This chapter assumes the background knowledge in section Basic TOP-C (Master-Slave) commands. ParGAP must be invoked through a script like
pargap.sh generated in ParGAP's bin during installation. In
addition, there must be a procgroup file in the current directory when
ParGAP is called, or alternatively, -p4pg FULL_PATH/procgroup may
be included on the pargap.sh command line. See Section Running ParGAP
for the details. You should find a sample procgroup file in ParGAP's
bin directory after installation of ParGAP. See the
section Invoking ParGAP with Remote Slaves for further details. Many of
the examples of this section can be found in ParGAP's examples
directory.
A master-slave computation is invoked when a ParGAP program issues the
command MasterSlave(). This command is an example of what is called
``collective communication'' in MPI (although the command is not part of
MPI). It is also sometimes called SPMD (Single Program, Multiple Data),
since all processes see the same code, although different processes may
execute different parts of the code. The MasterSlave() command must be
invoked on all processes before execution can begin. The following
trivial example does this. (Note that the final \ on a line that is
still inside a string allows continuation of a string to the next line.)
We illustrate these principles first in their simplest form, making all
variables global variables. Later, we introduce additional ParGAP
utilities that allow one to write in better style.
#Shared Data: none
#TaskInput: counter
#TaskOutput: counter^2 (square of counter)
#Task: compute counter^2 from counter
ParEval( "result := []" );
ParEval( "counter := 0; \
SubmitTaskInput := function() \
counter := counter + 1; \
if counter <= 10 then return counter; else return NOTASK; fi; \
end;");
ParEval( "DoTask := x->x^2" );
ParEval( "CheckTaskResultVers1 := function( input, output ) \
result[input] := [input, output]; \
return NO_ACTION; \
end;" );
ParEval( "MasterSlave( SubmitTaskInput, DoTask, CheckTaskResultVers1 )" );
Print(result);
By default, ParTrace = true (see ParTrace), causing the execution to
display each input, x, as it is passed from the master to a slave, and
each output, x^2, as it is passed from the slave back to the master.
This behavior can be turned off by setting: ParTrace := false; The
fourth argument of MasterSlave(), Print, is a dummy argument that is
never invoked in this example.
Note that the result list is filled in only on the local process, and was
never defined or modified on the slave processes. To remedy this
situation, we introduce the concept of shared data, a globally shared,
application-defined data structure. A central principle of the TOP-C
model in ParGAP is that any routine may ``read'' the shared data, but
it may be modified only by the application-defined routine,
UpdateSharedData(). Hence, if we wanted the result list to be
recorded on all processes (perhaps as a lookup table), we would now
write:
ParBroadcast(PrintToString("result = ", result));
Until now, we have been using global variables. It is better style to use local variables, where possible. We rewrite the above routine in the improved style:
#Shared Data: result [ result is shared among all processes ]
#TaskInput: counter
#TaskOutput: counter^2 (square of counter)
#Task: compute counter^2 from counter
#UpdateSharedData: record [counter, counter^2] at index, counter, of result
MSSquare := function( range ) # expects as input a range, like [1..10]
local counter, result,
SubmitTaskInput, DoTask, CheckTaskResultVers2, UpdateSharedData;
counter := range[1]; # Reset counter for use in SubmitTaskInput()
result := [];
SubmitTaskInput := function()
counter := counter + 1;
if counter <= range[Length(range)] then return counter;
else return NOTASK;
fi;
end;
DoTask := x->x^2;
CheckTaskResultVers2 := function( input, output )
return UPDATE_ACTION;
end;
UpdateSharedData := function( input, output )
result[input] := [input, output];
end;
MasterSlave( SubmitTaskInput, DoTask, CheckTaskResultVers2, UpdateSharedData );
return result;
end;
#ParSquare() is the main calling function; It must define MSSquare on
# all slaves before calling it in parallel.
ParSquare := function( range ) # expects as input a range, like [1..10]
ParEval( PrintToString( "MSSquare := ", MSSquare ) );
return ParEval( PrintToString( "MSSquare( ", range, ")" ) );
end;
5.3 ParInstallTOPCGlobalFunction() and TaskInputIterator() (ParSquare revisited)
This example can be written more compactly by using some of the convenience functions provided by ParGAP. Specifically, we would rewrite this as:
ParInstallTOPCGlobalFunction( "ParSquare", function( range )
local result;
result := [];
MasterSlave( TaskInputIterator( range ),
x->x^2,
function( input, output ) return UPDATE_ACTION; end,
function( input, output ) result[input] := [input, output]; end
);
return result;
end );
The usage above demonstrates the use of two utilities.
ParInstallTOPCGlobalFunction( string, function ) F
This defines gvar as a function on the master and on the slaves.
On each slave, the definition of gvar is given by function.
However, on the master, gvar is defined as a function that first
calls gvar on all slaves with the arguments originally passed
to gvar, and then on the master, function is called with the
original arguments. This is exactly the behavior that is wanted
in order to compress an invocation of MasterSlave() so that the
right things happen on both the master and on the slaves. This
is exactly what we saw in the previous definition of ParSquare,
above.
TaskInputIterator( collection ) F
This function provides the functionality of a common case of
SubmitTaskInput(), by turning it into a GAP iterator
(see iterators in the Reference Manual). Its meaning is best
understood from its definition:
TaskInputIterator := function( collection )
local iter;
iter := Iterator( collection );
return function()
if IsDoneIterator(iter) then return NOTASK;
else return NextIterator(iter);
fi;
end;
end;
Let us now write a matrix-matrix multiplication routine in this style. Since matrix multiplication for dimension n requires n3 operations, we can afford to spend n2 time doing any sequential work. (A finer analysis would also consider the number of slaves, k, resulting in up to k*n2 time to send all messages, depending on the MPI broadcast algorithm.) So, a sequential matrix multiplication program might be written as follows. (The style emphasizes clarity over efficiency.)
SeqMultMat := function(m1, m2) # sequential code
local i, j, k, n, m2t, sum, result;
n := Length(m1);
result := [];
m2t := TransposedMat(m2);
for i in [1..n] do
result[i] := [];
for j in [1..n] do
sum := 0;
for k in [1..n] do
sum := sum + m1[i][k]*m2t[j][k];
od;
result[i][j] := sum;
od;
od;
return result;
end;
We choose to define the task as the computation of a single row of the result matrix. This corresponds to the body of the outermost loop.
#Shared Data: m1, m2t, result (three matrices)
#TaskInput: i (row index of result matrix)
#TaskOutput: result[i] (row of result matrix)
#Task: Compute result[i] from i, m1, and m2
#UpdateSharedData: Given result[i] and i, modify result on all processes.
ParInstallTOPCGlobalFunction( "ParMultMat", function(m1, m2)
local i, n, m2t, result, DoTask, CheckTaskResult, UpdateSharedData;
n := Length(m1);
result := [];
m2t := TransposedMat(m2);
DoTask := function(i) # i is task input
local j, k, sum;
result[i] := [];
for j in [1..n] do
sum := 0;
for k in [1..n] do
sum := sum + m1[i][k]*m2t[j][k];
od;
result[i][j] := sum;
od;
return result[i]; # return task output, row_i
end;
# CheckTaskResult executes only on the master
CheckTaskResult := function(i, row_i) # task output is row_i
return UPDATE_ACTION; # Pass on output and input to UpdateSharedData
end;
# UpdateSharedData executes on the master and on all slaves
UpdateSharedData := function(i, row_i) # task output is row_i
result[i] := row_i;
end;
# We're done defining the task. Let's do it now.
MasterSlave( TaskInputIterator([1..n]), DoTask, CheckTaskResult,
UpdateSharedData );
# result is defined on all processes; return local copy of result
return result;
end );
5.5 DefaultCheckTaskResult (as illustrated by ParSemiEchelonMatrix)
Now that the basic principles of the TOP-C model are clear, we
investigate an example that requires most of the basic features of
ParGAP, including the use of IsUpToDate() and
REDO_ACTION. Recall the standard idiom for
CheckTaskResult(). These issues were discussed in
the section Efficient Parallelism in MasterSlave() using CheckTaskResult().
DefaultCheckTaskResult := function( taskOutput, taskInput ) if taskOutput = false then return NO_ACTION; elif not IsUpToDate() then return REDO_ACTION; else return UPDATE_ACTION; fi; end;
The next example is a parallelization of the function
SemiEchelonMat() (a form of Gaussian elimination) in the GAP
library, lib/matrix.gi. Unlike the previous examples, parallelizing
Gaussian elimination efficiently is a non-trivial undertaking. This
is because a naive parallelization has poor load balancing. A slave
executing a task in the middle will have to sift a row vector
through many previous row vectors, while a slave executing a task
toward the beginning or end will have little work to do. We will
begin with a naive parallelization based on the sequential code, and
then migrate the code in a natural manner toward a more efficient
form, by analyzing the inefficiencies and applying the TOP-C model.
The reader may wish to stop and read the original code in
lib/matrix.gi first. The logic of SemiEchelonMat() is to examine
each row vector of an input matrix, in order, reduce it by a list of
basis vectors stored in vectors, and then add the row to vectors.
Upon completion, the number of leading zeroes of the row vectors in
vectors may not increase monotonically, but each element of
vectors will have a unique number of leading zeroes. Some rows of
the input matrix may reduce to the zero matrix, in which case they are
not added to vectors.
For the reader's convenience, the original sequential code is reproduced here.
SemiEchelonMat := function( mat )
local zero, # zero of the field of <mat>
nrows, # number of rows in <mat>
ncols, # number of columns in <mat>
vectors, # list of basis vectors
heads, # list of pivot positions in 'vectors'
i, # loop over rows
j, # loop over columns
x, # a current element
nzheads, # list of non-zero heads
row; # the row of current interest
mat:= List( mat, ShallowCopy );
nrows:= Length( mat );
ncols:= Length( mat[1] );
zero:= Zero( mat[1][1] );
heads:= ListWithIdenticalEntries( ncols, 0 );
nzheads := [];
vectors := [];
for i in [ 1 .. nrows ] do
row := mat[i];
# Reduce the row with the known basis vectors.
for j in [ 1 .. Length(nzheads) ] do
x := row[nzheads[j]];
if x <> zero then
AddRowVector( row, vectors[ j ], - x );
fi;
od;
j := PositionNot( row, zero );
if j <= ncols then
# We found a new basis vector.
MultRowVector(row, Inverse(row[j]));
Add( vectors, row );
Add( nzheads, j);
heads[j]:= Length( vectors );
fi;
od;
return rec( heads := heads,
vectors := vectors );
end;
Although GAP's Gaussian elimination algorithm appears to be
awkward to parallelize (since the next row vector in vectors depends
on row reduction by all previous vectors, we will see how the original
code of SemiEchelonMat() can be modified in a natural manner to
produce useful parallelism. This illustrates the general TOP-C
paradigm for ``naturally'' parallelizing a sequential algorithm.
#Shared Data: vectors (basis vectors), heads, nzheads, mat (matrix)
#TaskInput: i (row index of matrix)
#TaskOutput: List of (1) j and (2) row i of matrix, mat, reduced by vectors
# j is the first non-zero element of row i
#Task: Compute reduced row i from mat, vectors, heads
#UpdateSharedData: Given i, j, reduced row i, add new basis vector
# to vectors and update heads[j] to point to it
ParInstallTOPCGlobalFunction( "ParSemiEchelonMat", function( mat )
local zero, # zero of the field of <mat>
nrows, # number of rows in <mat>
ncols, # number of columns in <mat>
vectors, # list of basis vectors
heads, # list of pivot positions in 'vectors'
i, # loop over rows
nzheads, # list of non-zero heads
DoTask, UpdateSharedData;
mat:= List( mat, ShallowCopy );
nrows:= Length( mat );
ncols:= Length( mat[1] );
zero:= Zero( mat[1][1] );
heads:= ListWithIdenticalEntries( ncols, 0 );
nzheads := [];
vectors := [];
DoTask := function( i ) # taskInput = i
local j, # loop over columns
x, # a current element
row; # the row of current interest
row := mat[i];
# Reduce the row with the known basis vectors.
for j in [ 1 .. Length(nzheads) ] do
x := row[nzheads[j]];
if x <> zero then
AddRowVector( row, vectors[ j ], - x );
fi;
od;
j := PositionNot( row, zero );
if j <= ncols then return [j, row]; # return taskOutput
else return fail; fi;
end;
UpdateSharedData := function( i, taskOutput )
local j, row;
j := taskOutput[1];
row := taskOutput[2];
# We found a new basis vector.
MultRowVector(row, Inverse(row[j]));
Add( vectors, row );
Add( nzheads, j);
heads[j]:= Length( vectors );
end;
MasterSlave( TaskInputIterator( [1..nrows] ), DoTask, DefaultCheckTaskResult,
UpdateSharedData );
return rec( heads := heads,
vectors := vectors );
end );
The next section describes how to make this code more efficient.
5.6 Caching slave task outputs (ParSemiEchelonMat revisited)
The code above is inefficient unless nrows >> ncols. This is
because if nrows is comparable to ncols, it will be rare for
DoTask() to return fail. If most slaves return a result distinct
from fail, then DefaultCheckTaskResult() will return an
UPDATE_ACTION upon receiving the output from the first slave, and it
will return a REDO_ACTION to all other slaves, until those slaves
execute UpdateSharedData(). The inefficiency arose because a
REDO_ACTION caused the original slave process to re-compute
DoTask() from the beginning.
In the case of a REDO_ACTION, we can fix this by taking advantage of
information that was already computed. To accomplish this, a global
variable should be defined on all slaves:
ParEval("globalTaskOutput := [[-1]]");
the routine DoTask() in the previous example should be modified to:
DoTask := function( i )
local j, # loop over columns
x, # a current element
row; # the row of current interest
if i = globalTaskOutput[1] then
# then this is a REDO_ACTION
row := globalTaskOutput[2]; # recover last row value
else row := mat[i];
fi;
# Reduce the row with the known basis vectors.
for j in [ 1 .. Length(nzheads) ] do
x := row[nzheads[j]];
if x <> zero then
AddRowVector( row, vectors[ j ], - x );
fi;
od;
j := PositionNot( row, zero );
# save row in case of new REDO_ACTION
globalTaskOutupt[1] := i;
globalTaskOutput[2] := row;
if j <= ncols then return [j, row]; # return taskOutput
else return fail; fi;
end;
(A perceptive reader will have noticed that it was not necessary to
also save and restore row from globalTaskOutput, since this can
be found again based on the saved variable value i. However, the
additional cost is small, and it illustrates potentially greater
generality of the method.)
The next section describes how to make this code more efficient.
5.7 Agglomerating tasks for efficiency (ParSemiEchelonMat revisited again)
A more efficient parallelization would partition the matrix into sets
of adjacent rows, and send an entire set as a single taskInput.
This would minimize the communication overhead, since the network latency
varies only slowly with message sizxe, but linearly with the number
of messages. To minimize network latency, one adds an extra parameter
to MasterSlave() in order to bundle, perhaps, up to 5 tasks at a time.
MasterSlave( TaskInputIterator( [1..n] ), DoTask, DefaultCheckTaskResult,
UpdateSharedData, 5 );
Now the task input will be a list of the next 5 tasks returned by
GetTaskInput(), or in this case by TaskInputIterator( [1..nrows] ).
If fewer than 5 tasks are produced before NOTASK is returned, then
the task input will be correspondingly shorter. If the first input
task is NOTASK (yielding a list of tasks of length 0), then this
will be interpreted as a traditional NOTASK. The task output
corresponding to this task input is whatever the application routine,
DoTask() produces as task output. The routine DoTask() will be
unchanged, and MasterSlave() will arrange to repeatedly call
DoTask(), once for each input task and produce a list of task
outputs.
Hence, this new variation requires us to rewrite
UpdateSharedData() in the obvious manner, to handle a list of input
and output tasks. Here is one solution to patch the earlier code.
TaskAgglomIndex V
This global variable is provided for use inside DoTask(). It allows
the application code to inquire about the index of the input task in
the full list of tasks created when agglomTask is used. The variable
is most useful in the case of a REDO_ACTION or
CONTINUATION_ACTION(), as illustrated below.
ParEval("globalTaskOutput := [[-1]]");
ParEval("globalTaskOutputs := []");
#Shared Data: vectors (basis vectors), heads, mat (matrix)
#TaskInput: i (row index of matrix)
#TaskOutput: List of (1) j and (2) row i of matrix, mat, reduced by vectors
# j is the first non-zero element of row i
#Task: Compute reduced row i from mat, vectors, heads
#UpdateSharedData: Given i, j, reduced row i, add new basis vector
# to vectors and update heads[j] to point to it
ParInstallTOPCGlobalFunction( "ParSemiEchelonMat", function( mat )
local zero, # zero of the field of <mat>
nrows, # number of rows in <mat>
ncols, # number of columns in <mat>
vectors, # list of basis vectors
heads, # list of pivot positions in 'vectors'
i, # loop over rows
nzheads, # list of non-zero heads
DoTask, UpdateSharedDataWithAgglom;
mat:= List( mat, ShallowCopy );
nrows:= Length( mat );
ncols:= Length( mat[1] );
zero:= Zero( mat[1][1] );
heads:= ListWithIdenticalEntries( ncols, 0 );
nzheads := [];
vectors := [];
DoTask := function( i )
local j, # loop over columns
x, # a current element
row; # the row of current interest
if IsBound(globalTaskOutputs[TaskAgglomIndex])
and i = globalTaskOutputs[TaskAgglomIndex][1] then
# then this is a REDO_ACTION
row := globalTaskOutputs[TaskAgglomIndex][2][2]; # recover last row value
else row := mat[i];
fi;
# Reduce the row with the known basis vectors.
for j in [ 1 .. Length(nzheads) ] do
x := row[nzheads[j]];
if x <> zero then
AddRowVector( row, vectors[ j ], - x );
fi;
od;
j := PositionNot( row, zero );
# save [input, output] in case of new REDO_ACTION
globalTaskOutputs[TaskAgglomIndex] := [ i, [j, row] ];
if j <= ncols then return [j, row]; # return taskOutput
else return fail; fi;
end;
# This version of UpdateSharedData() expects a list of taskOutput's
UpdateSharedDataWithAgglom := function( listI, taskOutputs )
local j, row, idx, tmp;
for idx in [1..Length( taskOutputs )] do
j := taskOutputs[idx][1];
row := taskOutputs[idx][2];
if idx > 1 then
globalTaskOutputs[1] := [-1, [j, row] ];
tmp := DoTask( -1 ); # Trick DoTask() into a REDO_ACTION
if tmp <> fail then
j := tmp[1];
row := tmp[2];
fi;
fi;
# We found a new basis vector.
MultRowVector(row, Inverse(row[j]));
Add( vectors, row );
Add( nzheads, j);
heads[j]:= Length( vectors );
od;
end;
MasterSlave( TaskInputIterator( [1..nrows] ), DoTask, DefaultCheckTaskResult,
UpdateSharedDataWithAgglom, 5 ); #taskAgglom set to 5 tasks
return rec( heads := heads,
vectors := vectors );
end );
Note that in this simple example, we were able to re-use most of the
code from the previous version, at the cost of adding an additional
global variable, globalTaskOutputs. In fact, the last DoTask() is
backward compatible to the first version of the code, for which
agglomTasks is not used. If we wanted to run the latest code
without agglomeration of tasks, it would suffice either to set
the taskAgglom parameter to 1, or else to remove it entirely and
replace UpdateSharedDataWithAgglom() by UpdateSharedData().
It is useful to experiment with the above code by substituting a
variable, taskAgglom, for the number 5, and trying it out with
remote slaves on your own network for different values of taskAgglom
and for different size matrices. You can call MasterSlaveStats()
to see the effect of different parameters. Suitable pseudo-random matrices
can be quickly generated via mat := PseudoRandom( GL( 30, 5 ) and similar
commands.
The paper Coo98 is suggested as further reading to see
a still more efficient parallel implementation
of ParSemiEchelonMatrix (also known as Gaussian elimination)
using the TOP-C model.
5.8 Raw MasterSlave (ParMultMat revisited)
Finally, we given an example of a variation of MasterSlave(), based
on a ``raw'' MasterSlave(). These versions are designed for the
common case of legacy code that contains deeply nested parentheses.
The taskInput may be generated inside several nested loops, making
it awkward and error-prone to produce a function, SubmitTaskInput(),
that will generate instances of taskInput in the appropriate
sequence.
Effectively, when we wish to return successive values from several deeply
nested loops, we are in the situation of programming the ``opposite of a
GAP iterator'' (see iterators in the Reference Manual).
We are already producing successive iterator values, and we wish to
``stuff them back into some iterator''. Until GAP develops such a
language construct :-) , the following example of a ``raws''
MasterSlave() demonstrates a solution.
Before studying this example, please review the sequential version,
SeqMultMat() near the beginning of section ParMultMat.
BeginRawMasterSlave( DoTask, CheckTaskResult, UpdateSharedData ) F
RawSubmitTaskInput( taskInput ) F
EndRawMasterSlave() F
Their use will be obvious in the next example. This time, we
parallelize SeqMultMat() by defining the task as the computation of a
single entry in the result matrix. Hence, the task will be the
computation of the appropriate inner product. For dimension n,
n2 tasks are now generated, and each task is generated inside a
doubly nested loop.
#Shared Data: m1, m2t, result (three matrices)
#TaskInput: [i,j] (indices of entry in result matrix)
#TaskOutput: result[i][j] (value of entry in result matrix)
#Task: Compute inner produce of row i of m1 by colum j of m1
( Note that column j of m1 is also row j of m2t, the transpose )
#UpdateSharedData: Given result[i][j] and [i,j], modify result everywhere
ParInstallTOPCGlobalFunction( "ParRawMultMat", function(m1, m2)
local i, j, k, n, m2t, sum, result, DoTask, CheckTaskResult, UpdateSharedData;
n := Length(m1);
m2t := TransposedMat(m2);
result := ListWithIdenticalEntries( Length(m2t), [] );
DoTask := function( input )
local i,j,k,sum;
i:=input[1]; j:=input[2];
sum := 0;
for k in [1..n] do
sum := sum + m1[i][k]*m2t[j][k];
od;
return sum;
end;
CheckTaskResult := function( input, output )
return UPDATE_ACTION;
end;
UpdateSharedData := function( input, output )
local i, j;
i := input[1]; j := input[2];
result[i][j] := output;
# result[i,j] := sum;
end;
BeginRawMasterSlave( DoTask, CheckTaskResult, UpdateSharedData );
for i in [1..n] do
result[i] := [];
for j in [1..n] do
RawSubmitTaskInput( [i,j] );
# sum := 0;
# for k in [1..n] do
# sum := sum + m1[i][k]*m2t[j][k];
# od;
# result[i][j] := sum;
od;
od;
EndRawMasterSlave();
return result;
end );
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