1. September 4, 1997
Parallel Processing (CS 676) Lecture 8: Grouping Data and Communicators in MPI
Jeremy R. Johnson
*Parts of this lecture was derived from chapters 6,7 in Pacheco
Parallel Processing

2. September 4, 1997
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
Parallel Processing

3. 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
{(t0,d0),…,(tn-1,dn-1)}
ti is an existing type
di is a displacement
Parallel Processing

4. 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)
Parallel Processing

5. 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)
Parallel Processing

6. Transpose float A[10][10]; /* stored in row-major order. */
/* Send 3rd 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. */
Parallel Processing

7. Transpose float A[10][10]; /* stored in row-major order. */
/* Send 3rd 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); }
Parallel Processing

8. 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);
Parallel Processing

9. 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 {(t0,d0),…,(tn-1,dn-1)}
Displacements do not matter
Type signatures {t0,…,tn-1} and {u0,…,um-1} must be compatible
n  m, ti = ui for i=0,…,n-1
For collective communications sending and receiving types must be identical
Parallel Processing

10. 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);
Parallel Processing

11. 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
Parallel Processing

12. 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
Parallel Processing

13. 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_FLOAT,buffer,HUGE,
MPI_Send(buffer,position,MPI_PACKED, 1,0,MPI_COMM_WORLD); }
Parallel Processing

14. 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,
MPI_UnPack(buffer,HUGE,&position, &column_subscripts, MPI_INT, }
Parallel Processing

15. 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
Parallel Processing

16. 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)
Parallel Processing

17. 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);
Parallel Processing

18. 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); }
Parallel Processing

19. 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. */
Parallel Processing

20. 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
Parallel Processing

21. 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)
Parallel Processing

22. 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);
Parallel Processing

23. 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)
Parallel Processing

24. 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)
Parallel Processing

25. 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]; }
Parallel Processing

26. 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 */
Parallel Processing

27. Cyclic Matrix Multiplication
/* Arrange processors in a circle, storing rows of A and B in each process. Ci.* = Ai,0 * B0,* + … + Ai,n-1 * Bn-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++) {
Ci,* = Ci,* + Ai,j*Bj,*
Clocal = Clocal + Alocal * Blocal;
send_recv(Blocal,dest,src); }
Parallel Processing

28. 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) block on processor (i,j)
Broadcast elements of A as k = 0,…,q-1
Cyclically rotate elements of B.
Parallel Processing

29. Example A00 B00 A00 B01 A00 B02 A11 B10 A11 B11 A11 B12 A22 B20
Parallel Processing

30. Fox’s Matrix Multiplication
/* Uses a block matrix allocation. Group processors in a q × q grid, where q = sqrt(p). Processor (i,j) stores Aij and initially Bij */
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; }
Parallel Processing

31. 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.
Parallel Processing

32. Example A00 B00 A01 B11 A02 B22 A11 B10 A12 B21 A10 B02 A22 B20
Parallel Processing

33. 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 Ai,i+j and initially Bi+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; }
Parallel Processing

34. 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
Parallel Processing

35. Analysis of Matrix Multiplication
Let A, B be n  n matrices, and C = A*B
Sequential cost
T(n)= an3+bn2+cn+d = (n3)
Least squares fit
T(n)  an3
Parallel Processing

36. Analysis of Parallel Matrix Multiplication (Allgather)
Let A, B be n  n matrices, and C = A*B
Let p = number of processors
Store ith block of n/p rows of A, B, and C on process i
Parallel computing time: (n3/p + plog(p) + n2log(p))
Computation time: p(n/p  n  n/p) = n3/p
Communication time: p(log(p)(+n2/p) [Allgather]
Parallel Processing

37. Analysis of Parallel Matrix Multiplication (Cyclic)
Let A, B be n  n matrices, and C = A*B
Let p = number of processors
Store ith block of n/p rows of A, B, and C on process i
Parallel computing time: (n3/p + p + n2)
Computation time: p(n/p  n/p  n ) = n3/p
Communication time: p(+n2/p)
Parallel Processing

38. Analysis of Parallel Matrix Multiplication (Fox)
Let A, B be n  n matrices, and C = A*B
Let p = q2 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:
(n3/p + qlog(q) + log(q)n2/q)
Computation time: q(n/q  n/q  n/q ) = n3/q2 = n3/p
Communication time: qlog(q)(+(n/q)2)
Parallel Processing

39. Analysis of Parallel Matrix Multiplication (Fox Variant)
Let A, B be n  n matrices, and C = A*B
Let p = q2 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:
(n3/p + q + n2/q)
Computation time: q(n/q  n/q  n/q ) = n3/q2 = n3/p
Communication time: q(+(n/q)2)
Parallel Processing