Evaluation Strategies, or simply Strategies, are a means for modularizing parallel code by separating the algorithm from the parallelism. Sometimes they require you to rewrite your algorithm, but once you do so, you will be able to parallelize it in different ways just by substituting a new Strategy.
Concretely, a Strategy
is a function in the Eval
monad that takes a
value of type a
and returns the same value:
type
Strategy
a
=
a
>
Eval
a
The idea is that a Strategy takes a data structure as input, traverses
the structure creating parallelism with rpar
and rseq
, and then
returns the original value.
Here’s a simple example: Let’s create a Strategy for pairs that
evaluates the two components of the pair in parallel. We want a
function parPair
with the following type:
parPair
::
Strategy
(
a
,
b
)
From the definition of the Strategy
type previously shown, we know that this
type is the same as (a,b) > Eval (a,b)
. So parPair
is a function
that takes a pair, does some computation in the Eval
monad, and
returns the pair again. Here is its definition:
parPair
::
Strategy
(
a
,
b
)
parPair
(
a
,
b
)
=
do
a'
<
rpar
a
b'
<
rpar
b
return
(
a'
,
b'
)
This is similar to the rpar/rpar pattern that we saw in
The Eval Monad, rpar, and rseq. The difference is that we’ve packaged it up
as a Strategy: It takes a data structure (in this case a pair),
creates some parallelism using rpar
, and then returns the same data structure.
We’ll see this in action in a moment, but first we need to know how to
use a Strategy. Using a Strategy consists of applying it to its
input and running the Eval
computation to get the output. We could
write that directly with runEval
; for example, to evaluate the pair
(fib 35, fib 36)
in parallel, we could write:
runEval
(
parPair
(
fib
35
,
fib
36
))
This works just fine, but it turns out to be much nicer to package up
the application of a Strategy into a function named using
:
using
::
a
>
Strategy
a
>
a
x
`
using
`
s
=
runEval
(
s
x
)
The using
function takes a value of type a
and a Strategy for a
,
and applies the Strategy to the value. We normally write using
infix, as its definition suggests. Here is the parPair
example
above rewritten with using
:
(
fib
35
,
fib
36
)
`
using
`
parPair
Why write it this way? Well, a Strategy returns the same value that it was passed, so we know that aside from its performance, the above code is equivalent to just:
(
fib
35
,
fib
36
)
So we’ve clearly separated the code that describes what the program
does (the pair) from the code that adds the parallelism (`using`
parPair
). Indeed, everywhere we see x `using` s
in our program,
we can delete the `using` s
part and the program should produce the
same result.^{[6]} Conversely, someone who is interested in
parallelizing the program can focus on modifying the Strategy without
worrying about breaking the program.
The example program strat.hs contains the parPair
example just shown;
try running it yourself with one and two processors to see it compute
the two calls to fib
in parallel.
The parPair
Strategy embodies a fixed policy: It always evaluates
the components of the pair in parallel, and always to weak head normal
form. If we wanted to do something different with a pair—fully
evaluate the components to normal form, for example—we would have to
write a completely new Strategy. A better way to factor things is to
write a parameterized Strategy, which takes as arguments the
Strategies to apply to the components of the data structure. Here is
a parameterized Strategy for pairs:
evalPair
::
Strategy
a
>
Strategy
b
>
Strategy
(
a
,
b
)
evalPair
sa
sb
(
a
,
b
)
=
do
a'
<
sa
a
b'
<
sb
b
return
(
a'
,
b'
)
This Strategy no longer has parallelism built in, so I’ve called it
evalPair
instead of parPair
.^{[7]} It takes
two Strategy
arguments, sa
and sb
, applies them to the
respective components of the pair, and then returns the pair.
Compared with parPair
, we are passing in the functions to apply to a
and b
instead of making fixed calls to rpar
. So to define
parPair
in terms of evalPair
, we can just pass rpar
as the
arguments:
parPair
::
Strategy
(
a
,
b
)
parPair
=
evalPair
rpar
rpar
This means we’re using rpar
itself as a Strategy
:
rpar
::
Strategy
a
The type of rpar
is a > Eval a
, which is equivalent to Strategy
a
; rpar
is therefore a Strategy for any type, with the effect of
starting the evaluation of its argument while the enclosing Eval
computation proceeds in parallel. (The rseq
operation is also a
Strategy
.)
But parPair
is still restrictive, in that the components of the pair
are always evaluated to weak head normal form. What if we wanted to
fully evaluate the components using force
, for example? We can make
a Strategy
that fully evaluates its argument:
rdeepseq
::
NFData
a
=>
Strategy
a
rdeepseq
x
=
rseq
(
force
x
)
But how do we combine rpar
with rdeepseq
to give us a single
Strategy that fully evaluates its argument in parallel? We need one
further combinator, which is provided by Control.Parallel.Strategies
:
rparWith
::
Strategy
a
>
Strategy
a
Think of rparWith s
as wrapping the Strategy s
in an rpar
.
Now we can provide a parameterized version of parPair
that
takes the Strategies to apply to the components:
parPair
::
Strategy
a
>
Strategy
b
>
Strategy
(
a
,
b
)
parPair
sa
sb
=
evalPair
(
rparWith
sa
)
(
rparWith
sb
)
And we can use parPair
to write a Strategy
that fully evaluates
both components of a pair in parallel:
parPair
rdeepseq
rdeepseq
::
(
NFData
a
,
NFData
b
)
=>
Strategy
(
a
,
b
)
To break down what happens when this Strategy
is applied to a pair:
parPair
calls evalPair
, and evalPair
calls rparWith rdeepseq
on each component of the pair. So the effect is that each component
will be fully evaluated to normal form in parallel.
When using these parameterized Strategies, we sometimes need a way
to say, “Don’t evaluate this component at all.” The Strategy
that
does no evaluation is called r0
:
r0
::
Strategy
a
r0
x
=
return
x
For example, we can write a Strategy
over a pair of pairs that
evaluates the first component (only) of both pairs in parallel.
evalPair
(
evalPair
rpar
r0
)
(
evalPair
rpar
r0
)
::
Strategy
((
a
,
b
),(
c
,
d
))
The first rpar
applies to a
and the first r0
to b
, while the
second rpar
applies to c
and the second r0
to d
.
In Chapter 2, we defined a function parMap
that would map a
function over a list in parallel. We can think of parMap
as a
composition of two parts:
And indeed, with Strategies, we can express it exactly this way:
parMap
::
(
a
>
b
)
>
[
a
]
>
[
b
]
parMap
f
xs
=
map
f
xs
`
using
`
parList
rseq
The parList
function is a Strategy on lists that evaluates the list
elements in parallel. To define parList
, we can take the same approach
that we took with pairs earlier and first define a parameterized
Strategy on lists, called evalList
:
evalList
::
Strategy
a
>
Strategy
[
a
]
evalList
strat
[]
=
return
[]
evalList
strat
(
x
:
xs
)
=
do
x'
<
strat
x
xs'
<
evalList
strat
xs
return
(
x'
:
xs'
)
Note that evalList
walks the list recursively, applying the Strategy
parameter strat
to each of the elements and building the result
list. Now we can define parList
in terms of evalList
, using
rparWith
:
parList
::
Strategy
a
>
Strategy
[
a
]
parList
strat
=
evalList
(
rparWith
strat
)
In fact, both evalList
and parList
are already provided by Control.Parallel
.Strategies
so you don’t have to define them yourself,
but it’s useful to see that their implementations are not mysterious.
As with parPair
, the parList
function is a parameterized Strategy. That is, it takes as an argument a Strategy on values of type a
and
returns a Strategy for lists of a
. So parList
describes a
family of Strategies on lists that evaluate the list elements in
parallel.
The parList
Strategy covers a wide range of uses for parallelism in typical Haskell programs; in many cases, a single parList
is all
that is needed to expose plenty of parallelism.
Returning to our Sudoku solver from Chapter 2 for a moment:
instead of our own handwritten parMap
, we could have used
parList
:
let
solutions
=
map
solve
puzzles
`
using
`
parList
rseq
Using rseq
as the Strategy for the list elements is enough here: The
result of solve
is a Maybe
, so evaluating it to
weak head normal form forces the solver to determine whether the puzzle has a solution.
This version has essentially the same performance as the version that used parMap
in Chapter 2.
Let’s look at a slightly more involved example. In the KMeans problem, the goal is to partition a set of data points into clusters. Figure 31 shows an example data set, and the circles indicate the locations of the clusters that the algorithm should derive. From the locations of the clusters, partitioning the points is achieved by simply finding the closest cluster to each point.
Finding an optimal solution to the problem is too expensive to be practical. However, there are several heuristic techniques that are fast, and even though they don’t guarantee an optimal solution, in practice, they give good results. The most wellknown heuristic technique for KMeans is Lloyd’s algorithm, which finds a solution by iteratively improving an initial guess. The algorithm takes as a parameter the number of clusters to find and makes an initial guess at the center of each cluster. Then it proceeds as follows:
The initial guess can be constructed by randomly assigning each point in the data set to a cluster and then finding the centroids of those clusters.
The algorithm works in any number of dimensions, but we will use two for ease of visualization.
A complete Haskell implementation can be found in the directory kmeans in the sample code.
A data point is represented by the type Point
, which is just a pair
of Double
s representing the x
and y
coordinates
respectively:^{[8]}
data
Point
=
Point
!
Double
!
Double
There are a couple of basic operations on Point
:
zeroPoint
::
Point
zeroPoint
=
Point
0
0
sqDistance
::
Point
>
Point
>
Double
sqDistance
(
Point
x1
y1
)
(
Point
x2
y2
)
=
((
x1

x2
)
^
2
)
+
((
y1

y2
)
^
2
)
We can make a zero point with zeroPoint
, and find the square of the distance between two points
with sqDistance
. The actual distance between the points would be
given by the square root of this value, but since we will only be
comparing distances, we can save time by comparing squared distances
instead.
Clusters are represented by the type Cluster
:
data
Cluster
=
Cluster
{
clId
::
Int
,
clCent
::
Point
}
A Cluster
contains its number (clId
) and its centroid (clCent
).
We will also need an intermediate type called PointSum
:
data
PointSum
=
PointSum
!
Int
!
Double
!
Double
A PointSum
represents the sum of a set of points; it contains the
number of points in the set and the sum of their x
and y
coordinates respectively. A PointSum
is constructed
incrementally, by repeatedly adding points using addToPointSum
:
addToPointSum
::
PointSum
>
Point
>
PointSum
addToPointSum
(
PointSum
count
xs
ys
)
(
Point
x
y
)
=
PointSum
(
count
+
1
)
(
xs
+
x
)
(
ys
+
y
)
A PointSum
can be turned into a Cluster
by computing the centroid.
The x
coordinate of the centroid is the sum of the x
coordinates
of the points in the cluster divided by the total number of points,
and similarly for the y
coordinate.
pointSumToCluster
::
Int
>
PointSum
>
Cluster
pointSumToCluster
i
(
PointSum
count
xs
ys
)
=
Cluster
{
clId
=
i
,
clCent
=
Point
(
xs
/
fromIntegral
count
)
(
ys
/
fromIntegral
count
)
}
The roles of the types Point
, PointSum
, and Cluster
in the
algorithm are as follows. The input is a set of points represented as
[Point]
, and an initial guess represented as [Cluster]
. The
algorithm will iteratively refine the clusters until convergence is
reached.
Cluster
to
which each Point
is closest. However, instead of collecting sets of
Point
s, we build up a PointSum
for each cluster. This is an
optimization that avoids constructing the intermediate data
structure and allows the algorithm to run in constant space. We’ll
represent the output of this step as Vector PointSum
.
Vector PointSum
is fed into step 2, which makes a Cluster
from each PointSum
, giving [Cluster]
.
The function assign
implements step 1 of the algorithm, assigning
points to clusters and building a vector of PointSum
s:
assign
::
Int
>
[
Cluster
]
>
[
Point
]
>
Vector
PointSum
assign
nclusters
clusters
points
=
Vector
.
create
$
do
vec
<
MVector
.
replicate
nclusters
(
PointSum
0
0
0
)
let
addpoint
p
=
do
let
c
=
nearest
p
;
cid
=
clId
c
ps
<
MVector
.
read
vec
cid
MVector
.
write
vec
cid
$!
addToPointSum
ps
p
mapM_
addpoint
points
return
vec
where
nearest
p
=
fst
$
minimumBy
(
compare
`
on
`
snd
)
[
(
c
,
sqDistance
(
clCent
c
)
p
)

c
<
clusters
]
Given a set of clusters and a set of points, the job of assign
is to
decide, for each point, which cluster is closest. For each cluster, we
build up a PointSum
of the points that were found to be closest to
it. The code has been carefully optimized, using mutable vectors from
the vector
package; the details aren’t important here.
The function makeNewClusters
implements step 2 of the algorithm:
makeNewClusters
::
Vector
PointSum
>
[
Cluster
]
makeNewClusters
vec
=
[
pointSumToCluster
i
ps

(
i
,
ps
@
(
PointSum
count
_
_
))
<
zip
[
0
..
]
(
Vector
.
toList
vec
)
,
count
>
0
]
Here we make a new Cluster
, using pointSumToCluster
, from each
PointSum
produced by assign
. There is a slight complication in
that we have to avoid creating a cluster with no points, because it
cannot have a centroid.
Finally step
combines assign
and makeNewClusters
to implement
one complete iteration:
step
::
Int
>
[
Cluster
]
>
[
Point
]
>
[
Cluster
]
step
nclusters
clusters
points
=
makeNewClusters
(
assign
nclusters
clusters
points
)
To complete the algorithm, we need a loop to repeatedly apply the step
function until convergence. The function kmeans_seq
implements this:
kmeans_seq
::
Int
>
[
Point
]
>
[
Cluster
]
>
IO
[
Cluster
]
kmeans_seq
nclusters
points
clusters
=
let
loop
::
Int
>
[
Cluster
]
>
IO
[
Cluster
]
loop
n
clusters

n
>
tooMany
=
do

putStrLn
"giving up."
return
clusters
loop
n
clusters
=
do
printf
"iteration %d
\n
"
n
putStr
(
unlines
(
map
show
clusters
))
let
clusters'
=
step
nclusters
clusters
points

if
clusters'
==
clusters

then
return
clusters
else
loop
(
n
+
1
)
clusters'
in
loop
0
clusters
tooMany
=
80
The first argument to loop
is the number of iterations completed
so far. If this figure reaches the limit tooMany
, then we bail
out (sometimes the algorithm does not converge).
After printing the iteration number and the current clusters
for diagnostic purposes, we calculate the next iteration by calling
the function step
. The arguments to step
are the number of
clusters, the current set of clusters, and the set of points.
If this iteration did not change the clusters, then the algorithm has converged, and we return the result. Otherwise, we do another iteration.
We compile this program in the same way as before:
$ cd kmeans $ ghc O2 threaded rtsopts eventlog kmeans.hs
The sample code comes with a program to generate some input data,
GenSamples.hs, which uses the normaldistribution
package to
generate a realistically clustered set of values. The data set is
large, so it isn’t included with the sample code, but you can generate
it using GenSamples
:
$ ghc O2 GenSamples.hs $ ./GenSamples 5 50000 100000 1010
This should generate a data set of about 340,000 points with 5
clusters in the file points.bin
.
Run the kmeans
program using the sequential algorithm:
$ ./kmeans seq
The program will display the clusters at each iteration and should converge after 65 iterations.
Note that the program displays its own running time at the end; this is because there is a significant amount of time spent reading in the sample data at the beginning, and we want to be able to calculate the parallel speedup for the portion of the runtime spent computing the KMeans algorithm only.
How can this algorithm be parallelized? One place that looks
profitable to parallelize is the assign
function because it is
essentially just a map
over the points, and indeed that is where we
will concentrate our efforts. The operations are too finegrained
here to use a simple parMap
or parList
as we did before; the
overhead of the parMap
will swamp the parallelism, so we need to
increase the size of the operations. One way to do that is to divide
the list of points into chunks, and process the chunks in parallel.
First we need some code to split a list into chunks:
split
::
Int
>
[
a
]
>
[[
a
]]
split
numChunks
xs
=
chunk
(
length
xs
`
quot
`
numChunks
)
xs
chunk
::
Int
>
[
a
]
>
[[
a
]]
chunk
n
[]
=
[]
chunk
n
xs
=
as
:
chunk
n
bs
where
(
as
,
bs
)
=
splitAt
n
xs
So we can split the list of points into chunks and map assign
over
the list of chunks. But what do we do with the results? We have a
list of Vector PointSum
s that we need to combine into a single
Vector PointSum
. Fortunately, PointSum
s can be added together:
addPointSums
::
PointSum
>
PointSum
>
PointSum
addPointSums
(
PointSum
c1
x1
y1
)
(
PointSum
c2
x2
y2
)
=
PointSum
(
c1
+
c2
)
(
x1
+
x2
)
(
y1
+
y2
)
And using this, we can combine vectors of PointSum
s:
combine
::
Vector
PointSum
>
Vector
PointSum
>
Vector
PointSum
combine
=
Vector
.
zipWith
addPointSums
We now have all the pieces to define a parallel version of step
:
parSteps_strat
::
Int
>
[
Cluster
]
>
[[
Point
]]
>
[
Cluster
]
parSteps_strat
nclusters
clusters
pointss
=
makeNewClusters
$
foldr1
combine
$
(
map
(
assign
nclusters
clusters
)
pointss
`
using
`
parList
rseq
)
The arguments to parSteps_strat
are the same as for step
, except
that the list of points is now a list of lists of points, that is, the
list of points divided into chunks by split
. We want to pass in the chunked data rather than call split
inside parSteps_strat
so
that we can do the chunking of the input data just once instead of
repeating it for each iteration.
The kmeans_strat
function below is our parallel version of
kmeans_seq
, the only differences being that we call split
to
divide the list of points into chunks () and we call parSteps_strat
instead of steps
():
kmeans_strat
::
Int
>
Int
>
[
Point
]
>
[
Cluster
]
>
IO
[
Cluster
]
kmeans_strat
numChunks
nclusters
points
clusters
=
let
chunks
=
split
numChunks
points

loop
::
Int
>
[
Cluster
]
>
IO
[
Cluster
]
loop
n
clusters

n
>
tooMany
=
do
printf
"giving up."
return
clusters
loop
n
clusters
=
do
printf
"iteration %d
\n
"
n
putStr
(
unlines
(
map
show
clusters
))
let
clusters'
=
parSteps_strat
nclusters
clusters
chunks

if
clusters'
==
clusters
then
return
clusters
else
loop
(
n
+
1
)
clusters'
in
loop
0
clusters
Note that the number of chunks doesn’t have to be related to the number of processors; as we saw earlier, it is better to produce plenty of sparks and let the runtime schedule them automatically, because this should enable the program to scale over a wide range of processors.
Next we’re going on an exploration of the performance of this parallel program. Along the way, we’ll learn several lessons about the kinds of things that can go wrong when parallelizing Haskell code, how to look out for them using ThreadScope, and how to fix them.
We’ll start by taking some measurements of the speedup for various numbers of cores. When running the program in parallel, we get to choose the number of chunks to divide the input into, and for these measurements I’ll use 64 (but we’ll revisit this in Granularity). The program is run in parallel like this:
$ ./kmeans strat 64 +RTS N2
strat
indicates that we want to use the Strategies version of
the algorithm, and 64 is the number of chunks to divide the input data
into. Here, I’m telling the GHC runtime to use two cores.
Here are the speedup results I get on my computer for the kmeans
program I showed
earlier.^{[9]} For each
measurement, I ran the program a few times and took the average
runtime.^{[10]}
We can see that speedup is quite good for two to three cores but starts to drop off at four cores. Still, a 2.6 speedup on 4 cores is reasonably respectable.
The ThreadScope profile gives us some clues about why the speedup might be less than we hope. The overall view of the fourcore run can be seen in Figure 32.
We can clearly see the sequential section at the start, where the program reads in the input data. But that isn’t a problem; remember that the program emits its own timing results, which begin at the parallel part of the run. The parallel section itself looks quite good; all cores seem to be running for the duration. Let’s zoom in on the beginning of the parallel section, as shown in Figure 33.
There’s a segment between 0.78s and 0.8s where, although parallel
execution has started, there is heavy GC activity. This is similar to
what we saw in Example: Parallelizing a Sudoku Solver, where the work of splitting
the input data into lines was overlapped with the parallel execution.
In the case of kmeans
, the act of splitting the data set into chunks
is causing the extra work.
The sequential version of the algorithm doesn’t need to split the data into
chunks, so chunking is a source of extra overhead in the parallel
version. This is one reason that we aren’t achieving full speedup.
If you’re feeling adventurous, you might want to see whether you can
avoid this chunking overhead by using Vector
instead of a list to
represent the data set, because Vector
s can be sliced in O(1)
time.
Let’s look at the rest of the parallel section in more detail (see Figure 34).
The parallel execution, which at first looked quite uniform, actually
consists of a series of humps when we zoom in. Remember that the
algorithm performs a series of iterations over the data set—these
humps in the profile correspond to the iterations. Each iteration is a separate
parallel segment, and between the iterations lies some sequential
execution. We expect a small amount of sequential execution
corresponding to makeNewClusters
, combine
, and the comparison
between the new and old clusters in the outer loop.
Let’s see whether the reality matches our expectations by zooming in on one of the gaps to see more clearly what happens between iterations (Figure 35).
There’s quite a lot going on here. We can see the parallel execution of the previous iteration tailing off, as a couple of cores run longer than the others. Following this, there is some sequential execution on HEC 3 before the next iteration starts up in parallel.
Looking more closely at the sequential bit on HEC 3, we can see some gaps where nothing appears to be happening at all. In the ThreadScope GUI, we can show the detailed events emitted by the RTS (look for the “Raw Events” tab in the lower pane), and if we look at the events for this section, we see:
0.851404792s HEC 3: stopping thread 4 (making a foreign call) 0.851405771s HEC 3: running thread 4 0.851406373s HEC 3: stopping thread 4 (making a foreign call) 0.851419669s HEC 3: running thread 4 0.851451713s HEC 3: stopping thread 4 (making a foreign call) 0.851452171s HEC 3: running thread 4 ...
The program is popping out to make several foreign calls during this
period. ThreadScope doesn’t tell us any more than this, but it’s
enough of a clue: A foreign call usually indicates some kind of I/O,
which should remind us to look back at what happens between iterations in the
kmeans_seq
function:
loop
n
clusters
=
do
printf
"iteration %d
\n
"
n
putStr
(
unlines
(
map
show
clusters
))
...
We’re printing some output. Furthermore, we’re doing this in the sequential part of the program, and Amdahl’s law is making us pay for it in parallel speedup.
Commenting out these two lines (in both kmeans_seq
and
kmeans_strat
, to be fair) improves the parallel speedup from 2.6 to
3.4 on my quadcore machine. It’s amazing how easy it is to make a
small mistake like this in parallel programming, but fortunately
ThreadScope helps us identify the problem, or at least gives us clues
about where we should look.
We can also use ThreadScope to visualize the creation and use of sparks
during the run of the program. Figure 36 shows the profile for
kmeans
running on four cores, showing the spark pool size over time for
each HEC (these graphs are enabled in the ThreadScope GUI from the
“Traces” tab in the left pane).
The figure clearly shows that as each iteration starts, 64 sparks are created on one HEC and then are gradually consumed. What is perhaps surprising is that the sparks aren’t always generated on the same HEC; this is the GHC runtime moving work behind the scenes as it tries to keep the load balanced across the cores.
There are more sparkrelated graphs available in ThreadScope, showing the rates of spark creation and conversion (running sparks). All of these can be valuable in understanding the performance characteristics of your parallel program.
Looking back at Figure 35, I remarked earlier that the parallel section didn’t finish evenly, with two cores running a bit longer than the others. Ideally, we would have all the cores running until the end to maximize our speedup.
As we saw in Example: Parallelizing a Sudoku Solver, having too few work items in our parallel program can impact the speedup, because the work items can vary in cost. To get a more even run, we want to create finegrained work items and more of them.
To see the effect of this, I ran kmeans
with various numbers of
chunks from 4 up to 512, and measured the runtime on 4 cores. The
results are shown in Figure 37.
We can see not only that having too few chunks is not good for the reasons given above, but also having too many can have a severe impact. In this case, the sweet spot is somewhere around 50100.
Why does having too many chunks increase the runtime? There are two reasons:
Recall the definition of parList
:
parList
::
Strategy
a
>
Strategy
[
a
]
parList
strat
=
evalList
(
rparWith
strat
)
And the underlying parameterized Strategy
on lists, evalList
:
evalList
::
Strategy
a
>
Strategy
[
a
]
evalList
strat
[]
=
return
[]
evalList
strat
(
x
:
xs
)
=
do
x'
<
strat
x
xs'
<
evalList
strat
xs
return
(
x'
:
xs'
)
As evalList
traverses the list applying the strategy strat
to the
list elements, it remembers each value returned by strat
(bound to
x'
), and constructs a new list from these values. Why? Well, one
answer is that a Strategy must return a data structure equal to the
one it was passed.
But do we really need to build a new list? After all, this means that
evalList
is not tailrecursive; the recursive call to
evalList
is not the last operation in the do
on its righthand
side, so evalList
requires stack space linear in the length of
the input list.
Couldn’t we just write a tailrecursive version of parList
instead?
Perhaps like this:
parList
::
Strategy
a
>
Strategy
[
a
]
parList
strat
xs
=
do
go
xs
return
xs
where
go
[]
=
return
()
go
(
x
:
xs
)
=
do
rparWith
strat
x
go
xs
After all, this is typecorrect and seems to call rparWith
on each
list element as required.
Unfortunately, this version of parList
has a serious problem: All the
parallelism it creates will be discarded by the garbage
collector. The omission of the result list turns out to be
crucial. Let’s take a look at the data structures that our original,
correct implementations of parList
and evalList
created
(Figure 38).
At the top of the diagram is the input list xs
: a linked list
of cells, each of which points to a list element (x1
, x2
, and so
forth). At the bottom of the diagram is the spark pool, the runtime
system data structure that stores references to sparks in the heap.
The other structures in the diagram are built by parList
(the
correct version, not the one I most recently showed). Each strat
box represents the
strategy strat
applied to an element of the original list, and
xs'
is the linked list of cells in the output list. The spark pool
contains pointers to each of the strat
boxes; these are the pointers
created by each call to rparWith
.
The GHC runtime regularly checks the spark pool for any entries that are not required by the program and removes them. It would be bad to retain entries that aren’t needed, because that could cause the program to hold on to memory unnecessarily, leading to a space leak. We don’t want parallelism to have a negative impact on performance.
How does the runtime know whether an entry is needed? The same way it
knows whether any item in memory is needed: There must be a pointer to
it from something else that is needed. This is the reason that parList
creates a new list xs'
. Suppose we did not build the new
list xs'
, as in the tailrecursive version of parList
above.
Then the only reference to each strat
box in the heap would be from
the spark pool, and hence the runtime would automatically sweep all
those references from the spark pool, discarding the parallelism.
So we build a new list xs'
to hold references to the strat
calls
that we need to retain.
The automatic discarding of unreferenced sparks has another benefit besides avoiding space leaks; suppose that under some circumstances the program does not need the entire list. If the program simply forgets the unused remainder of the list, the runtime system will clean up the unreferenced sparks from the spark pool and will not waste any further parallel processing resources on evaluating those sparks. The extra parallelism in this case is termed speculative, because it is not necessarily required, and the runtime will automatically discard speculative tasks that it can prove will never be required—a useful property!
Although the runtime system’s discarding of unreferenced sparks is
certainly useful in some cases, it can be tricky to work with because
there is no languagelevel support for catching mistakes. Fortunately,
the runtime system will tell us if it garbagecollects unreferenced
sparks. For example, if you use the tailrecursive parList
with the
Sudoku solver from Chapter 2, the +RTS s
stats will show
something like this:
SPARKS: 1000 (2 converted, 0 overflowed, 0 dud, 998 GC'd, 0 fizzled)
Garbagecollected sparks are reported as “GC’d.” ThreadScope will also indicate GC’d sparks in its spark graphs.
If you see that a large number of sparks are GC’d, it’s a good indication that sparks are being removed from the spark pool before they can be used for parallelism. Unless you are using speculation, a nonzero figure for GC’d sparks is probably a bad sign.
All the combinators in the Control.Parallel.Strategies
libraries
retain references to sparks correctly. These are the rules of thumb for not
shooting yourself in the foot:
using
to apply Strategies instead of runEval
; it encourages the right pattern, in which the program uses the results of applying the Strategy.
When writing your own Eval
monad code, this is wrong:
do
...
rpar
(
f
x
)
...
Equivalently, using rparWith
without binding the result is wrong.
However, this is OK:
do
...
y
<
rpar
(
f
x
)
...
y
...
And this might be OK, as long as y
is required by the program somewhere:
do
...
rpar
y
...
A common pattern in Haskell programming is to use a lazy list as a stream so that the program can consume input while simultaneously producing output and consequently run in constant space. Such programs present something of a challenge for parallelism; if we aren’t careful, parallelizing the computation will destroy the lazy streaming property and the program will require space linear in the size of the input.
To demonstrate this, we will use the sample program rsa.hs,
an implementation of RSA encryption and decryption. The program
takes two command line arguments: the first specifies which action to
take, encrypt
or decrypt
, and the second is either the filename of
the file to read, or the character 
to read from stdin
. The
output is always produced on stdout
.
The following example uses the program to encrypt the message "Hello World!"
:
$ echo 'Hello World!'  ./rsa encrypt  11656463941851871045300458781178110195032310900426966299882646602337646308966290 04616367852931838847898165226788260038683620100405280790394258940505884384435202 74975036125752600761230510342589852431747
And we can test that the program successfully decrypts the output,
producing the original text, by piping the output back into rsa
decrypt
:
$ echo "Hello World!"  ./rsa encrypt   ./rsa decrypt  Hello World!
The rsa
program is a stream transformer, consuming input and
producing output lazily. We can see this by looking at the RTS stats:
$ ./rsa encrypt /usr/share/dict/words >/dev/null +RTS s 8,040,128,392 bytes allocated in the heap 66,756,936 bytes copied during GC 186,992 bytes maximum residency (71 sample(s)) 36,584 bytes maximum slop 2 MB total memory in use (0 MB lost due to fragmentation)
The /usr/share/dict/words file is about 1 MB in size, but the program has a maximum residency (live memory) of 186,992 bytes.
Let’s try to parallelize the program. The program uses the lazy
ByteString
type from Data.ByteString.Lazy
to achieve streaming,
and the toplevel encrypt
function has this type:
encrypt
::
Integer
>
Integer
>
ByteString
>
ByteString
The two Integer
s are the key with which to encrypt the data. The
implementation of encrypt
is a beautiful pipeline composition:
encrypt
n
e
=
B
.
unlines

.
map
(
B
.
pack
.
show
.
power
e
n
.
code
)

.
chunk
(
size
n
)

We won’t delve into the details of the RSA implementation here, but if
you’re interested, go and look at the code in rsa.hs (it’s fairly
short). For the purposes of parallelism, all we need to know is that
there’s a map
on the second line, so that’s our target for
parallelization.
First, let’s try to use the parList
Strategy that we have seen before:
encrypt
n
e
=
B
.
unlines
.
withStrategy
(
parList
rdeepseq
)

.
map
(
B
.
pack
.
show
.
power
e
n
.
code
)
.
chunk
(
size
n
)
I’m using withStrategy
here, which is just a version of using
with the arguments flipped; it is slightly nicer in situations like
this. The Strategy is parList
, with rdeepseq
as the Strategy to
apply to the list elements (the list elements are lazy ByteStrings
, so we
want to ensure that they are fully evaluated).
If we run this program on four cores, the stats show something interesting:
6,251,537,576 bytes allocated in the heap 44,392,808 bytes copied during GC 2,415,240 bytes maximum residency (33 sample(s)) 550,264 bytes maximum slop 10 MB total memory in use (0 MB lost due to fragmentation)
The maximum residency has increased to 2.3 MB, because the parList
Strategy forces the whole spine of the list, preventing the program
from streaming in constant space. The speedup in this case was 2.2;
not terrible, but not great either. We can do better.
The Control.Parallel.Strategies
library provides a Strategy to solve
exactly this problem, called parBuffer
:
parBuffer
::
Int
>
Strategy
a
>
Strategy
[
a
]
The parBuffer
function has a similar type to parList
but takes an Int
argument as a buffer size. In contrast to parList
which eagerly
creates a spark for every list element, parBuffer
N creates
sparks for only the first N elements of the list, and then creates
more sparks as the result list is consumed. The effect is that there
will always be N sparks available until the end of the list is
reached.
The disadvantage of parBuffer
is that we have to choose a particular
value for the buffer size, and as with the chunk factor we saw
earlier, there will be a “best value” somewhere in the range.
Fortunately, performance is usually not too sensitive to this value,
and something in the range of 50500 is often good. So let’s see how
well this works:
encrypt
n
e
=
B
.
unlines
.
withStrategy
(
parBuffer
100
rdeepseq
)

.
map
(
B
.
pack
.
show
.
power
e
n
.
code
)
.
chunk
(
size
n
)
This programs achieves a speedup of 3.5 on 4 cores. Furthermore, it
runs in much less memory than the parList
version:
6,275,891,072 bytes allocated in the heap 27,749,720 bytes copied during GC 294,872 bytes maximum residency (58 sample(s)) 62,456 bytes maximum slop 4 MB total memory in use (0 MB lost due to fragmentation)
We can expect it to need more memory than the sequential version, which required only 2 MB, because we’re performing many computations in parallel. Indeed, a higher residency is common in parallel programs for the simple reason that they are doing more work, although it’s not always the case; sometimes parallel evaluation can reduce memory overhead by evaluating thunks that were causing space leaks.
ThreadScope’s spark pool graph shows that parBuffer
really does keep
a constant supply of sparks, as shown in Figure 39.
The spark pool on HEC 0 constantly hovers around 90100 sparks.
In programs with a multistage pipeline, interposing more calls to
withStrategy
in the pipeline can expose more parallelism.
When parallelizing KMeans in Parallelizing KMeans, we divided
the input data into chunks to avoid creating parallelism with
excessively fine granularity. Chunking is a common technique, so the
Control.Parallel.Strategies
library provides a version of parList
that has chunking built in:
parListChunk
::
Int
>
Strategy
a
>
Strategy
[
a
]
The first argument is the number of elements in each chunk; the list
is split in the same way as the chunk
function that we saw earlier
in the kmeans
example. You might find parListChunk
useful if you
have a list with too many elements to spark every one, or when the
list elements are too cheap to warrant a spark each.
The spark pool has a fixed size, and when the pool is full, subsequent
sparks are dropped and reported as overflowed
in the +RTS s
stats output. If you see some overflowed sparks, it is probably a
good idea to create fewer sparks; replacing parList
with
parListChunk
is a good way to do that.
Note that chunking the list incurs some overhead, as we noticed in the
earlier kmeans
example when we used chunking directly. For that
reason, in kmeans
we created the chunked list once and shared it
amongst all the iterations of the algorithm, rather than using
parListChunk
, which would chunk the list every time.
I mentioned at the beginning of this chapter that if we see an expression of this form:
x
`
using
`
s
We can delete `using` s
, leaving an equivalent program. For this to be true, the Strategy s
must obey the identity property; that is, the value it returns must be equal to the value it was passed. The operations provided by the Control.Parallel
.Strategies
library all satisfy this property, but unfortunately it isn’t possible to enforce it for arbitrary userdefined Strategies. Hence we cannot guarantee that x `using` s == x
, just as we cannot guarantee that all instances of Monad
satisfy the monad laws, or that all instances of Eq
are reflexive. These properties are satisfied by convention only; this is just something to be aware of.
There is one more caveat to this property. The
expression x `using` s
might be less defined than x
, because it
evaluates more structure of x
than the context does. What does
less defined mean? It means that the program containing x `using` s
might fail with an error when simply x
would not. A trivial
example of this is:
$
snd
(
1
`
div
`
0
,
"Hello!"
)
This program works and prints "Hello!"
, but:
$
snd
((
1
`
div
`
0
,
"Hello!"
)
`
using
`
rdeepseq
)
This program fails with divide by zero
. The original program didn’t
fail because the erroneous expression was never evaluated, but adding
the Strategy has caused the program to fully evaluate the pair, including
the division by zero.
This is rarely a problem in practice; if the Strategy evaluates more than the program would have done anyway, the Strategy is probably wasting effort and needs to be modified.
^{[6] }This comes with a couple of minor caveats that we’ll describe in The Identity Property.
^{[7] }The evalPair
function is
provided by Control.Parallel.Strategies
as evalTuple2
.
^{[8] }The actual implementation adds UNPACK
pragmas for efficiency, which I have omitted here for clarity.
^{[9] }A quadcore Intel i73770
^{[10] }To do this scientifically, you would need to be much more rigorous, but the goal here is just to optimize our program, so rough measurements are fine.