Multiplication of two 3 d sparse matrices using 1d arrays and linked lists

International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: 2321-2152 (Online)
Volume No.-1, Issue No.-2, May, 2013
RES Publication©2012 Page | 10
http://www.resindia.org
New Algorithms for Multiplication of Two 3D Sparse
Matrices Using 1D Arrays and Linked Lists
Abhishek Jain 1st
Department of Computer Engineering,
Poornima College of Engineering,
Jaipur, India
E-mail: aj.2170@gmail.com
Sandeep Kumar 2nd
Faculty of Engineering & Technology,
Jagan Nath University,
Jaipur, India
E-mail: sandeep.kumar@jagannathuniversity.org
Abstract: A basic algorithm of 3D sparse matrix multiplication (BASMM) is presented using one dimensional (1D) arrays which is used further
for multiplying two 3D sparse matrices using Linked Lists. In this algorithm, a general concept is derived in which we enter non- zeros elements
in 1st
and 2nd
sparse matrices (3D) but store that values in 1D arrays and linked lists so that zeros could be removed or ignored to store in
memory. The positions of that non-zero value are also stored in memory like row and column position. In this way space complexity is
decreased. There are two ways to store the sparse matrix in memory. First is row major order and another is column major order. But, in this
algorithm, row major order is used. Now multiplying those two matrices with the help of BASMM algorithm, time complexity also decreased.
For the implementation of this, simple c programming and concepts of data structures are used which are very easy to understand for everyone.
Keywords: sparse, non-zero values, dimension, depth, limit
I. INTRODUCTION
A matrix, in which maximum elements of matrix are zero, is
called sparse matrix. A matrix that is not sparse is called dense
matrix. [3]
Consider the sparse matrix given below:-
The natural method of sparse matrices in memory as two
dimensional arrays may not be suitable for sparse matrices
because it will waste our memory if we store „zero‟ in a block
by 2 bytes. So, we should store only „non-zero‟ elements. [3]
One of the basic methods for storing such a sparse matrix is to
store non-zero elements in a one-dimensional array and to
identify each array element with row and column indices as
shown in figure below.
There are two ways to represent the sparse matrix
1.1. Using array
1.2. Using linked list
In both representations, information about the non-zero
element is stored.
1.1. Using Array:-
Below figure shows representation of sparse matrix as an
array.
Note that the sparse matrix elements are stored in row major
order with zero elements removed. Any matrix, with larger
quantity of zeros then non-zeros is said to be sparse matrix.
1.2. Using Linked List:-
A sparse matrix is given below:-
Representation of sparse matrix in c is given below:-
struct matrix
{
int value, row, column;
struct matrix * next;
} *start;
The sparse matrix pictorial represented as:-[3]

Phases
The whole work is divided in two phases which are given
below:-
i. Multiplication of two 3D sparse matrices using 1D
array
ii. Multiplication of two 3D sparse matrices using linked
list
II. PHASE 1. PROPOSED ALGORITHM FOR
MULTIPLICATION OF TWO 3D SPARSE MATRICES
USING 1D ARRAY
If a matrix is a size of m x n x d, then here m is for number of
rows, n is for number of columns and d is for number of faces
or depth. Here take m=n=d=3.
Let us understand how non-zero values are stored in sparse
matrix. We use three arrays, one for row position, second for
column position and third for storing non-zero values for each
matrix. In a sparse matrix has size of m x n, then the maximum
number of non-zero values is calculated by limit = (d*m*n)/2.
Consider three matrices A, B and C. For A matrix to storing
row positions array is denoted by ar, to storing column
positions array is denoted by ac and to non-zero values array is
denoted by av. Similar for B and C Matrices.
Now, Firstly we take first non-zero value from A matrix and
note its position. Now we compare its column position with
row position of B matrix. If it matches, we multiply A matrix‟s
non-zero value with B matrix‟s non-zero value and store the
result in value array of C matrix, and store row position from
A matrix and column position from B matrix to row and
column position of C matrix respectively.
If it not matches, then we compare A‟s column position to the
next B‟s row position in array. This process repeated until all
elements of A matrix multiplied with B matrix. In this process,
we get some non-zero values of same positions, so we have to
add them to get final result.
limit1=maximum non-zero values in 1st sparse matrix (d1 x r1
x c1)
limit2=maximum non-zero values in 2nd sparse matrix (d2 x
r2 x c2)
k1=non-zero values in 1st matrix
k2 =non-zero values in 2nd matrix
k3= pointer for 3rd matrix of multiplication
t1=traversing pointer for 1st matrix
t2= traversing pointer for 2nd matrix
ad[limit1],ar[limit1], ac[limit1], av[limit1] – arrays of 1st
matrix for storing positions of depth,rows and columns and for
values.
bd[limit2],br[limit2], bc[limit2], bv[limit2] – arrays of 2nd
matrix for storing positions of depth, rows & columns and for
values.
cd[limit1*clo2],cr[limit1*col2], cc[limit1*col2],
cv[limit1*col2] – arrays of 3rd matrix for storing positions of
depth, rows & columns and for values after multiplication.
1. t1=0,t2=0;
2. while(t1<k1)
3. {
4. match=0;
5. while(t2<k2)
6. {
7. count=0;
8. if(br[t2]==ac[t1] && ad[t1]== bd[t2])
9. {
10. while(count<c2)
11. {
12. if( bc[t2]==count)
13. {
14. int f=0;
15. for(int y=0;y<k3;y++)
16. {
17. if( cd[y]== ad[t1] && cr[y] == ar[t1] && cc[y]
== bc[t2])
18. {
19. f=1;//flag
20. cv[y] = cv[y] + av[t1] * bv[t2];

21. break;
22. }
23. }
24. if(f==0)
25. {
26. cd[k3] = ad[t1]; //or ad[t2] becoz both are same
27. cr[k3] = ar[t1];
28. cc[k3] = bc[t2] ;
29. cv[k3] = av[t1] * bv[t2];
30. k3++;
31. count++;
32. }//while
33. }//if
34. if(match==c2)
35. {
36. break;
37. }
38. t2++;
39. }//while
40. t2=0;
41. t1++;
42. }//while
43. }
44. }//if
III. PHASE 2. PROPOSED ALGORITHM FOR
MULTIPLICATION OF TWO 3D SPARSE MATRICES
USING LINKED LISTS
This is totally different concept from multiplication of 2D or
3D sparse matrix multiplication using 1D array. Here we use
linked lists instead of 1D arrays to store information of sparse
matrices.
In this new approach, the structure of liked list is modified and
created two new structures of linked list to store information
about sparse matrix.
First structure is node1, which hold three information, first-
row position of non-zero values, second-down pointer which
store the address of next node of non-zero value and third-
right pointer which store the address of node2. Start pointers
store the address of first node of each matrix. Like start1 store
the address of A matrix, start2 store the address of B matrix
and start3 store the address of resultant C matrix.
Now, second structure node2, which also hold three
information, first- col to store column position of non-zero
element, second- value which store non-zero value and third-
next which store the address of next node which hold the non-
zero values in same row if it has. In this way, sparse matrix
information is totally re-organized in better structure like
matrix form in which non-zero values of same row are linked
with each other in one line and next row is linked separately in
next line but lined with previous row by down pointer.
In this way, multiplication algorithm is also similar to
traditional multiplication approach but in traditional approach,
multiplication loop depends of matrix size and in this new
approach multiplication loop depends on number of non-zero
values.
This node3 used for third dimension (depth d or faces). It has 3
parts, first- depth which store the depth number of face
number of sparse matrix, second- start pointer which store the
address of either start1 or start2 or start3. It means, this start
pointer store the address of first node of first face of first
sparse matrix, similarly for all. Third- dnext which store the
address of next node of node3 structure who hold the similar
information of previous node. Like, next or second node of
node3 hold the information as in depth part it store second
face, in start part it store the address of first node of second
face of first sparse matrix. And dnext node holds the
information of third node of node3 structure.
Similarly all these information is also logically arranged for B
and resultant C matrix.
Information of each face will be multiplied with corresponding
same face and results will also store in same corresponding
face.
struct node1
{
int row;
struct node1 * down;
struct node2 * right;
} *start1, *start2, *start3;
struct node2
{
int col;
int value;
struct node2 * next;
};
struct node3 // singly linked list for depth node
{
int depth;
struct node3 * start;
struct node3 * dnext;
} * A_depth, * B_depth, * C_depth ;

Multiplication Algorithm for 3D Sparse Matrix using
Linked Lists.
1. Repeat step 2 until dnext pointer becomes NULL for
A matrix
2. Repeat step 3 and 4 until down pointer becomes
NULL for A matrix
3. If right pointer not equals to NULL for A matrix, then
4. Repeat step 5 until next pointer becomes NULL for A
matrix
5. Repeat step 6 until dnext pointer becomes NULL for
B matrix
6. Repeat step 7 and 8 until down pointer becomes
NULL for B matrix
7. If right pointer not equals to NULL for B matrix, then
8. Repeat step 9 until next pointer becomes NULL for B
matrix
9. If col part of A matrix is equal to row part of B
matrix, then
(a) Multiply element from A matrix‟s value part with
element from B matrix‟s value and store the result to
C matrix‟s value part.
(b) Copy row position from row part of A matrix to row
part of C matrix.
(c) Copy column position from column part of B matrix
to column part of C matrix.
10. Exit.
IV. ANALYSIS OF EXISTING AND PROPOSED
ALGORITHM IN TERMS OF TIME COMPLEXITY
For 3D sparse matrix, if
1st matrix = d * R1 x C1 = 3 x 3 x 3
2nd matrix =d * R2 x C2 = 3 x 3 x 3
limit1 = maximum no. of non-zero elements in 1st matrix
= (d x R1 x C1)/2 = 13
limit2 = maximum no. of non-zero elements in 2nd matrix
= (d x R2 x C2)/2 = 13
k1 = actual no. of non-zero elements in 1st matrix
k2 = actual no. of non-zero elements in 2nd matrix
E1 - Fast Transpose and Mmult Algorithm (Using Arrays)
(Existing Algorithm)
A1 - Proposed Algorithm for 3d Sparse Matrix
Multiplication Using 1D Arrays
A2 - Proposed Algorithm for 3d Sparse Matrix
Multiplication Using Linked Lists
Table 1.Ddifferent time complexities of given algorithms
Algorithm
Time
Complexity
formula
E1 О(d*R1*C1*C2)
A1 O(d*K1*K2*C2)
A2 O(d*K1*K2)

Table 2. Calculating time complexities of all three algorithms on different K
values
Table3. Comparison between existing and proposed 1st
algorithm
Table4. Comparison between existing and proposed 2nd
algorithm
Graph : Comprasion among all three algorithms
V. CONCLUSION AND FUTURE WORK
Normally space and time complexity is greater if we store non-
zero values in 1D array but lesser if we stored those values in
linked list.
Now, all time complexity depends on number of comparisons,
multiplication steps and number of non-zero values in sparse
matrix. If we decrease number of comparisons then we can
decrease time complexity. if 1st
matrix is of d x R1 x C1 order,
K1 is the number of non-zero values in 1st
matrix, 2nd
matrix is
of d x R2 x C2 order, K2 is the number of non-zero values in
2nd
matrix and d is the depth or dimension of sparse matrix,
then time complexity for A1 algorithm for 3D sparse matrix
using 1D arrays is d*K1*K2*C2 and for A2 algorithm for 3D
sparse matrix using linked lists is d*K1*K2.
At present, both algorithms are suitable only for multiplication
for two 3D sparse matrices. But in future it could be
generalized for multidimensional arrays.
REFERENCES
[1] Ellis Horowitz and Sartaz Sahni – “Fundamentals of
data structures” (2012), Galgotia Booksource (Page
51-61)
[2] Indra Natarajan – “Data structure using C++” (2012),
Galgotia Booksource. (Page 11-164, 297-346)
[3] Hariom Pancholi- “Data Structures and Algorithms
using C” (5th
ed. 2012), Genus Publication. ISBN
978-93-80311-01-2 (Page 2.26-2.30, 4.78)
[4] Cormen – “Introduction to Algorithms (2nd
ed.)”,
Prentice Hall of India. (Page 527-548). ISBN- 978-
81-203-2141-0
K1=K2=K Time Complexity
K E1 A1 A2
1 81 9 3
2 81 36 12
3 81 81 27
4 81 144 48
5 81 225 75
6 81 324 108
7 81 441 147
8 81 576 192
9 81 729 243
10 81 900 300
11 81 1089 363
12 81 1296 432
13 81 1521 507
Condition Case
Time Complexity
Comparison
E1 A1
d*K1*K2 <
d*R1*C2
Best
Case
High Low
d*K1*K2 =
d*R1*C2
Average
Case
High High
d*K1*K2 >
d*R1*C2
Worst Case Low High
Condition Case
Time Complexity
Comparison
E1 A2
d*K1*K2 <
d*R1*C2
Best Case High Low
d*K1*K2 =
d*R1*C2
Average Case High Low
d*K1*K2 >
d*R1*C2 and
K1 < √
(R1*C1*C2)
Below
Average Case
Medium Low
d*K1*K2 >
d*R1*C2
Worst Case Low High

[5] R B Patel and M M S Rauthan – “Expert data
structure with C++ (2nd
ed.)”,ISBN-8187522-03-8,
(Page 262-264)
[6] Randolph E. Banky, Craig C. Douglasz – “Sparse
Matrix Multiplication Package (SMMP)” April 23,
2001. Advances in Computational Mathematics
[7] Golub, Gene H.; Van Loan, Charles F.(1996).-
“Matrix Computations” (3rd ed.). Baltimore: Johns
Hopkins. ISBN 978-0-8018-5414-9.
[8] Jess, J.A.G. –“ A Data Structure for Parallel L/U
Decomposition” Proc. IEEE Volume: C-31 , Issue: 3
Page(s): 231 – 239, ISSN : 0018-9340
[9] I. S. Duff - "A survey of sparse matrix
research", Proc. IEEE, vol. 65, pp. 500 -1977
[10]Stoer, Josef; Bulirsch, Roland (2002)-“Introduction to
Numerical Analysis” (3rd ed.). Berlin, New York:
Springer-Verlag. ISBN 978-0-387-95452-3.
[11]Pissanetzky, Sergio (1984)-“Sparse Matrix
Technology”, Academic Press.
[12]Reguly, I. ; Giles, M. – “Efficient sparse matrix-
vector multiplication on cache-based GPUs ” IEEE
Conference Publications, Year: 2012, Page(s): 1 – 12
[13]Daniel Krála, Pavel Neográdyb, Vladimir Kellöb-
“Simple sparse matrix multiplication algorithm”,
Computer Physics Communications, ELSEVIER,
Volume 85, Issue 2, February 1995, Pages 213–216,
[14]R. C. Agarwal ET AL-“A three dimensional approach
to parallel matrix multiplication”, IBM J. Res.
Develop. Vol. 39 No. 5 September 1995
AUTHOR’S BIOGRAPHIES
Abhishek Jain 1st
B.E. (CSE, SKIT College, Jaipur/UOR, 2009 Batch),
M.Tech (Pursuing) (CS, Jagan Nath University, Jaipur, 2011-13
Batch)
Sandeep Kumar 2nd
B.E. (CSE, ECK Kota/UOR, 2005 Batch)
M. Tech (ACEIT, Jaipur/RTU, 2011 Batch)
Ph. D. (Pursuing) (CSE, Jagan Nath University, Jaipur)

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