Parallel Programming: Techniques and Applications Using Networked Workstations and Parallel Computers Chapter 11: Numerical Algorithms Sec 11.2: Implementing Matrix Multiplication (c) Prentice-Hall Inc., All rights reserved. Modified 5/11/05 by T. O’Neil for 3460:4/577, Fall 2005, Univ. of Akron. Previous revision: 7/28/03. Barry Wilkinson and Michael Allen

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Implementing Matrix Multiplication Sequential Code Assume throughout that the matrices are square (n  n matrices). The sequential code to compute A  B could simply be for (i=0; i<n; i++) for (j=0; j<n; j++) { c[i][j] = 0; for (k=0; k<n; k++) c[i][j] = c[i][j] + a[i][k]* b[k][j]; } This algorithm requires n 3 multiplications and n 3 additions, leading to a sequential time complexity of O(n 3 ). Very easy to parallelize.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Parallel Code With n processors (and n  n matrices), can obtain: Time complexity of O(n 2 ) with n processors Each instance of inner loop independent and can be done by a separate processor. Cost optimal since O(n 3 )=n  O(n 2 ). Time complexity of O(n) with n 2 processors One element of A and B assigned to each processor. Cost optimal since O(n 3 )=n 2  O(n). Time complexity of O(log n) with n 3 processors By parallelizing the inner loop. Not cost-optimal since O(n 3 )  n 3  O(log n). O(log n) lower bound for parallel matrix multiplication.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Partitioning into Submatrices Suppose matrix divided into s 2 submatrices. Each submatrix has n/s  n/s elements. Using notation A p,q as submatrix in submatrix row p and submatrix column q: for (p=0; p<s; p++) for (q=0; q<s; q++) { /* clear elements of submatrix */ C p,q = 0; /* submatrix mult and */ /* add to accum submatrix */ for (r=0; r<m; r++) C p,q = C p,q + A p,r * B r,q ; }

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Partitioning into Submatrices (cont) for (p=0; p<s; p++) for (q=0; q<s; q++) { C p,q = 0; for (r=0; r<m; r++) C p,q = C p,q + A p,r * B r,q ; } The line C p,q =C p,q +A p,r *B r,q ; means multiply submatrix A p,r by B r,q using matrix multiplication and add to submatrix C p,q using matrix addition. Known as block matrix multiplication.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Block Matrix Multiplication

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Submatrix multiplication Matrices

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Submatrix multiplication Multiplying A 0,0  B 0,0 to obtain C 0,0

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Direct Implementation One processor needed to compute each element of C – n 2 processors would be needed. One row of elements of A and one column of elements of B needed. Some of same elements sent to more than one processor. Can use submatrices.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Performance Improvement Using tree construction n numbers can be added in log n steps using n processors. Computational time complexity of O(log n) using n 3 processors.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Recursive Implementation Apply same algorithm on each submatrix recursively. Excellent algorithm for a shared memory system because of locality of operations.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Recursive Algorithm mat_mult(A pp,B pp,s) { /* if submatrix has one elt multiply */ if (s==1) C = A*B; /* continue to make recursive calls */ else { s = s/2; /* no elts in each row/column */ P0 = mat_mult(A pp,B pp,s); P1 = mat_mult(A pq,B qp,s); P2 = mat_mult(A pp,B pq,s); P3 = mat_mult(A pq,B qq,s);

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Recursive Algorithm (cont) P4 = mat_mult(A qp,B pp,s); P5 = mat_mult(A qq,B qp,s); P6 = mat_mult(A qp,B pq,s); P7 = mat_mult(A qq,B qq,s); /* add submatrix products to form */ /* submatrices of final matrix */ C pp = P0 + P1; C pq = P2 + P3; C qp = P4 + P5; C qq = P6 + P7; } return(C);/* return final matrix */ }

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Two-dimensional array (mesh) interconnection network Also three-dimensional – used in some large high performance systems

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Mesh Implementations Cannon’s algorithm Fox’s algorithm Not in textbook but similar complexity Systolic array All involve using processors arranged in a mesh and shifting elements of the arrays through the mesh. Accumulate the partial sums at each processor.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Mesh Implementations Cannon’s Algorithm Uses a mesh of processors with wraparound connections (a torus) to shift the A elements (or submatrices) left and the B elements (or submatrices) up. 1.Initially processor P i,j has elements a i,j and b i,j (0  i,k<n). 2.Elements are moved from their initial position to an “aligned” position. The complete i th row of A is shifted i places left and the complete j th column of B is shifted j places upward. This has the effect of placing the element a i,j+i and the element b i+j,j in processor P i,j. These elements are a pair of those required in the accumulation of c i,j.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Mesh Implementations Cannon’s Algorithm (cont) 3.Each processor P i,j multiplies its elements. 4.The i th row of A is shifted one place left, and the j th column of B is shifted one place upward. This has the effect of bringing together the adjacent elements of A and B, which will also be required in the accumulation. 5.Each processor P i,j multiplies the elements brought to it and adds the result to the accumulating sum. 6.Steps 4 and 5 are repeated until the final result is obtained (n-1 shifts with n rows and n columns of elements).

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Movement of A and B elements

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Step 2 – Alignment of elements of A and B

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Step 4 – One-place shift of elements of A and B

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Cannon’s Algorithm Example Matrix elements from A in red, from B in blue

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Cannon’s Algorithm Example (cont) After realignment; accumulator values on right

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Cannon’s Algorithm Example (cont) After first shift

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Cannon’s Algorithm Example (cont) After second (final) shift

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Systolic array

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Pseudocode Each processor P i,j repeatedly performs the same algorithm (after c is initialized to zero): recv(&a, P i,j-1 );/* receive from left */ recv(&b, P i-1,j );/* receive from above */ c = c + a * b;/* accumulate value */ send(&a, P i,j+1 );/* send right */ send(&b, P i+1,j );/* send downwards */ which accumulates the required summations.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Matrix-Vector Multiplication

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Other Matrix Multiplication Methods Strassen’s method (1969) Suppose we wish to multiply n  n matrices A and B to produce C. Divide each of these into four n/2  n/2 submatrices as follows:

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Strassen’s Method (cont) A traditional recursive approach executes in O(n 3 ) time so no faster. Strassen discovered a different recursive approach that executes in O(n log 7 ) time: 1.Divide the input matrices into quarters as above. 2.Use O(n 2 ) scalar additions and subtractions to recursively compute seven matrix products Q 1 through Q 7. 3.Compute the desired submatrices of C by adding and/or subtracting various combinations of the Q i matrices, using O(n 2 ) scalar operations.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Strassen’s Method (cont) To this end set Q 1 = (A 11 + A 22 )·(B 11 + B 22 ) Q 2 = (A 21 + A 22 )·B 11 Q 3 = A 11 ·(B 12 – B 22 ) Q 4 = A 22 ·(B 21 – B 11 ) Q 5 = (A 11 + A 12 )·B 22 Q 6 = (A 21 – A 11 )·(B 21 + B 22 ) Q 7 = (A 12 – A 22 )·(B 21 + B 22 ) As indicated above can be done recursively.

Wilkinson & Allen, Parallel Programming: Techniques & Applications Using Networked Workstations & Parallel Computers, Strassen’s Method (cont) Now C 11 = Q 1 + Q 4 – Q 5 + Q 7 C 21 = Q 2 + Q 4 C 12 = Q 3 + Q 5 C 22 = Q 1 + Q 3 – Q 2 + Q 6