A NEW PARALLEL MATRIX MULTIPLICATION ALGORITHM ON HEX-CELL NETWORK (PMMHC) USING IMAN1 SUPERCOMPUTER

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 5, October 2017
DOI:10.5121/ijcsit.2017.9503 29
A NEW PARALLEL MATRIX MULTIPLICATION
ALGORITHM ON HEX-CELL NETWORK (PMMHC)
USING IMAN1 SUPERCOMPUTER
Enas Rawashdeh1
, Mohammad Qatawneh1
and Hussein A. Al Ofeishat2
1
Department of Computer Science, King Abdullah II School for Information
Technology, The University of Jordan,
2
Al-Balqa Applied University-Jordan.
ABSTRACT
A widespread attention has been paid in parallelizing algorithms for computationally intensive
applications. In this paper, we propose a new parallel Matrix multiplication on the Hex-cell
interconnection network. The proposed algorithm has been evaluated and compared with sequential
algorithm in terms of speedup, and efficiency using IMAN1, where a set of simulation runs, carried out on
different input data distributions with different sizes. Thus, simulation results supported the theoretical
analysis and meet the expectations in which they show good performance in terms of speedup and
efficiency.
KEYWORDS
Parallel processing, matrix multiplication, Interconnection Network, Hex-Cell.
1. INTRODUCTION
Matrix multiplication is commonly used in many areas like graph theory, residue-level protein
folding [4], numerical algorithms, digital image processing and others. Working with matrix
multiplication algorithm of huge matrices requires a lot of computation time where the
complexity time for sequential matrix multiplication algorithm is O (n3
), where n is the dimension
of the matrix. Because higher computational throughputs are required with the applications, many
parallel algorithms based on sequential algorithms are developed to improve the performance of
matrix multiplication algorithm. There a lot of improvement [7, 8] done on sequential algorithms
to follow the big requirements but still has shown a limitation in performance. For that, parallel
approaches have been examined and enhanced for decades.
In common parallel matrix multiplication algorithms used decomposition of matrices depends on
the number of processors available in the interconnection network [10, 9]. Each algorithms use
the matrices that decomposed into sub matrices (blocks). During execution process of matrix
multiplication, each processor calculates a partial multiplication result using the sub matrices that
are currently accessed by it. When the multiplication is completed, the coordinator processor
assembles and generates the complete matrix multiplication result.
The interconnection networks are the core of a parallel processing system which the system’s
processors are linked. Due to the big role played by the networks topology to improve the parallel
system’s performance, Several interconnection network topologies have been proposed for that
purpose; such as the tree, hypercube, mesh, ring, and Hex-Cell (HC) [1, 2, 5, 6, 11, 12, 14, 15,
18].

30
Among the wide variety of interconnection networks structures proposed for parallel computing
systems is Hex-Cell network which received much attention due to the attractive properties
inherited in their topology [1, 16, 17].
The proposed parallel matrix multiplication on the Hex-cell network is implemented by the
library Message Passing Interface MPI, where MPI processes are assigned to the cores. If the MPI
process is assigned to a core, then it will be parallel computation; but if more than one MPI
process is assigned to the same core, then it will be concurrent computation. Experimentation of
the proposed algorithm was conducted using IMAN1 supercomputer which is Jordan's first
supercomputer. The IMAN1 is available for use by academia and industry in Jordan and the
region.
The rest of the paper is organized as follows. Section 2 describes the definition of Hex-Cell
network. Section 3 presents the proposed algorithm. Section 4 provides an Analytical Evaluation.
Section 5 provides the performance results, and Section 6 summarizes and concludes the paper.
2.DEFINITION OF HEX-CELL NETWORK TOPOLOGY
Hex-Cell network is one of interconnection networks structures proposed for parallel computing
systems where the nodes are connected with each other in hexagonal topology. A Hex-Cell
network with depth d is denoted by HC(d) and can be constructed by using units of hexagon cells,
each of six nodes. A Hex-Cell network with depth d has d levels numbered from 1 to d, as shown
in Figure 1:
• Level 1 states the innermost level corresponding to one hexagon cell.
• Level 2 correlate with the six hexagon cells surrounding the hexagon at level 1.
• Level 3 correlate with the 12 hexagon cells surrounding the six hexagons at level 2.
The levels of Hex-Cell network with depth d are labeled from 1 to d. Each level i has Ni nodes,
representing processing elements and interconnected in a ring structure [1].
HC(3)HC(1) HC(2)
Figure 1. Hex-cell network in different level one, two and three [1].
The address of each node in the Hex-Cell topology is identified by (S,L,Y) where S denotes the
section number, L denotes the level number, and Y denotes the node number on that level labeled
from Y1,…, Yn; where n = ((2×L) - 1) [1].

31
A node with the address 1.1.1 is the first node that exists at the section number 1 and level
number 1, and address 6.1.1 is first node that exists at the section number 6 and level number 1,
as shown in Figure 2.
Figure 2. Hex-Cell addressing scheme by section [5].
3.PMMHC ALGORITHM
In this section, we propose a new Parallel Matrix Multiplication Algorithm on Hex-Cell Network
(PMMHC) as shown in Figure 4. The aim behind the parallelism of the matrix multiplication is to
make the algorithm runs faster and more efficient in comparison with the sequential one for very
large data matrices. It depends on partitioning matrices of size n into a set of partitions; each
partition is assigned to a separate processor to multiply sequentially using sequential matrix
multiplication. Thus, the number of partitions depends on the number of the available processors.
In this paper, we apply matrix multiplication on the Hex-cell interconnection network topology.
The hex-cell network [1] is divided into six sections as shown in Figure 2. The proposed
algorithm uses each section as ring topology and the root nodes of level 0 depend on one to all
personalized broadcast for child’s nodes. As shown in figure 4, the proposed work is assumed
that a matrices data is stored in the main coordinator processor (MC), which it will be partitioned,
multiply, and then combined at the main coordinator processor. And L0-HC nodes are level 0 ring
nodes of Hex-Cell network; L1-Ring coordinators are the root nodes of each ring section
correspond each one with the nodes of level 0.

32
Figure 3. PMMHC algorithm Hex-Cell section.
Input: Matrix A and B
Output: Matrix C on Hex-Cell using parallel Matrix Multiplication
Phase 1: Data Distribution Phase
Data Distribution in the Hex-Cell root nodes at L0.
1. MC (Main Coordinator) generates a set of blocks of Matrix A.
2. MC generates a set of blocks according to the Matrix B.
3. MC routes the Aik and Bkj values internally to all L0-HC nodes on L0-Ring.
4. For all processors in L0 (that received blocks of matrices in the previous steps), do the
following in parallel: Send the blocks of matrices A and B to the L1-RCs (L1-Ring
Coordinator) of the connected ring.
5. Wait until the coordinator who received the data will send an acknowledgment message.
6. Send a message for the MC informing that the process completed.
7. MC stops the process of distribution and announces the beginning of the next step.
L1-Ring Distribution of Data
8. For all ring coordinators L1-RCs, do the following in parallel:
9. Blocks of matrix A is partitioned into a number of horizontal stripes.
10. Blocks of matrix B is presented as a set of vertical stripes.
11. Send stripes for all processors in each Ring in L1.
12. Stop the process of distribution and announce the beginning of the next step.
Phase 2: Data Multiplication Phase
13. For all processors in each L1-Rings, do the following in parallel:
14. Multiply the stripes of matrix A with stripes of Matrix B (for each block) of data using
sequential matrix Multiplication. Where all processors perform .
Phase 3: Data Combining Phase
L1-Ring Data Combining
15. For all L1-RCs, do in parallel:
16. Combine the collected multiplication in one matrix.
Global Data Combining
17. For all level 1-Ring coordinators (L1-RCs) in the Hex-Cell interconnection, do the following
in parallel:
18. Send the multiplication matrix to the Hex-Cell root nodes at L0.
Combining Data in the Hex-Cell root nodes
19. MC combines the collected matrices correctly from L0-HC roots nodes in matrix C.
Figure 4. The PMMHC algorithm

33
The parallel matrix multiplication on Hex-Cell interconnection network in Figure 4 is illustrated
in more details as follows:
Phase 1: Data Distribution Phase.
Assume I×K matrix A and a K×J matrix B, and the whole matrices Aik and Bkj stored on MC
(main coordinator). The distribution phase is composed of three steps as follows (see Figure 4):
• Data Distribution in the Main Ring (Lines 1-4 in Figure 4). The MC starts the process of
data decomposition the initial matrices A and B. We assume all the matrices are square of n×n
size, the number of vertical blocks and the number of horizontal blocks are the same and are
equal to q and the size of all block is equal to v×v, v=n/q.
• Global Distribution of Data (Lines 5-7 in Figure 4). The nodes (in parallel) start sending the
partitions through the optical links. As shown in Figure 5, the nodes {N0, N1, N2, N3, N4, N5}
will send their partitions to their directly connected neighbors; rings {R0, R1, R2, R3, R4, R5},
respectively. It is important to note that each node in the main group (L0-Ring) receives an
acknowledgement message from its neighbor in the other ring after the process is completed.
Consequently, each node in the main ring (L0-Ring) sends a message to MC telling that the
process was completed. When MC receives messages from all the processors, who
participated in the global distribution steps, it announces the beginning of the next step, which
is the ring distribution.
• Ring Distribution of Data (Lines 8-12 in Figure 3). In this step each L1-RC makes blocks of
matrix A as a number of horizontal stripes, and matrix B is presented as a set of vertical
stripes. The stripe size should be equal to v=n/p (assuming that n is divisible by p), as it will
make possible to provide equal distribution of the computational load among the processors.
Phase 2: Multiplication Phase (Lines 13-14 in Figure 3).
• All the elementary processors (nodes) in the interconnection network apply sequential matrix
multiplies a stripes of A by stripes of B .The processor computes it’s part of the product to
produce a block of rows of C, as
For i from 1 to n:
For j from 1 to p:
Let sum = 0
For k from 1 to m:
Set sum ← sum + Aik × Bkj
Set Cij ← sum
Phase 3: Data Combining Phase.
Combining phase is parallelized by reversing the order of steps in the distribution phase as
follows:
• Level 1-Ring Data Combining (Lines 15-16 in Figure.3). The aim of this step is to combine
all the result of multiplication for the Hex-Cell sections via electronic links. This is done by
first collecting the multiplication from the elementary processors of L1-Ring for each section
on hex-cell network and stores the first combined partitions in the RCs of each section.
• Global Data Combining (Lines 17-18 in Figure 3). All RCs in the whole section of
interconnection network will send their chunks of multiplication data via optical links to their
corresponding processors in the main Ring (L0-Ring).

34
• Combining Data in the Main Ring (Lines 19 in Figure 3). MC collects the whole set of
data multiplication by combines all partitions in one matrix called C.
Figure 5. Nodes on ring topology in proposed algorithm
4.ANALYTICAL EVALUATION
This section provides the analytical evaluation of the proposed (PMMHC) parallel Matrix
multiplication on Hex-Cell interconnection network. Three performance metrics are used to
evaluate the algorithm, namely: Run time complexity, speedup and efficiency.
4.1 Run time complexity
Time complexities of distribution phase in PMMHC is the same as complexity of One to all
personalized in L0-Ring and L1-Ring with the different ( ) chunk for each processor, and the time
complexities of combining phase in PMMHC the same as complexity of All- to one personalized
for L1-Ring and L0-Ring, So, the total Time communication in the matrix multiplication on Hex-
Cell network is:
Time Complexity of Computation for each processor will multiply elements using sequential
Matrix multiplication= ( . So, the total Time complexity of the proposed algorithm is:
4.2 Speedup
Speedup is one of the performance metrics used in the evaluation of parallel algorithms in
general. It evaluates the performance of a parallel algorithm in comparison with its sequential
counterpart [3]. The speedup of the PMMHC network is shown in Equation 1.
(1)

35
4.3 Efficiency
The efficiency is another performance metric that is widely used to assess the performance
improvement in parallel algorithms in general. Its value represents an indicator on how much do
the processors being utilized [3]. The efficiency of the PMMHC network is shown in Equation 2.
Efficiency = (2)
5.EXPERIMENTAL RESULTS AND PERFORMANCE EVALUATION
In this section, the results of different simulation runs over different data distributions are
presented. Table 1 show the results of speedup for different datasets in which you can observe
that in general, the result of speedup is better with large matrices to multiply. IMAN1 Zaina
cluster is used to conduct our experiments and open MPI library is used in our implementation of
the following parallel matrix multiplication algorithms; and the experimental runs on a dual quad
core intel xeon Cpu with smp, 16 gb ram, where the software specification is conducted on
scientific linux 6.4 with open mpi 1.5.4, C and C++ compiler.
Table 1 shows architectural information about the Hex-Cell interconnection network. Also, it
shows information about the expected size of the input data that can be assigned for each group in
a lucky-case partitioning, when applying the parallel matrix multiplication on the Hex-Cell
interconnection network.
Table 1. Experimental results of proposed algorithm
Matrix
Size
2 processors 4 processors 8 processors 16 processors 32 processors
Time Speed up Time Speed Up Time Speed Up Time Speed Up Time Speed Up
500 2.7654 0.312829 1.4121 0.612633 0.9814 0.881495 0.824 1.049878 1.021 0.8473065
1000 7.9574 1.404818 6.0367 1.851789 2.5272 4.423353 2.628 4.253691 2.769 4.0370892
2000 77.629 1.44340 27.47 4.078998 22.543 4.970505 17.92 6.252795 5.256 21.318512
3000 189.21 1.733187 67.23 4.877932 54.698 5.995528 69.32 4.730862 33.78 9.7070625
4000 531.09 1.169383 182.54 3.402255 83.826 7.408772 81.82 7.590415 49.25 12.610107
5000 622.36 1.815913 198.63 5.689735 108.32 10.43345 88.17 12.81787 79.14 14.280417
Figure 5 shows the speedup for the proposed algorithm according to different matrices sizes. All
results are performed on a different number of processors. Where with the data size increases, the
run time increases due to the increased number of multiplication and the increased time required
for data combining. The size of data assigned to each processor plays a primary role in obtaining
the highest speedup values. This means that the ratio between the data size and the number of
processors can be considered as an indicator of whether we can obtain a high speedup value or
not.

36
Figure 5. Number of processors versus Speed Up
6. CONCLUSIONS
In this paper, we present a parallel matrix multiplication on Hex-Cell interconnection network.
The proposed parallel matrix multiplication algorithm was simulated over different number of
processors, with different sizes of matrices, where the algorithm comprises three phases to be
applied on the Hex-Cell interconnection network. These phases are the distribution phase, the
multiplication phase using the sequential matrix multiplication, and finally the combining phase.
Actually, these phases can be easily modified to suit other application that requires massive data
to be manipulated. However, the parallel matrix multiplication on Hex-Cell interconnection
network shows higher performance in comparison with its sequential version on a single
processor.
As a part of our future work, we aim to conduct a comparative study by applying the matrix
multiplication over different interconnection networks. We also aim to extend this study by
applying sorting algorithms on Hex-Cell interconnection such as merge sort and quick sort.
ACKNOWLEDGEMENTS
We thank Eng. Zaid Abudayyeh for assistance with to accomplish this research.
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A NEW PARALLEL MATRIX MULTIPLICATION ALGORITHM ON HEX-CELL NETWORK (PMMHC) USING IMAN1 SUPERCOMPUTER

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