Matrix Computation & Communication Ratio in Parallel Computing

1. Matrix Addition and Multiplication Computation/Communication Ratio ITCS 4/5145 Parallel Computing UNC-Charlotte, B. Wilkinson, Jan 21, 2014. slides11A.ppt

2. Matrices — A Review An n xm matrix

3. Matrix Addition Sequential code to compute A + B could simply be for (i = 0; i < n; i++) for (j = 0; j < n; j++) { c[i][j] = a[i][j] * b[i][j]; } Requires n2multiplications and n2additions Sequential time complexity of O(n2). Very easy to parallelize as each result independent although may not get speedup in message-passing environment.

4. Matrix Addition, C = A + B Add corresponding elements of each matrix to form elements of result matrix. Given elements of A as ai,jand elements of B as bi,j, each element of C computed as: Add A B C Could have one process(or) compute one or more C elements.

5. Workpool/master slave implementation Slaves Return one element of C C A B Send one element of A and B to slave Master Compute node Each slave process doing just one addition not a good partition of work. Could try each slave doing more work - next. Source/sink

6. Each process adding a pair of complete rows Adds one row of A with one row of B to create one row of C (rather than each process adding single elements) Add A B C

7. Workpool implementation Slaves (one for each row) Return one row of C C A B Send one row of A and B to slave Master Compute node Although slaves do more, need more data sent! Source/sink

8. Matrix Multiplication Sequential code to compute A x B could simply be for (i = 0; i < n; i++) // for each row of A for (j = 0; j < n; j++) { // for each column of B c[i][j] = 0; for (k = 0; k < n; k++) c[i][j] = c[i][j] + a[i][k] * b[k][j]; } Requires n3 multiplications and n3 additions Sequential time complexity of O(n3). Very easy to parallelize as each result independent

9. Matrix Multiplication, C = A * B Multiplication of two matrices, A and B, produces matrix C whose elements, ci,j(0 <= i < n, 0 <= j < m), computed as follows: where A is an n x l matrix and B is an l x m matrix.

10. Workpool implementation Slaves (one for each element of result) Return one element of C C A Send one row of A and one column of B to slave B Master Compute node Following example 3 x 3 arrays and 9 slaves Source/sink

11. Usually size of matrices (n) much larger than number of processors (p). So divide matrix into s2 submatrices. Each submatrix has n/s x n/s elements. One processor produces each submatrix result (p = s2). Block Matrix Multiplication for (p = 0; p < s; p++) for (q = 0; q < s; q++) { Cp,q = 0; /* clear elements of submatrix*/ for (r = 0; r < m; r++) /* submatrix multiplication */ Cp,q = Cp,q + Ap,r * Br,q; /*add to accum. submatrix*/ } Can be applied to all parallelization methods. Means multiply submatrix Ap,r and Br,q using matrix multiplication and add to submatrix Cp,q using matrix addition.

12. Workpool implementation Slaves (one for each element of result) Return s x s submatrix C C A Send s rows of A and s column of B to slave B Master Compute node Source/sink

13. Computation/Communication Ratio where tcomp is computation time and tcomm is communication time. Normally communication very costly Typically approximately linear cost with increasing message data. data Startup time Time

14. Computation/Communication Ratio Could write as time complexities. Suppose: Computation = O(n2) Communication = O(n) (i.e. linear) where n is number of data items By increasing n, eventually n can be found when computation can dominate communication – that is good.

15. Matrix Addition With pairs of rows added by each process: With communication linear with data Computation = O(n) Communication = O(n) Not good in this context as communication constant large

16. Matrix Multiplication ? Discussion

17. Questions