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- 1. 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]
- 2. 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 | 11 http://www.resindia.org 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];
- 3. 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 | 12 http://www.resindia.org 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 ;
- 4. 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 | 13 http://www.resindia.org 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)
- 5. 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 | 14 http://www.resindia.org 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
- 6. 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 | 15 http://www.resindia.org [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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