# MPI topic: Collectives

$\newcommand\inv{^{-1}}\newcommand\invt{^{-t}} \newcommand\bbP{\mathbb{P}} \newcommand\bbR{\mathbb{R}} \newcommand\defined{ \mathrel{\lower 5pt \hbox{{\equiv\atop\mathrm{\scriptstyle D}}}}}$ 3.1 : Working with global information
3.1.1 : Practical use of collectives
3.1.2 : Synchronization
3.1.3 : Collectives in MPI
3.2 : Reduction
3.2.1 : Reduce to all
3.2.2 : Reduction of distributed data
3.2.3 : Reduce in place
3.2.4 : Reduction operations
3.3 : Rooted collectives: broadcast, reduce
3.3.1 : Reduce to a root
3.3.2 : Reduce in place
3.4 : Rooted collectives: gather and scatter
3.4.1 : Example
3.4.2 : Allgather
3.5 : All-to-all
3.5.1 : All-to-all as data transpose
3.5.2 : All-to-all-v
3.6 : Reduce-scatter
3.6.1 : Examples
3.7 : Barrier
3.8 : Variable-size-input collectives
3.8.1 : Example of Gatherv
3.8.2 : Example of Allgatherv
3.8.3 : Variable all-to-all
3.9 : Scan operations
3.9.1 : Exclusive scan
3.9.2 : Use of scan operations
3.10 : MPI Operators
3.10.1 : Pre-defined operators
3.10.2 : User-defined operators
3.10.3 : Local reduction
3.11 : Non-blocking collectives
3.11.1 : Non-blocking barrier
3.12 : Performance of collectives
3.13 : Collectives and synchronization
3.14 : Implementation and performance of collectives

# 3.1 Working with global information

Top > Working with global information

If all processes have individual data, for instance the result of a local computation, you may want to bring that information together, for instance to find the maximal computed value or the sum of all values. Conversely, sometimes one processor has information that needs to be shared with all. For this sort of operation, MPI has collectives .

There are various cases, illustrated in figure  1 ,

which you can (sort of) motivated by considering some classroom activities:

• The teacher tells the class when the exam will be. This is a broadcast : the same item of information goes to everyone.

• After the exam, the teacher performs a gather operation to collect the invidivual exams.

• On the other hand, when the teacher computes the average grade, each student has an individual number, but these are now combined to compute a single number. This is a reduction .

• Now the teacher has a list of grades and gives each student their grade. This is a scatter operation, where one process has multiple data items, and gives a different one to all the other processes.

This story is a little different from what happens with MPI processes, because these are more symmetric; the process doing the reducing and broadcasting is no different from the others. Any process can function as the root process in such a collective.

Exercise

How would you realize the following scenarios with MPI collectives?

• Let each process compute a random number. You want to print the maximum of these numbers to your screen.

• Each process computes a random number again. Now you want to scale these numbers by their maximum.

• Let each process compute a random number. You want to print on what processor the maximum value is computed.

## 3.1.1 Practical use of collectives

Top > Working with global information > Practical use of collectives

Collectives are quite common in scientific applications. For instance, if one process reads data from disc or the commandline, it can use a broadcast or a gather to get the information to other processes. Likewise, at the end of a program run, a gather or reduction can be used to collect summary information about the program run.

However, a more common scenario is that the result of a collective is needed on all processes.

Consider the computation of the standard deviation : $$\sigma = \sqrt{\frac1N \sum_i^N (x_i-\mu) } \qquad\hbox{where}\qquad \mu = \frac{\sum_i^Nx_i}N$$ and assume that every processor stores just one $x_i$ value.

1. The calculation of the average $\mu$ is a reduction, since all the distributed values need to be added.

2. Now every process needs to compute $x_i-\mu$ for its value $x_i$, so $\mu$ is needed everywhere. You can compute this by doing a reduction followed by a broadcast, but it is better to use a so-called allreduce operation, which does the reduction and leaves the result on all processors.

3. The calculation of $\sum_i(x_i-\mu)$ is another sum of distributed data, so we need another reduction operation. Depending on whether each process needs to know $\sigma$, we can again use an allreduce.

For instance, if $x,y$ are distributed vector objects, and you want to compute $$y- (x^ty)x$$ which is part of the Gramm-Schmidt algorithm; see  Eijkhout:IntroHPC . Now you need the inner product value on all processors. You could do this by writing a reduction followed by a broadcast:

// compute local value
localvalue = innerproduct( x[ localpart], y[ localpart ] );
// compute inner product on the root
Reduce( localvalue, reducedvalue, root );
// send root value to all other from the root

or combine the last two steps in an allreduce, Surprisingly, an allreduce operation takes as long as a rooted reduction (see  Eijkhout:IntroHPC for details), and therefore half the time of a reduction followed by a broadcast.

## 3.1.2 Synchronization

Top > Working with global information > Synchronization

Collectives are operations that involve all processes in a communicator. A collective is a single call, and it blocks on all processors. That does not mean that all processors exit the call at the same time: because of implementational details and network latency they need not be synchronized in their execution. However, semantically we can say that a process can not finish a collective until every other process has at least started the collective. In addition to these collective operations, there are operations that are said to be collective on their communicator', but which do not involve data movement. Collective then means that all processors must call this routine; not to do so is an error that will manifest itself in hanging' code. One such example is MPI_Win_fence .

## 3.1.3 Collectives in MPI

Top > Working with global information > Collectives in MPI

We will now explain the MPI collectives in the following order.

• [Allreduce] We use the allreduce as an introduction to the concepts behind collectives; section  . As explained above, this routines servers many practical scenarios.

• [Broadcast and reduce] We then introduce the concept of a root in the reduce (section  ) and broadcast (section  ) collectives.

• [Gather and scatter] The gather/scatter collectives deal with more than a single data item.

There are more collectives or variants on the above.

• If you want to gather or scatter information, but the contribution of each processor is of a different size, there are variable' collectives; they have a  v in the name (section  ).

• Sometimes you want a reduction with partial results, where each processor computes the sum (or other operation) on the values of lower-numbered processors. For this, you use a scan collective (section  ).

• If every processor needs to broadcast to every other, you use an all-to-all operation (section  ).

• A barrier is an operation that makes all processes wait until every process has reached the barrier (section  ).

Finally, there are some advanced topics in collectives.

• Non-blocking collectives; section  .

• User-defined reduction operators; section  .

# 3.2 Reduction

Top > Reduction

## 3.2.1 Reduce to all

Top > Reduction > Reduce to all

Above we saw a couple of scenarios where a quantity is reduced, with all proceses getting the result. The MPI call for this is:

C:
int MPI_Allreduce(const void* sendbuf,
void* recvbuf, int count, MPI_Datatype datatype,
MPI_Op op, MPI_Comm comm)

Semantics:
IN sendbuf: starting address of send buffer (choice)
IN count: number of elements in send buffer (non-negative integer)
IN datatype: data type of elements of send buffer (handle)
IN op: operation (handle)
IN comm: communicator (handle)

Fortran:
MPI_Allreduce(sendbuf, recvbuf, count, datatype, op, comm, ierror)
TYPE(*), DIMENSION(..), INTENT(IN) :: sendbuf
TYPE(*), DIMENSION(..) :: recvbuf
INTEGER, INTENT(IN) :: count
TYPE(MPI_Datatype), INTENT(IN) :: datatype
TYPE(MPI_Op), INTENT(IN) :: op
TYPE(MPI_Comm), INTENT(IN) :: comm
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

Python native:
recvobj = MPI.Comm.allreduce(self, sendobj, op=SUM)
Python numpy:
MPI.Comm.Allreduce(self, sendbuf, recvbuf, Op op=SUM)


Example: we give each process a random number, and sum these numbers together. The result should be approximately $1/2$ times the number of processes.


// allreduce.c
float myrandom,sumrandom;
myrandom = (float) rand()/(float)RAND_MAX;
// add the random variables together
MPI_Allreduce(&myrandom,&sumrandom,
1,MPI_FLOAT,MPI_SUM,comm);
// the result should be approx nprocs/2:
if (procno==nprocs-1)
printf("Result %6.9f compared to .5\n",sumrandom/nprocs);


For Python we illustrate both the native and the numpy variant. In the numpy variant we create an array for the receive buffer, even though only one element is used.


## allreduce.py
random_number = random.randint(1,nprocs*nprocs)
print "[%d] random=%d" % (procid,random_number)

max_random = comm.allreduce(random_number,op=MPI.MAX)
if procid==0:
print "Python native:\n  max=%d" % max_random

myrandom = np.empty(1,dtype=np.int)
myrandom[0] = random_number
allrandom = np.empty(nprocs,dtype=np.int)

comm.Allreduce(myrandom,allrandom[:1],op=MPI.MAX)


Exercise

Let each process compute a random number, and compute the sum of these numbers using the MPI_Allreduce routine.

(The operator is MPI_SUM for C/Fortran, or MPI.SUM for Python.)

Each process then scales its value by this sum. Compute the sum of the scaled numbers and check that it is 1.

## 3.2.2 Reduction of distributed data

Top > Reduction > Reduction of distributed data

One of the more common applications of the reduction operation is the inner product computation. Typically, you have two vectors $x,y$ that have the same distribution, that is, where all processes store equal parts of $x$ and $y$. The computation is then

local_inprod = 0;
for (i=0; i<localsize; i++)
local_inprod += x[i]*y[i];
MPI_Allreduce( &local_inprod, &global_inprod, 1,MPI_DOUBLE ... )


## 3.2.3 Reduce in place

Top > Reduction > Reduce in place

By default MPI will not overwrite the original data with the reduction result, but you can tell it to do so using the MPI_IN_PLACE specifier:


// allreduceinplace.c
int nrandoms = 500000;
float *myrandoms;
myrandoms = (float*) malloc(nrandoms*sizeof(float));
for (int irand=0; irand
 This has the advantage of saving half the memory. 3.2.4 Reduction operations Top > Reduction > Reduction operations The following is the list of predefined MPI_OP values. MPI typemeaningapplies to \indexmpidef{MPI_MAX} maximuminteger, floating point \indexmpidef{MPI_MIN} minimum \indexmpidef{MPI_SUM} suminteger, floating point, complex, multilanguage types \indexmpidef{MPI_PROD} product \indexmpidef{MPI_LAND} logical andC integer, logical \indexmpidef{MPI_LOR} logical or \indexmpidef{MPI_LXOR} logical xor \indexmpidef{MPI_BAND} bitwise andinteger, byte, multilanguage types \indexmpidef{MPI_BOR} bitwise or \indexmpidef{MPI_BXOR} bitwise xor \indexmpidef{MPI_MAXLOC} max value and location \indexmpishow{MPI_DOUBLE_INT} and such \indexmpidef{MPI_MINLOC} min value and location {|lll|} The MPI_MAXLOC operation yields both the maximum and the rank on which it occurs. However, to use it the input should be an array of real/int structs, where the int is the rank of the number. For use in reductions and scans it is possible to define your own operator. MPI_Op_create( MPI_User_function *func, int commute, MPI_Op *op); 3.3 Rooted collectives: broadcast, reduce Top > Rooted collectives: broadcast, reduce In some scenarios there is a certain process that has a privileged status. One process can generate or read in the initial data for a program run. This then needs to be communicated to all other processes. At the end of a program run, often one process needs to output some summary information. This process is called the root process, and we will now consider routines that have a root. 3.3.1 Reduce to a root Top > Rooted collectives: broadcast, reduce > Reduce to a root In the broadcast operation a single data item was communicated to all processes. Reduction operations go the other way: each process has a data item, and these are all brought together into a single item. Here are the essential elements of a reduction operation: MPI_Reduce( senddata, recvdata..., operator, root, comm ); There is the original data, and the data resulting from the reduction. It is a design decision of MPI that it will not by default overwrite the original data. The send data and receive data are of the same size and type: if every processor has one real number, the reduced result is again one real number. There is a reduction operator. Popular choices are MPI_SUM , MPI_PROD and MPI_MAX , but complicated operators such as finding the location of the maximum value exist. You can also define your own operators; section  . There is a root process that receives the result of the reduction. Since the non-root processes do not receive the reduced data, they can actually leave the receive buffer undefined. // reduce.c float myrandom = (float) rand()/(float)RAND_MAX, result; int target_proc = nprocs-1; // add all the random variables together MPI_Reduce(&myrandom,&result,1,MPI_FLOAT,MPI_SUM, target_proc,comm); // the result should be approx nprocs/2: if (procno==target_proc) printf("Result %6.3f compared to nprocs/2=%5.2f\n", result,nprocs/2.); C: int MPI_Reduce( const void* sendbuf, void* recvbuf, int count, MPI_Datatype datatype, MPI_Op op, int root, MPI_Comm comm) Fortran: MPI_Reduce(sendbuf, recvbuf, count, datatype, op, root, comm, ierror) TYPE(*), DIMENSION(..), INTENT(IN) :: sendbuf TYPE(*), DIMENSION(..) :: recvbuf INTEGER, INTENT(IN) :: count, root TYPE(MPI_Datatype), INTENT(IN) :: datatype TYPE(MPI_Op), INTENT(IN) :: op TYPE(MPI_Comm), INTENT(IN) :: comm INTEGER, OPTIONAL, INTENT(OUT) :: ierror Python: native: comm.reduce(self, sendobj=None, recvobj=None, op=SUM, int root=0) numpy: comm.Reduce(self, sendbuf, recvbuf, Op op=SUM, int root=0) Exercise Write a program where each process computes a random number, and process 0 finds and prints the maximum generated value. Let each process print its value, just to check the correctness of your program. (See  for a discussion of random number generation.) Collective operations can also take an array argument, instead of just a scalar. In that case, the operation is applied pointwise to each location in the array. Exercise Create on each process an array of length 2 integers, and put the values $1,2$ in it on each process. Do a sum reduction on that array. Can you predict what the result should be? Code it. Was your prediction right? 3.3.2 Reduce in place Top > Rooted collectives: broadcast, reduce > Reduce in place Instead of using a send and a receive buffer in the reduction, it is possible to avoid the send buffer by putting the send data in the receive buffer. We see this mechanism in section  for the allreduce operation. For the rooted call MPI_Reduce , it is similarly possible to use the value in the receive buffer on the root. However, on all other processes, data is placed in the send buffer and the receive buffer is null or ignored as before. This example sets the buffer values through some pointer cleverness in order to have the same reduce call on all processes. // reduceinplace.c float mynumber,result,*sendbuf,*recvbuf; mynumber = (float) procno; int target_proc = nprocs-1; // add all the random variables together if (procno==target_proc) { sendbuf = (float*)MPI_IN_PLACE; recvbuf = &result; result = mynumber; } else { sendbuf = &mynumber; recvbuf = NULL; } MPI_Reduce(sendbuf,recvbuf,1,MPI_FLOAT,MPI_SUM, target_proc,comm); // the result should be nprocs*(nprocs-1)/2: if (procno==target_proc) printf("Result %6.3f compared to n(n-1)/2=%5.2f\n", result,nprocs*(nprocs-1)/2.); In Fortran the code is less elegant because you can not do these address calculations: // reduceinplace.F90 call random_number(mynumber) target_proc = ntids-1; ! add all the random variables together if (mytid.eq.target_proc) then result = mytid call MPI_Reduce(MPI_IN_PLACE,result,1,MPI_REAL,MPI_SUM,& target_proc,comm,err) else mynumber = mytid call MPI_Reduce(mynumber,result,1,MPI_REAL,MPI_SUM,& target_proc,comm,err) end if ! the result should be ntids*(ntids-1)/2: if (mytid.eq.target_proc) then write(*,'("Result ",f5.2," compared to n(n-1)/2=",f5.2)') & result,ntids*(ntids-1)/2. end if 3.3.3 Broadcast Top > Rooted collectives: broadcast, reduce > Broadcast The broadcast call has the following structure: MPI_Bcast( data..., root , comm); The root is the process that is sending its data. Typically, it will be the root of a broadcast tree. The comm argument is a communicator: for now you can use MPI_COMM_WORLD . Unlike with send/receive there is no message tag, because collectives are blocking, so you can have only one collective active at a time. The data in a broadcast (or any other MPI operation for that matter) is specified as A buffer. In C this is the address in memory of the data. This means that you broadcast a single scalar as MPI_Bcast ( &value, ... ) , but an array as MPI_Bcast ( array, ... ) . The number of items and their datatype. The allowable datatypes are such things as MPI_INT and MPI_FLOAT for C, and MPI_INTEGER and MPI_REAL for Fortran, or more complicated types. See section  for details. Python note In python it is both possible to send objects, and to send more C-like buffers. The two possibilities correspond (see section  ) to different routine names; the buffers have to be created as numpy objects. Example: in general we can not assume that all processes get the commandline arguments, so we broadcast them from process 0. // init.c if (procno==0) { if ( argc==1 || // the program is called without parameter ( argc>1 && !strcmp(argv[1],"-h") ) // user asked for help ) { printf("\nUsage: init [0-9]+\n"); MPI_Abort(comm,1); } input_argument = atoi(argv[1]); } MPI_Bcast(&input_argument,1,MPI_INT,0,comm); C: int MPI_Bcast( void* buffer, int count, MPI_Datatype datatype, int root, MPI_Comm comm) Fortran: MPI_Bcast(buffer, count, datatype, root, comm, ierror) TYPE(*), DIMENSION(..) :: buffer INTEGER, INTENT(IN) :: count, root TYPE(MPI_Datatype), INTENT(IN) :: datatype TYPE(MPI_Comm), INTENT(IN) :: comm INTEGER, OPTIONAL, INTENT(OUT) :: ierror Python native: rbuf = MPI.Comm.bcast(self, obj=None, int root=0) Python numpy: MPI.Comm.Bcast(self, buf, int root=0) Exercise If you give a commandline argument to a program, that argument is available as a character string as part of the argv,argc pair that you typically use as the arguments to your main program. You can use the function atoi to convert such a string to integer. Write a program where process 0 looks for an integer on the commandline, and broadcasts it to the other processes. Initialize the buffer on all processes, and let all processes print out the broadcast number, just to check that you solved the problem correctly. In python we illustrate the native and numpy variants. In the native variant the result is given as a function return; in the numpy variant the send buffer is reused. ## bcast.py # first native if procid==root: buffer = [ 5.0 ] * dsize buffer = comm.bcast(obj=buffer,root=root) if not reduce( lambda x,y:x and y, [ buffer[i]==5.0 for i in range(len(buffer)) ] ): print "Something wrong on proc %d: native buffer <<%s>>" \ % (procid,str(buffer)) # then with NumPy buffer = np.arange(dsize, dtype=np.float64) if procid==root: for i in range(dsize): buffer[i] = 5.0 comm.Bcast( buffer,root=root ) if not all( buffer==5.0 ): print "Something wrong on proc %d: numpy buffer <<%s>>" \ % (procid,str(buffer)) \tiny \begin{multicols} {3} Initial:\\ \nobreak \input jordan1 Step 1:\\ \nobreak \input jordan2 Step 2:\\ \nobreak \input jordan3 Step 3:\\ \nobreak \input jordan4 Step 4:\\ \nobreak \input jordan5 Step 5:\\ \nobreak \input jordan6 Step 6:\\ \nobreak \input jordan7 Step 7:\\ \nobreak \input jordan8 Step 8:\\ \nobreak \input jordan9 Step 9:\\ \nobreak \input jordan10 Step 10:\\ \nobreak \input jordan11 Step 11:\\ \nobreak \input jordan12 Step 12:\\ \nobreak \input jordan13 Step 13:\\ \nobreak \input jordan14 Step 14:\\ \nobreak \input jordan15 Step 15:\\ \nobreak \input jordan16 Step 16:\\ \nobreak \input jordan17 Step 17:\\ \nobreak \input jordan18 Step 18:\\ \nobreak \input jordan19 Finished:\\ \nobreak \input jordan20 \end{multicols} For the following exercise, study figure  . Exercise The Gauss-Jordan algorithm for solving a linear system with a matrix $A$ (or computing its inverse) runs as follows: {\small for pivot $k=1,…,n$\\ let the vector of scalings $\ell^{(k)}_i=A_{ik}/A_{kk}$\\ for row $r\not=k$\\ for column $c=1,…,n$\\ $A_{rc}\leftarrow A_{rc} - \ell^{(k)}_r A_{rc}$\\ } where we ignore the update of the righthand side, or the formation of the inverse. Let a matrix be distributed with each process storing one column. Implement the Gauss-Jordan algorithm as a series of broadcasts: in iteration $k$ process $k$ computes and broadcasts the scaling vector $\{\ell^{(k)}_i\}_i$. Replicate the right-hand side on all processors. Exercise Add partial pivoting to your implementation of Gauss-Jordan elimination. Change your implementation to let each processor store multiple columns, but still do one broadcast per column. Is there a way to have only one broadcast per processor? 3.4 Rooted collectives: gather and scatter Top > Rooted collectives: gather and scatter In the MPI_Scatter operation, the root spreads information to all other processes. The difference with a broadcast is that it involves individual information from/to every process. Thus, the gather operation typically has an array of items, one coming from each sending process, and scatter has an array, with an individual item for each receiving process; see figure  . These gather and scatter collectives have a different parameter list from the broadcast/reduce. The broadcast/reduce involves the same amount of data on each process, so it was enough to have a buffer, datatype, and size. In the gather/scatter calls you have a large buffer on the root, with a datatype and size specification, and a smaller buffer on each process, with its own type and size specification. Of course, since we're in SPMD mode, even non-root processes have the argument for the send buffer, but they ignore it. For instance: int MPI_Scatter (void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm) The sendcount is not, as you might expect, the total length of the sendbuffer; instead, it is the amount of data sent to each process. Exercise Let each process compute a random number. You want to print the maximum value and on what processor it is computed. What collective(s) do you use? Write a short program. In the gather and scatter calls, each processor has $n$ elements of individual data. There is also a root processor that has an array of length $np$, where $p$ is the number of processors. The gather call collects all this data from the processors to the root; the scatter call assumes that the information is initially on the root and it is spread to the individual processors. The prototype for MPI_Gather has two count' parameters, one for the length of the individual send buffers, and one for the receive buffer. However, confusingly, the second parameter (which is only relevant on the root) does not indicate the total amount of information coming in, but rather the size of each contribution. Thus, the two count parameters will usually be the same (at least on the root); they can differ if you use different MPI_Datatype values for the sending and receiving processors. C: int MPI_Gather( const void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm) Semantics: IN sendbuf: starting address of send buffer (choice) IN sendcount: number of elements in send buffer (non-negative integer) IN sendtype: data type of send buffer elements (handle) OUT recvbuf: address of receive buffer (choice, significant only at root) IN recvcount: number of elements for any single receive (non-negative integer, significant only at root) IN recvtype: data type of recv buffer elements (significant only at root) (handle) IN root: rank of receiving process (integer) IN comm: communicator (handle) Fortran: MPI_Gather (sendbuf, sendcount, sendtype, recvbuf, recvcount, recvtype, root, comm, ierror) TYPE(*), DIMENSION(..), INTENT(IN) :: sendbuf TYPE(*), DIMENSION(..) :: recvbuf INTEGER, INTENT(IN) :: sendcount, recvcount, root TYPE(MPI_Datatype), INTENT(IN) :: sendtype, recvtype TYPE(MPI_Comm), INTENT(IN) :: comm INTEGER, OPTIONAL, INTENT(OUT) :: ierror Python: MPI.Comm.Gather (self, sendbuf, recvbuf, int root=0) Here is a small example: // gather.c // we assume that each process has a value "localsize" // the root process collectes these values if (procno==root) localsizes = (int*) malloc( (nprocs+1)*sizeof(int) ); // everyone contributes their info MPI_Gather(&localsize,1,MPI_INT, localsizes,1,MPI_INT,root,comm); This will also be the basis of a more elaborate example in section  . The MPI_IN_PLACE option can be used for the send buffer on the root; the data for the root is then assumed to be already in the correct location in the receive buffer. The MPI_Scatter operation is in some sense the inverse of the gather: the root process has an array of length $np$ where $p$ is the number of processors and $n$ the number of elements each processor will receive. int MPI_Scatter (void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm) 3.4.1 Example Top > Rooted collectives: gather and scatter > Example In some applications, each process computes a row or column of a matrix, but for some calculation (such as the determinant) it is more efficient to have the whole matrix on one process. You should of course only do this if this matrix is essentially smaller than the full problem, such as an interface system or the last coarsening level in multigrid. Figure  pictures this. Note that conceptually we are gathering a two-dimensional object, but the buffer is of course one-dimensional. You will later see how this can be done more elegantly with the subarray' datatype; section  . 3.4.2 Allgather Top > Rooted collectives: gather and scatter > Allgather The MPI_Allgather routine does the same gather onto every process: each process winds up with the totality of all data; figure  . C: int MPI_Allgather(const void *sendbuf, int sendcount, MPI_Datatype sendtype, void *recvbuf, int recvcount, MPI_Datatype recvtype, MPI_Comm comm) int MPI_Iallgather(const void *sendbuf, int sendcount, MPI_Datatype sendtype, void *recvbuf, int recvcount, MPI_Datatype recvtype, MPI_Comm comm, MPI_Request *request) Fortran: MPI_ALLGATHER(SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, COMM, IERROR) SENDBUF (*), RECVBUF (*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNT, RECVTYPE, COMM, INTEGER IERROR MPI_IALLGATHER(SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, COMM, REQUEST, IERROR) SENDBUF(*), RECVBUF (*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNT, RECVTYPE, COMM INTEGER REQUEST, IERROR C++ Syntax Parameters: sendbuf : Starting address of send buffer (choice). sendcount: Number of elements in send buffer (integer). sendtype: Datatype of send buffer elements (handle). recvbuf: Starting address of recv buffer (choice). recvcount: Number of elements received from any process (integer). recvtype: Datatype of receive buffer elements (handle). comm; Communicator (handle). recvbuf: Address of receive buffer (choice). request: Request (handle, non-blocking only). This routine can be used in the simplest implementation of the dense matrix-vector product to give each processor the full input; see  Eijkhout:IntroHPC . Some cases look like an all-gather but can be implemented more efficiently. Suppose you have two distributed vectors, and you want to create a new vector that contains those elements of the one that do not appear in the other. You could implement this by gathering the second vector on each processor, but this may be prohibitive in memory usage. Exercise Can you think of another algorithm for taking the set difference of two distributed vectors. Hint: look up bucket-brigade algorithm' in  [Eijkhout:IHPSClulu] . What is the time and space complexity of this algorithm? Can you think of other advantages beside a reduction in workspace? 3.5 All-to-all Top > All-to-all The all-to-all operation can be seen as a collection of simultaneous broadcasts or simultaneous gathers. The parameter specification is much like an allgather, with a separate send and receive buffer, and no root specified. As with the gather call, the receive count corresponds to an individual receive, not the total amount. Unlike the gather call, the send buffer now obeys the same principle: with a send count of 1, the buffer has a length of the number of processes. int MPI_Alltoallv (void *sendbuf, int sendcnt, MPI_Datatype sendtype, void *recvbuf, int recvcnt, MPI_Datatype recvtype, MPI_Comm comm) 3.5.1 All-to-all as data transpose Top > All-to-all > All-to-all as data transpose The all-to-all operation can be considered as a data transpose. For instance, assume that each process knows how much data to send to every other process. If you draw a connectivity matrix of size $P\times P$, denoting who-sends-to-who, then the send information can be put in rows: $$\forall_i\colon C[i,j]>0\quad\hbox{if process i sends to process j}.$$ Conversely, the columns then denote the receive information: $$\forall_j\colon C[i,j]>0\quad\hbox{if process j receives from process i}.$$ Exercise In the initial stage of radix sorting , each process considers how many elements to send to every other process. Use MPI_Alltoall to derive from this how many elements they will receive from every other process. On a larger scale, the typical application for the all-to-all operation is in the FFT algorithm, where it can take tens of percents of the running time. 3.5.2 All-to-all-v Top > All-to-all > All-to-all-v Exercise The actual data shuffle of a radix sort can be done with MPI_Alltoallv . Finish the code of exercise  . 3.6 Reduce-scatter Top > Reduce-scatter There are several MPI collectives that are functionally equivalent to a combination of others. You have already seen MPI_Allreduce which is equivalent to a reduction followed by a broadcast. Often such combinations can be more efficient than using the individual calls; see  Eijkhout:IntroHPC . Here is another example: MPI_Reduce_scatter is equivalent to a reduction on an array of data (meaning a pointwise reduction on each array location) followed by a scatter of this array to the individual processes. We can look at reduce-scatter as a limited form of the all-to-all data transposition discussed above (section  ). Suppose that the matrix $C$ contains only $0/1$, indicating whether or not a messages is send, rather than the actual amounts. If a receiving process only needs to know how many messages to receive, rather than where they come from, it is enough to know the column sum, rather than the full column (see figure  ). One important example of this command is the sparse matrix-vector product ; see  Eijkhout:IntroHPC for background information. Each process contains one or more matrix rows, so by looking at indices the process can decide what other processes it needs data from. The problem is for a process to find out what other processes it needs to send data to. Using MPI_Reduce_scatter the process goes as follows: Each process creates an array of ones and zeros, describing who it needs data from. The reduce part of the reduce-scatter yields an array of requester counts; after the scatter each process knows how many processes request data from it. Next, the sender processes need to find out what elements are requested from it. For this, each process sends out arrays of indices. The big trick is that each process now knows how many of these requests will be coming in, so it can post precisely that many MPI_Irecv calls, with a source of MPI_ANY_SOURCE . The MPI_Reduce_scatter command is equivalent to a reduction on an array of data, followed by a scatter of that data to the individual processes. To be precise, there is an array recvcounts where recvcounts[i] gives the number of elements that ultimate wind up on process  i . The result is equivalent to doing a reduction with a length equal to the sum of the recvcounts[i] values, followed by a scatter where process  i receives recvcounts[i] values. (Since the amount of data to be scattered depends on the process, this is in fact equivalent to MPI_Scatterv rather than a regular scatter.) Semantics: MPI_REDUCE_SCATTER ( sendbuf, recvbuf, recvcounts, datatype, op, comm) MPI_Reduce_scatter_block ( sendbuf, recvbuf, recvcount, datatype, op, comm) Input parameters: sendbuf: starting address of send buffer (choice) recvcount: element count per block (non-negative integer) recvcounts: non-negative integer array (of length group size) specifying the number of elements of the result distributed to each process. datatype: data type of elements of send and receive buffers (handle) op: operation (handle) comm: communicator (handle) Output parameters: recvbuf: starting address of receive buffer (choice) C: int MPI_Reduce_scatter (const void* sendbuf, void* recvbuf, const int recvcounts[], MPI_Datatype datatype, MPI_Op op, MPI_Comm comm) F: MPI_Reduce_scatter(sendbuf, recvbuf, recvcounts, datatype, op, comm, ierror) TYPE(*), DIMENSION(..), INTENT(IN) :: sendbuf TYPE(*), DIMENSION(..) :: recvbuf INTEGER, INTENT(IN) :: recvcounts(*) TYPE(MPI_Datatype), INTENT(IN) :: datatype TYPE(MPI_Op), INTENT(IN) :: op TYPE(MPI_Comm), INTENT(IN) :: comm INTEGER, OPTIONAL, INTENT(OUT) :: ierror Py: comm.Reduce_scatter(sendbuf, recvbuf, recvcounts=None, Op op=SUM) For instance, if all recvcounts[i] values are 1, the sendbuffer has one element for each process, and the receive buffer has length 1. 3.6.1 Examples Top > Reduce-scatter > Examples An important application of this is establishing an irregular communication pattern. Assume that each process knows which other processes it wants to communicate with; the problem is to let the other processes know about this. The solution is to use MPI_Reduce_scatter to find out how many processes want to communicate with you, and then wait for precisely that many messages with a source value of MPI_ANY_SOURCE . // reducescatter.c // record what processes you will communicate with int *recv_requests; // find how many procs want to communicate with you MPI_Reduce_scatter (recv_requests,&nsend_requests,counts,MPI_INT, MPI_SUM,comm); // send a msg to the selected processes for (int i=0; i0) MPI_Isend(&msg,1,MPI_INT, /*to:*/ i,0,comm, mpi_requests+irequest++); // do as many receives as you know are coming in for (int i=0; i Use of MPI_Reduce_scatter to implement the two-dimensional matrix-vector product. Set up separate row and column communicators with MPI_Comm_split , use MPI_Reduce_scatter to combine local products. \cverbatimsnippet[examples/mpi/c/mvp2d.c]{mvp2d} 3.7 Barrier Top > Barrier A barrier is a routine that blocks all processes until they have all reached the barrier call. Thus it achieves time synchronization of the processes. C: int MPI_Barrier( MPI_Comm comm ) Fortran2008: MPI_BARRIER(COMM, IERROR) Type(MPI_Comm),intent(int) :: COMM INTEGER,intent(out) :: IERROR Fortran 95: MPI_BARRIER(COMM, IERROR) INTEGER :: COMM, IERROR Input parameter: comm : Communicator (handle) Output parameter: Ierror : Error status (integer), Fortran only This call's simplicity is contrasted with its usefulness, which is very limited. It is almost never necessary to synchronize processes through a barrier: for most purposes it does not matter if processors are out of sync. Conversely, collectives (except the new non-blocking ones; section  ) introduce a barrier of sorts themselves. 3.8 Variable-size-input collectives Top > Variable-size-input collectives In the gather and scatter call above each processor received or sent an identical number of items. In many cases this is appropriate, but sometimes each processor wants or contributes an individual number of items. Let's take the gather calls as an example. Assume that each processor does a local computation that produces a number of data elements, and this number is different for each processor (or at least not the same for all). In the regular MPI_Gather call the root processor had a buffer of size $nP$, where $n$ is the number of elements produced on each processor, and $P$ the number of processors. The contribution from processor $p$ would go into locations $pn,…,(p+1)n-1$. For the variable case, we first need to compute the total required buffer size. This can be done through a simple MPI_Reduce with MPI_SUM as reduction operator: the buffer size is $\sum_p n_p$ where $n_p$ is the number of elements on processor $p$. But you can also postpone this calculation for a minute. The next question is where the contributions of the processor will go into this buffer. For the contribution from processor $p$ that is $\sum_{q<p}n_p,…\sum_{q\leq p}n_p-1$. To compute this, the root processor needs to have all the $n_p$ numbers, and it can collect them with an MPI_Gather call. We now have all the ingredients. All the processors specify a send buffer just as with MPI_Gather . However, the receive buffer specification on the root is more complicated. It now consists of: outbuffer, array-of-outcounts, array-of-displacements, outtype and you have just seen how to construct that information. For example, in an MPI_Gatherv call each process has an individual number of items to contribute. To gather this, the root process needs to find these individual amounts with an MPI_Gather call, and locally construct the offsets array. Note how the offsets array has size ntids+1 : the final offset value is automatically the total size of all incoming data. See the example below. There are various calls where processors can have buffers of differing sizes. In MPI_Scatterv the root process has a different amount of data for each recipient. In MPI_Gatherv , conversely, each process contributes a different sized send buffer to the received result; MPI_Allgatherv does the same, but leaves its result on all processes; MPI_Alltoallv does a different variable-sized gather on each process. int MPI_Scatterv (void* sendbuf, int *sendcounts, int *displs, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm) C: int MPI_Gatherv( const void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, const int recvcounts[], const int displs[], MPI_Datatype recvtype, int root, MPI_Comm comm) Semantics: IN sendbuf: starting address of send buffer (choice) IN sendcount: number of elements in send buffer (non-negative integer) IN sendtype: data type of send buffer elements (handle) OUT recvbuf: address of receive buffer (choice, significant only at root) IN recvcounts: non-negative integer array (of length group size) containing the number of elements that are received from each process (significant only at root) IN displs: integer array (of length group size). Entry i specifies the displacement relative to recvbuf at which to place the incoming data from process i (significant only at root) IN recvtype: data type of recv buffer elements (significant only at root) (handle) IN root: rank of receiving process (integer) IN comm: communicator (handle) Fortran: MPI_Gatherv(sendbuf, sendcount, sendtype, recvbuf, recvcounts, displs, recvtype, root, comm, ierror) TYPE(*), DIMENSION(..), INTENT(IN) :: sendbuf TYPE(*), DIMENSION(..) :: recvbuf INTEGER, INTENT(IN) :: sendcount, recvcounts(*), displs(*), root TYPE(MPI_Datatype), INTENT(IN) :: sendtype, recvtype TYPE(MPI_Comm), INTENT(IN) :: comm INTEGER, OPTIONAL, INTENT(OUT) :: ierror Python: Gatherv(self, sendbuf, [recvbuf,counts], int root=0) int MPI_Allgatherv (void *sendbuf, int sendcount, MPI_Datatype sendtype, void *recvbuf, int *recvcounts, int *displs, MPI_Datatype recvtype, MPI_Comm comm) 3.8.1 Example of Gatherv Top > Variable-size-input collectives > Example of Gatherv We use MPI_Gatherv to do an irregular gather onto a root. We first need an MPI_Gather to determine offsets. // gatherv.c // we assume that each process has an array "localdata" // of size "localsize" // the root process decides how much data will be coming: // allocate arrays to contain size and offset information if (procno==root) { localsizes = (int*) malloc( (nprocs+1)*sizeof(int) ); offsets = (int*) malloc( nprocs*sizeof(int) ); } // everyone contributes their info MPI_Gather(&localsize,1,MPI_INT, localsizes,1,MPI_INT,root,comm); // the root constructs the offsets array if (procno==root) { offsets[0] = 0; for (int i=0; i ## gatherv.py # implicitly using root=0 globalsize = comm.reduce(localsize) if procid==0: print "Global size=%d" % globalsize collecteddata = np.empty(globalsize,dtype=np.int) counts = comm.gather(localsize) comm.Gatherv(localdata, [collecteddata, counts]) 3.8.2 Example of Allgatherv Top > Variable-size-input collectives > Example of Allgatherv Prior to the actual gatherv call, we need to construct the count and displacement arrays. The easiest way is to use a reduction. // allgatherv.c MPI_Allgather ( &my_count,1,MPI_INT, recv_counts,1,MPI_INT, comm ); int accumulate = 0; for (int i=0; i In python the receive buffer has to contain the counts and displacements arrays. ## allgatherv.py my_count = np.empty(1,dtype=np.int) my_count[0] = mycount comm.Allgather( my_count,recv_counts ) accumulate = 0 for p in range(nprocs): recv_displs[p] = accumulate; accumulate += recv_counts[p] global_array = np.empty(accumulate,dtype=np.float64) comm.Allgatherv( my_array, [global_array,recv_counts,recv_displs,MPI.DOUBLE] ) 3.8.3 Variable all-to-all Top > Variable-size-input collectives > Variable all-to-all int MPI_Alltoallv (void *sendbuf, int *sendcnts, int *sdispls, MPI_Datatype sendtype, void *recvbuf, int *recvcnts, int *rdispls, MPI_Datatype recvtype, MPI_Comm comm) 3.9 Scan operations Top > Scan operations The MPI_Scan operation also performs a reduction, but it keeps the partial results. That is, if processor $i$ contains a number $x_i$, and $\oplus$ is an operator, then the scan operation leaves $x_0\oplus\cdots\oplus x_i$ on processor $i$. This type of operation is often called a prefix operation ; see Eijkhout:IntroHPC . The MPI_Scan routine is an inclusive scan operation. C: int MPI_Scan(const void* sendbuf, void* recvbuf, int count, MPI_Datatype datatype, MPI_Op op, MPI_Comm comm) IN sendbuf: starting address of send buffer (choice) OUT recvbuf: starting address of receive buffer (choice) IN count: number of elements in input buffer (non-negative integer) IN datatype: data type of elements of input buffer (handle) IN op: operation (handle) IN comm: communicator (handle) Fortran: MPI_Scan(sendbuf, recvbuf, count, datatype, op, comm, ierror) TYPE(*), DIMENSION(..), INTENT(IN) :: sendbuf TYPE(*), DIMENSION(..) :: recvbuf INTEGER, INTENT(IN) :: count TYPE(MPI_Datatype), INTENT(IN) :: datatype TYPE(MPI_Op), INTENT(IN) :: op TYPE(MPI_Comm), INTENT(IN) :: comm INTEGER, OPTIONAL, INTENT(OUT) :: ierror Python: res = Intracomm.scan( sendobj=None,recvobj=None,op=MPI.SUM) res = Intracomm.exscan( sendobj=None,recvobj=None,op=MPI.SUM) The MPI_Op operations do not return an error code. In python native mode the result is a function return value. ## scan.py mycontrib = 10+random.randint(1,nprocs) myfirst = 0 mypartial = comm.scan(mycontrib) sbuf = np.empty(1,dtype=np.int) rbuf = np.empty(1,dtype=np.int) sbuf[0] = mycontrib comm.Scan(sbuf,rbuf) 3.9.1 Exclusive scan Top > Scan operations > Exclusive scan Often, the more useful variant is the exclusive scan C: int MPI_Exscan(const void *sendbuf, void *recvbuf, int count, MPI_Datatype datatype, MPI_Op op, MPI_Comm comm) int MPI_Iexscan(const void *sendbuf, void *recvbuf, int count, MPI_Datatype datatype, MPI_Op op, MPI_Comm comm, MPI_Request *request) Fortran: MPI_EXSCAN(SENDBUF, RECVBUF, COUNT, DATATYPE, OP, COMM, IERROR) SENDBUF(*), RECVBUF(*) INTEGER COUNT, DATATYPE, OP, COMM, IERROR MPI_IEXSCAN(SENDBUF, RECVBUF, COUNT, DATATYPE, OP, COMM, REQUEST, IERROR) SENDBUF(*), RECVBUF(*) INTEGER COUNT, DATATYPE, OP, COMM, REQUEST, IERROR Input Parameters sendbuf: Send buffer (choice). count: Number of elements in input buffer (integer). datatype: Data type of elements of input buffer (handle). op: Operation (handle). comm: Communicator (handle). Output Parameters recvbuf: Receive buffer (choice). request: Request (handle, non-blocking only). with the same prototype. The result of the exclusive scan is undefined on processor 0 ( None in python), and on processor 1 it is a copy of the send value of processor 1. In particular, the MPI_Op need not be called on these two processors. Exercise The exclusive definition, which computes $x_0\oplus x_{i-1}$ on processor $i$, can easily be derived from the inclusive operation for operations such as MPI_PLUS or MPI_MULT . Are there operators where that is not the case? 3.9.2 Use of scan operations Top > Scan operations > Use of scan operations The MPI_Scan operation is often useful with indexing data. Suppose that every processor $p$ has a local vector where the number of elements $n_p$ is dynamically determined. In order to translate the local numbering $0… n_p-1$ to a global numbering one does a scan with the number of local elements as input. The output is then the global number of the first local variable. Exercise Do you use MPI_Scan or MPI_Exscan for this operation? How would you describe the result of the other scan operation, given the same input? Exclusive scan examples: // exscan.c int my_first=0,localsize; // localsize = ..... result of local computation .... // find myfirst location based on the local sizes err = MPI_Exscan(&localsize,&my_first, 1,MPI_INT,MPI_SUM,comm); CHK(err); ## exscan.py localsize = 10+random.randint(1,nprocs) myfirst = 0 mypartial = comm.exscan(localsize,0) It is possible to do a segmented scan . Let $x_i$ be a series of numbers that we want to sum to $X_i$ as follows. Let $y_i$ be a series of booleans such that $$\begin{cases} X_i=x_i&\hbox{if y_i=0}\\ X_i=X_{i-1}+x_i&\hbox{if y_i=1} \end{cases}$$ (This is the basis for the implementation of the sparse matrix vector product as prefix operation; see Eijkhout:IntroHPC .) This means that $X_i$ sums the segments between locations where $y_i=0$ and the first subsequent place where $y_i=1$. To implement this, you need a user-defined operator $$\begin{pmatrix} X\\ x\\ y \end{pmatrix} = \begin{pmatrix} X_1\\ x_1\\ y_1 \end{pmatrix} \bigoplus \begin{pmatrix} X_2\\ x_2\\ y_2 \end{pmatrix} \colon \begin{cases} X=x_1+x_2&\hbox{if y_2==1}\\ X=x_2&\hbox{if y_2==0} \end{cases}$$ This operator is not communitative, and it needs to be declared as such with MPI_Op_create ; see section  3.10 MPI Operators Top > MPI Operators MPI operators are used in reduction operators. Most common operators, such as sum or maximum, have been built into the MPI library, but it is possible to define new operators. 3.10.1 Pre-defined operators Top > MPI Operators > Pre-defined operators Pre-defined operators are listed in section  . 3.10.2 User-defined operators Top > MPI Operators > User-defined operators In addition to predefined operators, MPI has the possibility of user-defined operators to use in a reduction or scan operation. Semantics: MPI_OP_CREATE( function, commute, op) [ IN function] user defined function (function) [ IN commute] true if commutative; false otherwise. [ OUT op] operation (handle) C: int MPI_Op_create(MPI_User_function *function, int commute, MPI_Op *op) Fortran: MPI_OP_CREATE( FUNCTION, COMMUTE, OP, IERROR) EXTERNAL FUNCTION LOGICAL COMMUTE INTEGER OP, IERROR The function needs to have the following prototype: typedef void MPI_User_function ( void *invec, void *inoutvec, int *len, MPI_Datatype *datatype); FUNCTION USER_FUNCTION( INVEC(*), INOUTVEC(*), LEN, TYPE) <type> INVEC(LEN), INOUTVEC(LEN) INTEGER LEN, TYPE The function has an array length argument  len , to allow for pointwise reduction on a a whole array at once. The inoutvec array contains partially reduced results, and is typically overwritten by the function. For example, here is an operator for finding the smallest non-zero number in an array of nonnegative integers: // reductpositive.c void reduce_without_zero(void *in,void *inout,int *len,MPI_Datatype *type) { // r is the already reduced value, n is the new value int n = *(int*)in, r = *(int*)inout; int m; if (n==0) { // new value is zero: keep r m = r; } else if (r==0) { m = n; } else if (n You can query the commutativity of an operator: Semantics: MPI_Op_commutative(op, commute) IN op : handle OUT commute : true/false C: int MPI_Op_commutative(MPI_Op op, int *commute) Fortran: MPI_OP_COMMUTATIVE( op, commute) TYPE(MPI_Op), INTENT(IN) :: op LOGICAL, INTENT(OUT) :: commute INTEGER, OPTIONAL, INTENT(OUT) :: ierror A created MPI_Op can be freed again: int MPI_Op_free(MPI_Op *op) This sets the operator to MPI_OP_NULL . 3.10.3 Local reduction Top > MPI Operators > Local reduction The application of an MPI_Op can be performed with the routine MPI_Reduce_local . Using this routine and some send/receive scheme you can build your own global reductions. Note that this routine does not take a communicator because it is purely local. Semantics: MPI_REDUCE_LOCAL( inbuf, inoutbuf, count, datatype, op) Input parameters: inbuf: input buffer (choice) count: number of elements in inbuf and inoutbuf buffers (non-negative integer) datatype: data type of elements of inbuf and inoutbuf buffers (handle) op: operation (handle) Input/output parameters: inoutbuf: combined input and output buffer (choice) C: int MPI_Reduce_local (void* inbuf, void* inoutbuf, int count, MPI_Datatype datatype, MPI_Op op) Fortran: MPI_REDUCE_LOCAL(INBUF, INOUBUF, COUNT, DATATYPE, OP, IERROR) INBUF(*), INOUTBUF(*) INTEGER :: COUNT, DATATYPE, OP, IERROR 3.11 Non-blocking collectives Top > Non-blocking collectives Above you have seen how the Isend' and Irecv' routines can overlap communication with computation. This is not possible with the collectives you have seen so far: they act like blocking sends or receives. However, there are also non-blocking collectives . These have roughly the same calling sequence as their blocking counterparts, except that they output an MPI_Request . You can then use an MPI_Wait call to make sure the collective has completed. Such operations can be used to increase efficiency. For instance, computing $$y \leftarrow Ax + (x^tx)y$$ involves a matrix-vector product, which is dominated by computation in the sparse matrix case, and an inner product which is typically dominated by the communication cost. You would code this as MPI_Iallreduce( .... x ..., &request); // compute the matrix vector product MPI_Wait(request); // do the addition This can also be used for 3D FFT operations  [Hoefler:case-for-nbc] . Occasionally, a non-blocking collective can be used for non-obvious purposes, such as the MPI_Ibarrier in  [Hoefler:2010:SCP] . The same calling sequence as the blocking counterpart, except for the addition of an MPI_Request parameter. For instance MPI_Ibcast : int MPI_Ibcast( void *buffer, int count, MPI_Datatype datatype, int root, MPI_Comm comm, MPI_Request *request) Semantics int MPI_Iallreduce( const void *sendbuf, void *recvbuf, int count, MPI_Datatype datatype, MPI_Op op, MPI_Comm comm, MPI_Request *request) Input Parameters sendbuf : starting address of send buffer (choice) count : number of elements in send buffer (integer) datatype : data type of elements of send buffer (handle) op : operation (handle) comm : communicator (handle) Output Parameters recvbuf : starting address of receive buffer (choice) request : communication request (handle) Semantics int MPI_Iallgather( const void *sendbuf, int sendcount, MPI_Datatype sendtype, void *recvbuf, int recvcount, MPI_Datatype recvtype, MPI_Comm comm, MPI_Request *request) Input Parameters sendbuf : starting address of send buffer (choice) sendcount : number of elements in send buffer (integer) sendtype : data type of send buffer elements (handle) recvcount : number of elements received from any process (integer) recvtype : data type of receive buffer elements (handle) comm : communicator (handle) Output Parameters recvbuf : address of receive buffer (choice) request : communication request (handle) 3.11.1 Non-blocking barrier Top > Non-blocking collectives > Non-blocking barrier Probably the most surprising non-blocking collective is the non-blocking barrier MPI_Ibarrier . The way to understand this is to think of a barrier not in terms of temporal synchronization, but state agreement: reaching a barrier is a sign that a process has attained a certain state, and leaving a barrier means that all processes are in the same state. The ordinary barrier is then a blocking wait for agreement, while with a non-blocking barrier: Posting the barrier means that a process has reached a certain state; and the request being fullfilled means that all processes have reached the barrier. C: int MPI_Ibarrier(MPI_Comm comm, MPI_Request *request) Input Parameters comm : communicator (handle) Output Parameters request : communication request (handle) Fortran2008: MPI_Ibarrier(comm, request, ierror) Type(MPI_Comm),intent(int) :: comm TYPE(MPI_Request),intent(out) :: request INTEGER,intent(out) :: ierror We can use a non-blocking barrier to good effect, utilizing the idle time that would result from a blocking barrier. In the following code fragment processes test for completion of the barrier, and failing to detect such completion, perform some local work. // ibarriertest.c for ( ; ; step++) { int barrier_done_flag=0; MPI_Test(&barrier_request,&barrier_done_flag,MPI_STATUS_IGNORE); if (barrier_done_flag) { break; } else { int flag; MPI_Status status; MPI_Iprobe ( MPI_ANY_SOURCE,MPI_ANY_TAG, comm, &flag, &status ); } } 3.12 Performance of collectives Top > Performance of collectives It is easy to visualize a broadcast as in figure  : see figure  . the root sends all of its data directly to every other process. While this describes the semantics of the operation, in practice the implementation works quite differently. The time that a message takes can simply be modeled as $$\alpha +\beta n,$$ where $\alpha$ is the latency , a one time delay from establishing the communication between two processes, and $\beta$ is the time-per-byte, or the inverse of the bandwidth , and $n$ the number of bytes sent. Under the assumption that a processor can only send one message at a time, the broadcast in figure  would take a time proportional to the number of processors. Exercise What is the total time required for a broadcast involving $p$ processes? Give $\alpha$ and $\beta$ terms separately. One way to ameliorate that is to structure the broadcast in a tree-like fashion. This is depicted in figure  . Exercise How does the communication time now depend on the number of processors, again $\alpha$ and $\beta$ terms separately. What would be a lower bound on the $\alpha,\beta$ terms? The theory of the complexity of collectives is described in more detail in Eijkhout:IntroHPC ; see also  [Chan2007Collective] . 3.13 Collectives and synchronization Top > Collectives and synchronization Collectives, other than a barrier, have a synchronizing effect between processors. For instance, in MPI_Bcast( ....data... root); MPI_Send(....); the send operations on all processors will occur after the root executes the broadcast. Conversely, in a reduce operation the root may have to wait for other processors. This is illustrated in figure  , which gives a TAU trace of a reduction operation on two nodes, with two six-core sockets (processors) each. We see that\footnote {This uses mvapich version 1.6; in version 1.9 the implementation of an on-node reduction has changed to simulate shared memory.}: In each socket, the reduction is a linear accumulation; on each node, cores zero and six then combine their result; after which the final accumulation is done through the network. We also see that the two nodes are not perfectly in sync, which is normal for MPI applications. As a result, core 0 on the first node will sit idle until it receives the partial result from core 12, which is on the second node. \footnotesize \leftskip=\unitindent \rightskip=\unitindent Code: {\tiny \begin{lstlisting} switch(rank) { case 0: MPI_Bcast(buf1, count, type, 0, comm); MPI_Send(buf2, count, type, 1, tag, comm); break; case 1: MPI_Recv(buf2, count, type, MPI_ANY_SOURCE, tag, comm, status); MPI_Bcast(buf1, count, type, 0, comm); MPI_Recv(buf2, count, type, MPI_ANY_SOURCE, tag, comm, status); break; case 2: MPI_Send(buf2, count, type, 1, tag, comm); MPI_Bcast(buf1, count, type, 0, comm); break; } \end{lstlisting} } The most logical execution is:\par\medskip However, this ordering is allowed too:\par\medskip Which looks from a distance like:\par\medskip In other words, one of the messages seems to go back in time'. While collectives synchronize in a loose sense, it is not possible to make any statements about events before and after the collectives between processors: ...event 1... MPI_Bcast(....); ...event 2.... Consider a specific scenario: switch(rank) { case 0: MPI_Bcast(buf1, count, type, 0, comm); MPI_Send(buf2, count, type, 1, tag, comm); break; case 1: MPI_Recv(buf2, count, type, MPI_ANY_SOURCE, tag, comm, status); MPI_Bcast(buf1, count, type, 0, comm); MPI_Recv(buf2, count, type, MPI_ANY_SOURCE, tag, comm, status); break; case 2: MPI_Send(buf2, count, type, 1, tag, comm); MPI_Bcast(buf1, count, type, 0, comm); break; } Note the MPI_ANY_SOURCE parameter in the receive calls on processor 1. One obvious execution of this would be: The send from 2 is caught by processor 1; Everyone executes the broadcast; The send from 0 is caught by processor 1. However, it is equally possible to have this execution: Processor 0 starts its broadcast, then executes the send; Processor 1's receive catches the data from 0, then it executes its part of the broadcast; Processor 1 catches the data sent by 2, and finally processor 2 does its part of the broadcast. This is illustrated in figure  . 3.14 Implementation and performance of collectives Top > Implementation and performance of collectives In this section we will consider how collectives can be implemented in multiple ways, and the performance implications of such decisions. You can test the algorithms described here using SimGrid (section  ). 3.14.1 Broadcast Top > Implementation and performance of collectives > Broadcast Naive broadcast Write a broadcast operation where the root does an MPI_Send to each other process. What is the expected performance of this in terms of $\alpha,\beta$? Run some tests and confirm. Simple ring Let the root only send to the next process, and that one send to its neighbour. This scheme is known as a bucket brigade ; see also section  . What is the expected performance of this in terms of $\alpha,\beta$? Run some tests and confirm. Pipelined ring In a ring broadcast, each process needs to receive the whole message before it can pass it on. We can increase the efficiency by breaking up the message and sending it in multiple parts. (See figure  .) This will be advantageous for messages that are long enough that the bandwidth cost dominates the latency. Assume a send buffer of length more than 1. Divide the send buffer into a number of chunks. The root sends the chunks successively to the next process, and each process sends on whatever chunks it receives. What is the expected performance of this in terms of $\alpha,\beta$? Why is this better than the simple ring? Run some tests and confirm. Recursive doubling Collectives such as broadcast can be implemented through recursive doubling , where the root sends to another process, then the root and the other process send to two more, those four send to four more, et cetera. However, in an actual physical architecture this scheme can be realized in multiple ways that have drastically different performance. First consider the implementation where process 0 is the root, and it starts by sending to process 1; then they send to 2 and 3; these four send to 4--7, et cetera. If the architecture is a linear array of procesors, this will lead to contention : multiple messages wanting to go through the same wire. (This is also related to the concept of bisecection bandwidth .) In the following analyses we will assume wormhole routing : a message sets up a path through the network, reserving the necessary wires, and performing a send in time independent of the distance through the network. That is, the send time for any message can be modeled as $$T(n)=\alpha+\beta n$$ regardless source and destination, as long as the necessary connections are available. Exercise Analyze the running time of a recursive doubling broad cast as just described, with wormhole routing. Implement this broadcast in terms of blocking MPI send and receive calls. If you have SimGrid available, run tests with a number of parameters. The alternative, that avoids contention, is to let each doubling stage divide the network into separate halves. That is, process 0 sends to $P/2$, after which these two repeat the algorithm in the two halves of the network, sending to $P/4$ and $3P/4$ respectively. Exercise Analyze this variant of recursive doubling. Code it and measure runtimes on SimGrid. Exercise Revisit exercise and replace the blocking calls by non-blocking MPI_Isend  / MPI_Irecv calls. Make sure to test that the data is correctly propagated. 
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