STORAGE SCHEME FOR SPARSE MATRICES

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1 EE87 STORAGE SCHEME FOR SPARSE MATRICES Pei Xu University of Minnesota xuxx088 at umn.edu

2 Outline Motivation Co-ordinate Storage (COO) Compressed Row Storage (CSR) Compressed Column Storage (CSC) Linked List Other Storage Schemes Sparse Matrices in Matlab References

3 Motivation Why we need special storage scheme for sparse matrices? Suppose a system with 000 buses with an average connectivity of per bus Then, only (+) x 000 = 000 non-zero entries in the Y matrix who has 000 x 000 = 0 6 entries in total. => If store all entries in the matrix: ( )/0 6 = 99. units are used to store zero entries Most of resources are wasted on computing 0 x 0 = 0 and 0 + n = n

4 Motivation Why we need special storage scheme for sparse matrices? Minimize storage Store only non-zero elements Simplify computations Work upon only non-zero elements

5 Co-ordinate Storage (COO) A = A b b b b a b c d e f or b b b b T A = value a b c d e f row column N N N N or T N N N N In order to calculate N, search all elements to find those whose row is or whose column is. Inefficient

6 Compressed Row Storage (CSR) A = A b b b b b a b e = d N N N N N c f g order 6 7 value a b c d e f g column row_begin 7 Put the elements in a same row together. Use row_begin to identify the order of the number with which a row begins. When using CSR, it is easy to find all elements in a row.

7 Compressed Row Storage (CSR)

8 Compressed Row Storage (CSR) Y = Value [ - - -] Column [ ] First [ ] DIAG

9 Compressed Row Storage (CSR) A = b b b b b a A = b e d c f g N N N N N order 6 7 value a b c d e f g column row_begin 7 Search all elements to find those whose column Is, in order to calculate N. Inefficient

10 Compressed Row Storage (CSR) A = a b e X d c f g Inefficient order 6 7 value a b c d e f g column row_begin 7 order value a x b c d e f g column row_begin 6 8

11 Compressed Column Storage (CSC) A = b b b b b T a A = b e d N N N c f T N N g order 6 7 value a b e d c f g row col_begin 7 Put elements in a same column together. Use col_begin identifies the order of the number that begins in a column. It is easy to find all elements in a column.

12 Linked List A = a b e d c f g Row Row Row Row Row Column Column Column Column Column a b c d e f g Order Row Col Val Next_in_row Next_in_col a b c d e 6 f 7 g 6 6

13 Linked List A = a b e d c f g Order Row Col Val Next_in_row Next_in_col a b c 6 d e 6 6 f 7 g Order Row Col Val Next_in_row Next_in_col a b c 6 d e 6 6 f 7 g

14 Linked List A = a b e d c x f Order Row Col Val N_in_r g N_in_c a b c 6 d e 6 6 f 7 g => Row Row Row Row Row Column Column Column Column Column a b c d x e f g Order Row Col Val N_in_r N_in_c a b c 8 d 8 8 x 6

15 Comparison Space Usage Rate Find elements in a row Find elements in a column COO CSR CSC Linked List +N/NNZ < +N/NNZ < Excellent Good x Excellent Good Add Elements Excellent x x Good Space Usage Rate: the average units the scheme needs to store non-zero elements N: no. of elements in a row or column x : Inefficient NNR: no. of non-zero elements

16 Comparison Space Usage Rate Find elements in a row Find elements in a column COO CSR CSC Linked List +N/NNZ < +N/NNZ < Excellent Good x Excellent Good Add Elements Excellent x x Good Space Usage Rate: the average units the scheme needs to store non-zero elements N: no. of elements in a row or column x : Inefficient NNR: no. of non-zero elements

17 Other Storage Schemes Compressed Diagonal Storage Jagged Diagonal Storage List of Lists Storage Yale Storage Block Compressed Row Storage Skyline Storage

18 Sparse Matrices in Matlab CSR is used in Matlab >> tic; Y*b; disp(numstr(toc)) >> >> tic; b*y; disp(numstr(toc)) >> help sparse sparse Create sparse matrix. S = sparse(x) converts a sparse or full matrix to sparse form by squeezing out any zero elements.

19 References [] Soman, Shreevardhan Arunchandra, Khaparde, S.A., Pandit and Shubha, Computational Methods for Large Sparse Power Systems Analysis An Object Oriented Approach. 00 [] Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst. Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide. 000

20 THANK YOU Pei Xu University of Minnesota xuxx088 at umn.edu

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