
300000
60000
250000
50000
Execution Time (ms)
Execution Time (ms)
TMA(4) Multiplication
G2A Multiplication
200000
150000
100000
TMA(8) Multiplication
G2A Multiplication
40000
30000
20000
10000
50000
0
0
3
0
20
40
60
80
Length of Dimensions
100
120
4
5
6
7
8
9
Length of Dimensions
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.
10
11
12
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