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Parallel Programming 0024

Parallel Programming 0024. Matrix Multiplication Spring Semester 2010. Outline. Discussion of last assignment Presentation of new assignment Introduction to matrix multiplication Issues in parallelizing matrix multiplication Performance measurements Questions/Comments?.

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Parallel Programming 0024

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  1. Parallel Programming0024 Matrix Multiplication Spring Semester 2010

  2. Outline • Discussion of last assignment • Presentation of new assignment • Introduction to matrix multiplication • Issues in parallelizing matrix multiplication • Performance measurements • Questions/Comments?

  3. Discussion of Homework 4

  4. Questions to be answered • Is the parallel version faster? • How many threads give the best performance? • What is the influence of the CPU model/CPU frequency?

  5. - RUNNABLE - TIMED_WAITING- TERMINATED - NEW - BLOCKED - WAINTING new Thread() NEW thread.start() RUNNABLE synchronized BLOCKED sleep() TIMED_WAITING join() WAITING* interrupt() TERMINATED

  6. Presentation of Homework 5

  7. Matrix multiplication • Problem: Given two matrices A, B of size N * N. Compute the matrix product C = A * B with • Cij = Sum(Aik * Bkj) (0 <= k < N) • A, B elements are double-precision floating point numbers (“double”) • Assume that A and B are dense matrices • Sparse matrices have many zero elements • Only the non-zero elements are stored • Dense matrices have mostly non-zero elements • Each matrix requires N2 storage cells

  8. Parallel matrix multiplication Which operations can be done in parallel? x =

  9. Programming matrix multiplication • Java code for the loop nest is easy. • double[][] a = new double[N][N]; • double[][] b = new double[N][N]; • double[][] c = new double[N][N]; • for (i=0; i<N; i++) { • for (j=0; j<N; j++) { • a[i][j] = rand.nextDouble(); • b[i][j] = rand.nextDouble(); • c[i][j] = 0.0; • } • } • for (i=0; i<N; i++) { • for (j=0; j<N; j++) { • for (k=0; k<N; k++) { • c[i][j] += a[i][k]*b[k][j]; • } • } • }

  10. Parallel matrix multiplication • Data partitioning based on • Input matrix A • Input matrix B • Output matrix C

  11. Parallel matrix multiplication • Data partitioning based on • Input matrix A • Input matrix B • Output matrix C • We assume that all threads can read inputs A and B • Start with partitioning of output matrix C • No need to use synchronized !

  12. Parallel matrix multiplication • Each thread computes its share of the output C • Partition C by columns x = T0 T1 Tx T… T…

  13. Two threads • One thread computes columns 0 .. N/2, the other columns N/2+1 .. N-1 x = T0 T1

  14. Two threads Thread 0 for (i=0; i<N; i++) { for (j=0; j<N; j++) { for (k=0; k<N/2; k++) { c[i][j] += a[i][k]*b[k][j]; } } } Thread 1 for (i=0; i<N; i++) { for (j=0; j<N; j++) { for (k=N/2; k<N; k++) { c[i][j] += a[i][k]*b[k][j]; } } }

  15. Other aspects • Partition C by columns or by rows? T0 T1 x = T…

  16. Other aspects • What should be the order of the loops? • for (i=0; i<N; i++) { • for (j=0; j<N; j++) { • for (k=0; k<N; k++) { • c[i][j] += a[i][k]*b[k][j]; • } • } • } • Or? • for (k=0; i<N; i++) { • for (i=0; j<N; j++) { • for (j=0; k<N; k++) { • c[i][j] += a[i][k]*b[k][j]; • } • } • } • Or?

  17. # of threads/matrix size 1 2 4 8 16 32 64 … 1024? 100 x 200 x … 10,000? Performance Measurement

  18. Any Questions? • synchronized • Thread, Runnable • - wait(), notify(), notifyAll() • - Thread States

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