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Vol. 8 No. 5 August 2010 International Journal of Computer Science and Information Security Publication August 2010, Volume 8 No. 5 (Download Full Journal) (Archive)
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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 5, August 2010
Feasibility Analysis of using Cannon Method
to Perform Block Rhotrix Multiplication.
Abdullahi M. 1, Ezugwu A. E. 2, Ahmed B. I. 3, Tukur S. M. 4
Department of Mathematics,
Ahmadu Bello University,
Zaria, Nigeria.
1 muham08@yahoo.com, 2 code_abs@yahoo.com, 3 barroonia@yahoo.co.uk, 4 smtexpert@yahoo.com
result of the rhomboid nature of the arrangement of
Abstract-This paper introduces the some mathematical arrays, which can simply
concept of parallel rhotrix – rhotrix be seen as the combination of matrices. A rhotrix R
multiplication, algorithm design and analysis is called a real rhotrix if all its entries belong to the
set of Real Numbers. On the other hand a rhotrix R
based on the idea of parallelization of is called an integer rhotrix if all its entries belong to
matrix–matrix multiplication with Cannon the set of Integer Numbers.
algorithm. Rhotrix is a matrix like array of n- dimensional
form of rhomboid nature. The major difference
Keywords: Rhotrix, rhotrix blocking, alignment between matrix and Rhotrix is the arrangement
operation, shift operation, main entries, heart and the introduction of concept of a heart (h).
entries.
a
1. INTRODUCTION
b h( R ) d
Rhotrices multiplication was first introduced e
and defined by Ajibade[1], and later Sani [2]
gave an alternative row-column multiplication Fig 1: Rhotrix Representation
of higher dimensional rhotrices. Their work was
Rhotrix can be split into two forms of matrices:
basically an idea that offers exercises for
the main entries and the heart as shown in the
mathematical enrichment with sets of objects
figure below.
which are in some ways between 2-dimensional
vectors and 2x2-dimensional matrices as in [1], a d
and for those objects which are in some ways b e h( R )
between nxn-dimensional matrices and (2n-1) x
(2n-1) dimensional matrices as in [3]. where h(R) is the heart of the rhotrix.
In this paper our interest is based on n- Fig 2: Rhotrix main entries and the heart
dimensional rhotrix operation and its
multiplication. The name rhotrix come up as a
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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 5, August 2010
2. HIGHER-DIMENSIONAL RHOTRICES We define the main rows and columns of rhotrix Rn
depicted in figure 3 as:
Ajibade[1] further added that, the dimension of
rhotrices can be expanded in size but would at ,1 at 1,1 . . . a1,1
always be of odd dimension, he further proved at ,2 at 1,2 . . . a1,2
that an n-dimensional rhotrix Rn will have |Rn|
at ,3 at 1,3 . . . a1,3
entries where |Rn| = (1/2)(n2+1). n-dimensional
rhotrix can be represented as shown in figure 3. . . . . . .
. . . . . .
a11
at ,t at 1,t . . . a1,t
a21 c11 a12 and
a31 c21 a 22 c12 a31
. . . . . . . a1,1 . . . a1,t 1 a1,t
Rn . . . . . . . . . a2,1 . . . a2,t 1 a2,t
. . . . . . . a3,1 . . . a3,t 1 a3,t
a tt-2 ct 1t 2 a t-1t-1 ct 2t 1 a t-2t . . . . . .
a t-1t-2 c t-1t-1 a t-2t-1 . . . . . .
a tt at ,1 . . . at ,t 1 at ,t
Figure 3: General representation of n- Respectively, the rows and columns of Rn, i.e. those
dimensional rhotrix (adopted from [3]) containing the values clk are similarly defined
likewise.
Where t=(n+1)/2, so that if n=3, then t=2. If
n=5, t=3 and if n=7, t=4. In this case the 3rd, 5th 3. MULTIPLICATION OPERATION OF
and 7th dimensional rhotrices are given in RHOTRICES
figure4:
Multiplication operation denoted by “o” can be
a11 defined in many ways, but in this paper we
R3 a21 c11 a12 restrict our idea based on B. Sani‟s [3] concept
a22 of row-column multiplication of high
dimensional rhotrices. Representing the n-
a11 dimensional rhotrix in figure 3 above by Rn =
a12 c11 a12 (aij,clk) where aij and clk represent the aij and clk
R5 a31 c12 a22 c12 a13 elements respectively, with i,j=1,2,3,4,……,t
a31 c22 a23 and l,k=1,2,3,4,…..t-1.
a33 The general idea of multiplying any two
a11 rhotrices of the same dimension still remain the
a21 c11 a12 same with that which is obtainable in the matrix
row-column multiplication method where the
a31 c21 a22 c12 a13
set of elements aij multiply each other and
R7 a41 c31 a32 c22 a23 c13 a14
elements clk also do likewise. Thus, the
a42 c32 a33 c23 a24 multiplication of any two rhotrices say Rn and
a43 c33 a34 Qn can defined as
a44 Rn Qn = (ai1 j1,cl1k1 ) (bi 2 j 2, dl 2k 2 ) =
t t
1
b)
a
( , cd )
(
2
i2
j1 1 j l k k
i i2 2l l 2
1
j 2 2 12
k 11
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Vol. 8, No. 5, August 2010
Figure 6a: Blocking process of an eight dimensional rhotrix with main entries a and hearts d
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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 5, August 2010
Figure 6b: Blocking process of an eight dimensional rhotrix with main entries b and hearts e
The blocking process from figure 6a and 6b above was obtained in similar manner as that of matrix blocking
method performed using cannon algorithm. The eight dimensional rhotrix above was blocked into sixteen
partitions comprising of rhotrix main entries and the heart elements denoted by a, b and d, e. Those elements
outside the blocks are mapped to their corresponding elements with the same index number thereby allowing
them to belong to the same partitioned block.
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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 5, August 2010
Two rhotrices R (A) and R (B) are considered to illustrate multiplication as shown below:
Figure 7a: Example of blocking process of an eight dimensional rhotrix, R (A)
Figure 7b: Example of blocking process of an eight dimensional rhotrix R (B)
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1 2 2 5 2 3 7 2 Vol. 8, No. 5, August 2010 of Computer Science and Information Security,
(IJCSIS) International Journal
3 1
3
3
1
3
1
2
Computation at Process P00, C00 and H00
2 3 2 2 6 1 7 5
1 7 1
4
6
1
3
2
After the first alignment
A
3 2 1 7 3 7 5 3 1 2 1 2 3 12
C00 A00 B00
1
4 1
5
3
1
2
4
3 1 1 5 4 11
2 1 1 5 2 1 3 1 3 1 2 3 8 12
1 H 00 D00 E00
2
5 4
1
7 1
1 7 1 2 3 16 24
Figure 8a: Block of matrices of main entries of After the first shift
rhotrix R (A)
3 12 2 5 3 1
C00 C00 A01 B10
3 1 1 4 1 2 1 4 11 3 3 2 2
7 1
1 4
2
5 4
3 12 16 12 19 24
5
2 1 4 7 1 1 4 11 15 9 19 20
D 1 3 4 1
1
2 2
8 12 1 4 4 3
H 00 H 00 D00 E00
7 5 1 5 1 1 3 16 24 1 4 4 1
3
5 7 7
1 2 3
8 12 20 7 28 19
1
2 2 3 3 3 3 16 24 20 7 36 31
Figure 8b: Block of matrices of heart entries of After the second shift
rhotrix R (A)
1 2 1 2 2 1 3 1 19 24 2 3 3 1
C00 C00 A02 B20
1 5
1 3
1
2
4
3
19 20 1 3 3 2
3 1 1 2 6 2 4 2 19 24 15 8 34 32
2 2 4 4
1
1 5
6
19 20 12 7 31 27
B
3 1 3 1 5 3 3 1
3
2 1 1
2 1 1
1
28 19 1 2 1 3
H 00 H 00 D02 E20
36 31 2 5 1 1
1 2 1 3 1 5 1 1
2 1 4 4 4 1 1 5 28 19 3 5 31 24
36 31 7 11 43 42
Figure 8c: Block of matrices of main entries of
rhotrix R (B) After the third shift
2 3 2 11 2 2 34 32 7 2 1 2
C00 C00 A03 B30
3
2 3 1
4
3 1
31 27 1 2 2 1
4 3 2 5 1 4 2
7 34 32 11 16 45 48
E 4 1 2
7 2 3
31 27 5 4 36 31
1 3 4 3 2 1 2
1
1
2 2 4 2
4
31 24 1
H 00 H 00 D03 E30 2 3
42 4
2
3 2 1 3 1 3 43
31 24 2 3 33 27
Figure 8d: Block of matrices of heart entries of 43 42 8 12 51 54
rhotrix R (B)
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4. BLOCK RHOTRIX OPERATIONS (IJCSIS) International Journal of Computer Science and Information Security,
6. COMMUNICATION STEPS IN
Vol. 8, No. 5, August 2010
CANNON'S ALGORITHM
Rhotrix computations involving scalar algebraic
operations on the rhotrix elements can be Applying the block row-column operation using
expressed in terms of identical operations on the cannon‟s algorithm method for rhotrices
submatrices of the original rhotrix. with main entries Ai,j , Bi,j and with
corresponding hearts Di,j , Ei,j , we consider
In this view, an n x n rhotrix A can be regarded cannon‟s matrix multiplication algorithm from
as a q x q array of blocks Ai,j (0 ≤ i, j < q) such the point of view of process Pi,j. The processes
that each block is an (n/q) x (n/q) subrhotrix. are organized into a 2-D mesh, and each process
We perform q3 rhotrix multiplications, each has a block of A, B, D and E needed for
involving (n/q) x (n/q) matrices requiring (n/q)3 computing C and H respectively. Figure 9a
additions and multiplications. depicts the first block multiplication step. After
each block multiplication process Pi,j sends its
5. RHOTRIX-RHOTRIX block of A to the process on its left and receives
MULTIPLICATION: CANNON’S a new block of A from the process on its right.
ALGORITHM PATTERN Similarly, it sends its block of B to the process
above it and receives a new block of B from the
Consider n dimensional rhotrices An and
process below it. Figure 9b, 9c and 9d shows
Bn partitioned into p blocks Ai,j and Bi,j
the second, third and final block rhotrix
(0 ≤ i, j < √p ) each.
multiplication respectively. Summing the results
Process Pi,j initially stores Ai,j ,Bi,j and of all block rhotrix multiplications yields Ci,j
Di,j, Ei,j and computes block Ci,j and and Hi,j.
Hi,j,of the resulting rhotrix.
Blocking and Job Assignment Operation
Computing subrhotrix Ci,j requires all A. First Alignment
subrhotrices Ai,k, Bk,j and Di,k, Ek,j for 0 ≤
k <√p.
Every process in the ith row requires all
subrhotrices, the all-to-all broadcast can
be avoided by scheduling the
computations of the processes of the ith
row such that, at any given time, each
process is using a different block Ai,k
and Di,k by systematically rotating these
blocks among the processes after every
submatrix multiplication so that every
process gets a fresh Ai,k, and Di,k after
each rotation.
If an identical schedule is applied to the Figure 9a: First block rhotrix multiplication step. A, B
columns of B and E, then no process and D, E after initial alignment
holds more than one block of each
matrix at any time.
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B. Second Alignment (IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 5, August 2010
First, align the blocks of A, D and B, E in such a way
that each process multiplies its local submatrices:
Shift sub matrices Ai,j and Di,j to the left
(with wraparound) by i steps.
Shift sub matrices Bi,j and Ei,j up (with
wraparound) by j steps.
Perform local block matrix multiplication.
Next, each block of A, D moves one step left
and each block of B, E moves one step up
(again with wraparound).
Figure 9b: Final block rhotrix multiplication step.
Sub-rhotrix location after first shift Perform next block multiplication; add to partial
result, repeat until all the √p blocks have been
C. Third Alignment multiplied.
Figure 9c: Final block rhotrix multiplication
step.Sub-rhotrix location after second shift
Figure 10a row-column block rhotrix multiplication
requiring all-gather operation
D. Final Alignment
Figure 9d: Final block rhotrix multiplication step
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Figure 10b row-column block rhotrix multiplication
ISSN 1947-5500
requiring all-gather communication
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 5, August 2010
7. BLOCK RHOTRIX ALGORITHM
REFERENCES
The block rhotrix multiplication algorithm for n x n
matrices with a block size of (n/q) x (n/q) [1] Ajibade, A. O.,2003, the concept of
rhotrix in mathematical enrichment,
procedure RHOTRIX_MULT (A, B, C, D, E, International Journal of Mathematical
H) Education in Science and Technology,
175-179.
begin
[2] B. Sani, 2006, the row-column
for i := 0 to t - 1 do
multiplication of high dimensional
for j := 0 to t - 1 do rhotrices, International Journal of
Mathematical Education in Science and
begin Technology, 1-6.
{Initialize all elements of Ci,j
[3] B. Sani, 2006, Conversion of a
and Hk,j to zero}
rhotrix to a „coupled matrix‟ ,
C[i, j] := 0; International Journal of Mathematical
Education in Science and Technology,
H[i, j] := 0; 1-7.
for k := 0 to t - 1 do
{Receive message from slave Abdullahi Mohd received his B.Tech.
process (worker) i,j} Degree from Federal University of
Technology, Minna. Nigeria, in 2006.
Ci,j := Ci,j + Ai,k x Bk,j; He is currently a staff in training
Ahmadu Bllo University, Zaria. His
Hi,j := Hi,j + Di,k x Ek,j; area of interest is High Performance Computing.
endfor;
endfor; Ezugwu E. A. received his BSc. Degree
from Ahmadu Bello University, Zaria.
{Write C and H to output file} Nigeria, in 2006. He is currently a
master’s degree student in same
end RHOTRIX_MULT
University. His area of interest is
Software Engineering.
8. CONCLUSION
In this paper we discussed the basic concept of
higher dimensional rhotrices multiplication and Barroon Isma’eel Ahmad received
parallel algorithms for multiplying two n- the B.Sc Degree from Usmanu
dimensional rhotrices R(A) and R(B) has been Danfodiyo University, Sokoto.
proposed. The parallel rhotrix multiplication Nigeria, in 2006. He is currently a
staff in training Ahmadu Bllo
algorithms proposed is based on the
University, Zaria. His interests are
conventional serial algorithm. The idea of Mobile Communication and Location Aware Application
matrix operations has been used in solving Development.
product of n x n rhotrices.
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