1. Parallel Processing1 Parallel Processing (CS 676) Lecture: Grouping Data and Communicators in MPI Jeremy R. Johnson *Parts of this lecture was derived from chapters 6,7 in Pacheco

2. Parallel Processing2 Introduction Objective: To introduce MPI commands for creating types and communicators. To discuss performance models and considerations in MPI and ways of reducing communication. Topics –MPI Datatypes and packing Revised version of Get_Data Matrix transposition –Creating communicators Topologies Grids –Matrix Multiplication Fox’s algorithm –Performance Model

3. Parallel Processing3 Derived Types Due to latency of communication it is usually a good idea to package up several elements into a single message MPI_Send and MPI_Recv, allow message to be given by a start address, basic type, and a count. –This allows multiple data elements to be sent in one message –Requires elements to be of the same type –Must be contiguous A generalized type –{(t 0,d 0 ),…,(t n-1,d n-1 )} –t i is an existing type –d i is a displacement

4. Parallel Processing4 Functions for Creating MPI Types int MPI_Type_struct(int count, int block_lengths[], MPI_Aint displacements[], MPI_Datatype typelist[], MPI_Datatype new_mpi_t) MPI_Address( void* location, MPI_Aint* address) int MPI_Type_commit(MPI_Datatype* new_mpi_t)

5. Parallel Processing5 Other Derived Datatype Constructors int MPI_Type_vector(int count, int block_length, int stride, MPI_Datatype element_type, MPI_Datatype new_mpi_t) int MPI_Type_contiguous(int count, MPI_Datatype old_type, MPI_Datatype new_mpi_t) int MPI_Type_indexed(int count, int block_lengths[], int displacements[], MPI_Datatype old_type, MPI_Datatype new_mpi_t)

6. Parallel Processing6 Transpose float A[10][10]; /* stored in row-major order. */ /* Send 3 rd row of A from process 0 to process 1. */ If (my_rank == 0) { MPI_Send(&(A[2][0]), 10, MPI_FLOAT, 1, 0, MPI_COMM_WORLD); } else { MPI_Recv(&(A[2][0]), 10, MPI_FLOAT, 0, 0, MPI_COMM_WORLD, &status); } /* Doesn’t work for columns, since not contiguous. */

7. Parallel Processing7 Transpose float A[10][10]; /* stored in row-major order. */ /* Send 3 rd column of A from process 0 to process 1. */ MPI_Type_vector(10, 1, 10, MPI_FLOAT, &column_mpi_t); MPI_Type_commit(&column_mpi_t); If (my_rank == 0) { MPI_Send(&(A[0][2]), 1, column_mpi_t, 1, 0, MPI_COMM_WORLD); } else { MPI_Recv(&(A[0][2]), 1, column_mpi_t, 0, 0, MPI_COMM_WORLD, &status); }

8. Parallel Processing8 Upper Triangular Matrix float A[n][n]; /* Complete matrix */ float T[n][n]; /* Upper triangle. */ int displacements[n]; int block_lengths[n]; MPI_Datatype index_mpi_t; for(i = 0; i < n; i++) { block_lengths[i] = n-i; displacements[i] = (n+1)*i; } MPI_Type_indexed(n, block_lengths, displacments, MPI_FLOAT, &index_mpi_t); MPI_Type_commit(&index_mpi_t); if (my_rank == 0) { MPI_Send(A, 1, index_mpi_t, 1, 0, MPI_COMM_WORLD); else MPI_Recv(T, 1, index_mpi_t, 0, 0, MPI_COMM_WORLD, &status);

9. Parallel Processing9 Type Matching When can a receiving process match the data sent by a sending process? –MPI_Send(message, send_count, send_mpi_t, 1, 0, MPI_COMM_WORLD) –MPI_Recv(message, recv_count, recv_mpi_t, 0, 0,MPI_COMM_WORLD,&status) Given a derived type {(t 0,d 0 ),…,(t n-1,d n-1 )} –Displacements do not matter –Type signatures {t 0,…,t n-1 } and {u 0,…,u m-1 } must be compatible –n  m, t i = u i for i=0,…,n-1 –For collective communications sending and receiving types must be identical

10. Parallel Processing10 Type Matching Example For type column_mpi_t (column of 10  10 array of floats) –{(MPI_FLOAT,0), (MPI_FLOAT,10*sizeof(float)), – (MPI_FLOAT,20*sizeof(float)),…,(MPI_FLOAT,90*sizeof(float))} –Signature is {MPI_FLOAT,…,MPI_FLOAT}, MPI_FLOAT 10 times –Example: Can send column to row float A[10][10]; /* stored in row-major order. */ If (my_rank == 0) MPI_Send(&(A[0][0]), 1, column_mpi_t, 1, 0, MPI_COMM_WORLD); else if (my_rank == 0) MPI_Recv(&(A[0][0]),10, MPI_FLOAT, 0, 0, MPI_COMM_WORLD, &status);

11. Parallel Processing11 Pack and Unpack MPI_Pack and MPI_Unpack allow a user to copy non-contiguous memory locations into a contiguous buffer and to copy a contiguous buffer into non-contiguous memory locations int MPI_Pack(void* pack_data, int in_count, MPI_Datatype datatype, void* buffer, int buffer_size, int* position, MPI_Comm comm) –On input copy data starting at location &buffer + position –On output position points to first location in the buffer after pack_data int MPI_Unpack(void* buffer, int size, int* position,void* unpack_data, int count,MPI_Datatype datatype, MPI_Comm comm) –Data starting at location &buffer + *position is copied into memory referenced by unpack_data –count data elements of type datatype are copied into unpack_data –position is updated to point to location in buffer after the data just copied Messages constructed with MPI_Pack should be communicated with datatype argument MPI_PACKED

12. Parallel Processing12 Deciding which Method to Use Creating a derived type has overhead associated with it. Depends on number of times type will be used. Can avoid system buffering with Pack/Unpack Can use variable length messages with Pack/Unpack

13. Parallel Processing13 Variable Length Messages float* entries; int* column_subscripts; int nonzeros; int position; int row_number; char buffer[HUGE]; if (my_rank == 0) { position = 0; MPI_Pack(&nonzeros,1,MPI_INT,buffer,HUGE,&position,MPI_COMM_WORLD); MPI_Pack(row_number,1,MPI_INT,buffer,HUGE,&position,MPI_COMM_WORLD); MPI_Pack(entries,nonzeros,MPI_FLOAT,buffer,HUGE, &position,MPI_COMM_WORLD); MPI_Pack(column_subscripts,nonzeros,MPI_INT,buffer,HUGE, &position,MPI_COMM_WORLD); MPI_Send(buffer,position,MPI_PACKED, 1,0,MPI_COMM_WORLD); }

14. Parallel Processing14 Variable Length Messages (cont) else { MPI_Recv(buffer,HUGE,MPI_PACKED, 0,0, MPI_COMM_WORLD, &status); position = 0; MPI_UnPack(buffer,HUGE,&position, &nonzeros, MPI_INT,MPI_COMM_WORLD); MPI_UnPack(buffer,HUGE,&position, &row_number, MPI_INT, MPI_COMM_WORLD); entries = (float *) malloc(nonzeros*sizeof(float)); column_subscripts = (int *) malloc(nonzeros*sizeof(int)); MPI_UnPack(buffer,HUGE,&position, entries, MPI_FLOAT,nonzeros, MPI_COMM_WORLD); MPI_UnPack(buffer,HUGE,&position, &column_subscripts, MPI_INT, nonzeros, MPI_COMM_WORLD); }

15. Parallel Processing15 Communicators A mechanism to treat a subset of processes as a universe for communication (both point-to-point and collective) Types –intra-communicator –inter-communicator Components –group (ordered collection of processes) –context (unique identifier) –optional additional information such as topology Create new communicators from existing communicators

16. Parallel Processing16 Working with Groups, Contexts, and Communicators MPI_Comm_group(MPI_Comm comm, MPI_Group* group) MPI_Group_incl(MPI_Group old_group, int new_group_size, int ranks_in_old_group[]; MPI_Group* new_group) MPI_Comm_create(MPI_Comm old_comm, MPI_Group group, MPI_Comm* new_com)

17. Parallel Processing17 Creating a Communicator /* Create communicator out of first row of q^2 processes organized in a q  q grid in row-major order. */ MPI_Group group_world; MPI_Group first_row_group; MPI_Comm first_row_comm; int* process_ranks; process_ranks = (int*) malloc(q*sizeof(int)); for (proc = 0; proc < q; proc++) process_ranks[proc] = proc; MPI_Comm_Group(MPI_COMM_WORLD,&group_world); MPI_Group_incl(group_world,q,process_ranks,&first_row_group); MPI_Comm_create(MPI_COMM_WORLD,first_row_group,&first_row_comm);

18. Parallel Processing18 Using a Communicator /* Broadcast first block to all processes in the same row. */ int my_rank_in_first_row; float* A00; if (my_rank < q) { MPI_Comm_rank(first_row_comm,&my_rank_in_first_row); A_00 = (float *) malloc(n_bar*n_bar*sizeof(float)); if (my_rank_in_first_row == 0) { /* initialize A_00 */} MPI_Bcast(A_00,n_bar*n_bar,MPI_FLOAT,0,first_row_comm); }

19. Parallel Processing19 MPI_Comm_split int MPI_Comm_split(MPI_Comm old_comm, int split_key, int rank_key, MPI_Comm new_comm) MPI_Comm my_row_comm; int my_row; /* my_rank is in MPI_COMM_WORLD, q*q = p */ my_row = my_rank/q; MPI_Comm_split(MPI_COMM_WORLD,my_row,my_rank,&my_row_comm); /* Creates q new communicators. Processes with the same value of split_key form a new group. The rank in the new communicator is determined by rank_key. Order is preserved. If the same rank_key is used, then the choice is arbitrary. */

20. Parallel Processing20 Topologies Communicators can have attributes. One such attribute is a topology. A topology is a mechanism for associating a different addressing scheme with processes belonging to a group. Provides a virtual interconnection organization of processes that may be convenient for a particular algorithm. Types –Cartesian (grid) –Graph

21. Parallel Processing21 Working with Cartesian Topologies MPI_Cart_create(MPI_Comm old_comm, int number_of_dims, int dim_sizes[], int wrap_around[], int reorder, MPI_Comm* cart_comm) MPI_Cart_rank(MPI_Comm comm, int rank, int number_of_dims, int* rank) MPI_Cart_coords(MPI_Comm comm, int rank, int number_of_dims, int coordinates[]) MPI_Cart_sub(MPI_Comm cart_comm, int free_coords[], MPI_Comm* new_comm)

22. Parallel Processing22 Creating a Cartesian Topology /* create communicator with 2D grid topology. */ MPI_Comm grid_comm; int dim_sizes[2]; int wrap_around[2]; int reorder = 1; dim_sizes[0] = dim_sizes[1] = q; wrap_around[0] = wrap_around[1] = 1; MPI_Cart_create(MPI_COMM_WORLD, 2, dim_sizes,wrap_around,reorder, &grid_comm);

23. Parallel Processing23 Creating a Sub-Cartesian Topology int free_coords[2]; MPI_Comm row_comm; /* create communicator for each row of grid_comm */ free_coords[0] = 0; free_coords[1] = 1; MPI_Cart_sub(grid_comm, free_coords, &row_comm) /* create communicator for each column of grid_comm */ free_coords[0] = 1; free_coords[1] = 0; MPI_Cart_sub(grid_comm, free_coords, &col_comm)

24. Parallel Processing24 Cartesian Addressing int coordinates[2]; int my_grid_rank; MPI_Comm_rank(grid_comm, &my_grid_rank); MPI_Cart_coords(grid_comm, my_grid_rank,2, coordinates); /* inverse operation */ MPI_Cart_rank(grid_comm, coordinates, &my_grid_rank)

25. Parallel Processing25 Matrix Multiplication Let A, B be n  n matrices, and C = A*B void Serial_matrix_mult(MATRIX_T A, MATRIX_T B, MATRIX_T C, int n) { int i,j,k; for (i=0; i< n; i++) for (j=0; j< n;j++) { C[i][j] = 0.0; for (k=0; k < n;k++) C[i][j] = C[i][j] + A[i][k]*B[k][j]; }

26. Parallel Processing26 Parallel Matrix Multiplication /* distribute matrices by rows. */ void Parallel_matrix_mult(MATRIX_T A, MATRIX_T B, MATRIX_T C, int n) { for each column of B { Allgather(column); Compute dot product of my row of A with column; } /* can distribute matrices by blocks of rows. Also B could be distributed by * columns */

27. Parallel Processing27 Cyclic Matrix Multiplication /* Arrange processors in a circle, storing rows of A and B in each process. C i.* = A i,0 * B 0,* + … + A i,n-1 * B n-1,* */ void Parallel_matrix_mult(MATRIX_T A, MATRIX_T B, MATRIX_T C, int n) { i = rank; Blocal = ith row of B; Alocal = ith row of A; Clocal = 0; /* ith row of C */ dest = (i+1) % n; src = (i-1) % n; for (k=0;k<n;k++) { C i,* = C i,* + A i,i+kmod n * B i+k mod n,* send_recv(Blocal,dest,src); }

28. Parallel Processing28 Fox’s Matrix Multiplication Let A, B be p  p matrices, and C = A*B Organize processors into a q  q grid, q = sqrt(p) Store (i,j) block on processor (i,j) Broadcast elements of A as k = 0,…,q-1 Cyclically rotate elements of B.

29. Parallel Processing29 Example A00 B00 A00 B01 A00 B02 A11 B10 A11 B11 A11 B12 A22 B20 A22 B21 A22 B22 A01 B10 A01 B11 A01 B12 A12 B20 A12 B21 A12 B22 A20 B00 A20 B01 A20 B02 A02 B20 A02 B21 A02 B22 A10 B00 A10 B01 A10 B02 A21 B10 A21 B11 A21 B12

30. Parallel Processing30 Fox’s Matrix Multiplication /* Uses a block matrix allocation. Group processors in a q × q grid, where q = sqrt(p). Processor (i,j) stores A ij and initially B ij */ void Parallel_matrix_mult(MATRIX_T A, MATRIX_T B, MATRIX_T C, int n) { i = my process row; j = my process column; dest = ((i-1) % q,j); src = ((i+1) % q,j); for (stage=0;stage < q; stage++) { k_bar = (i + stage) mod q; Broadcast A[i,k_bar] across process row i; C[i,j] = C[i,j] + A[i,k_bar]*B[k_bar,j]; Send B[k_bar,j] to dest; Receive B[(k_bar+1) mod q,j] from source; }

31. Parallel Processing31 Variant of Fox’s Matrix Multiplication Let A, B be q  q matrices, and C = A*B Organize processors into a sqrt(p)  sqrt(p) grid Store (i,j+i mod q) block of A and (i+j mod q,j) block of B on processor (i,j) Cyclically rotate rows of A to the left. Cyclically rotate columns of B upward.

32. Parallel Processing32 Example A00 B00 A01 B11 A02 B22 A11 B10 A12 B21 A10 B02 A22 B20 A20 B01 A21 B12 A01 B10 A02 B21 A00 B02 A12 B20 A10 B01 A11 B12 A20 B00 A21 B11 A22 B22 A02 B20 A00 B01 A01 B12 A10 B00 A11 B11 A12 B22 A21 B10 A22 B21 A20 B02

33. Parallel Processing33 Variant of Fox’s Matrix Multiplication /* Uses a block matrix allocation. Group processors in a q × q grid, where q = sqrt(p). Processor (i,j) stores A i,i+j and initially B i+i,j */ void Parallel_matrix_mult(MATRIX_T A, MATRIX_T B, MATRIX_T C, int n) { i = my process row; j = my process column; coldest = ((i-1) % q,j); colsrc = ((i+1) % q,j); rowdest = (i,(j-1) % q); rowsrc = (i,(j+1) % q); for (stage=0;stage < q; stage++) { k_bar = (i +j + stage) mod q; C[i,j] = C[i,j] + A[i,k_bar]*B[k_bar,j]; Send_Recv A[i,k_bar] to/from rowdest,rowsrc; Send_Recv B[k_bar,j] to/from coldest, colsrc; }

34. Parallel Processing34 Performance Model Communication cost: C(n) =  +  n –  = latency –1/  = bandwidth Empirically determine  and  by measuring time to send/recv messages with different lengths –Least squares fit

35. Parallel Processing35 Analysis of Matrix Multiplication Let A, B be n  n matrices, and C = A*B Sequential cost –T(n)= an 3 +bn 2 +cn+d =  (n 3 ) –Least squares fit –T(n)  an 3

36. Parallel Processing36 Analysis of Parallel Matrix Multiplication (Allgather) Let A, B be n  n matrices, and C = A*B Let p = number of processors Store i th block of n/p rows of A, B, and C on process i Parallel computing time:  (n 3 /p + plog(p) + n 2 log(p)) Computation time: p(n/p  n  n/p) = n 3 /p Communication time: p(log(p)(  +  n 2 /p) [Allgather]

37. Parallel Processing37 Analysis of Parallel Matrix Multiplication (Cyclic) Let A, B be n  n matrices, and C = A*B Let p = number of processors Store i th block of n/p rows of A, B, and C on process i Parallel computing time:  (n 3 /p + p + n 2 ) Computation time: p(n/p  n/p  n ) = n 3 /p Communication time: p(  +  n 2 /p)

38. Parallel Processing38 Analysis of Parallel Matrix Multiplication (Fox) Let A, B be n  n matrices, and C = A*B Let p = q 2 number of processors organized in a q  q grid Store (i,j) th n/q  n/q block of A, B, and C on process (i,j) Parallel computing time: –  (n 3 /p + qlog(q) + log(q)n 2 /q) Computation time: q(n/q  n/q  n/q ) = n 3 /q 2 = n 3 /p Communication time: qlog(q)(  +  (n/q) 2 )

39. Parallel Processing39 Analysis of Parallel Matrix Multiplication (Fox Variant) Let A, B be n  n matrices, and C = A*B Let p = q 2 number of processors organized in a q  q grid Store (i,j) th n/q  n/q block of A, B, and C on process (i,j) Parallel computing time: –  (n 3 /p + q + n 2 /q) Computation time: q(n/q  n/q  n/q ) = n 3 /q 2 = n 3 /p Communication time: q(  +  (n/q) 2 )