In the previous two chapters, we looked at the Eval
monad and
Strategies, which work in conjunction with lazy evaluation to express
parallelism. A Strategy consumes a lazy data structure and evaluates
parts of it in parallel. This model has some advantages: it allows
the decoupling of the algorithm from the parallelism, and it allows
parallel evaluation strategies to be built compositionally. But
Strategies and Eval
are not always the most convenient or effective
way to express parallelism. We might not want to build a lazy data
structure, for example. Lazy evaluation brings the nice modularity
properties that we get with Strategies, but on the flip side, lazy
evaluation can make it tricky to understand and diagnose performance.
In this chapter, we’ll explore another parallel programming model, the
Par
monad, with a different set of tradeoffs. The goal of the Par
monad is to be more explicit about granularity and data dependencies,
and to avoid the reliance on lazy evaluation, but without sacrificing
the determinism that we value for parallel programming.
In this programming model, the programmer has to give more
detail but in return gains more control. The Par
monad has some
other interesting benefits; for example, it is implemented entirely as
a Haskell library and the implementation can be readily modified to
accommodate alternative scheduling strategies.
The interface is based around a monad called, unsurprisingly, Par
:
newtype
Par
a
instance
Applicative
Par
instance
Monad
Par
runPar
::
Par
a
>
a
A computation in the Par
monad can be run using runPar
to produce
a pure result. The purpose of Par
is to introduce parallelism, so
we need a way to create parallel tasks:
fork
::
Par
()
>
Par
()
The Par
computation passed as the argument to fork
(the “child”)
is executed in parallel with the caller of fork
(the “parent”).
But fork
doesn’t return anything to the parent, so you might be
wondering how we get the result back if we start a parallel
computation with fork
. Values can be passed between Par
computations
using the IVar
type^{[11]} and its operations:
data
IVar
a
 instance Eq
new
::
Par
(
IVar
a
)
put
::
NFData
a
=>
IVar
a
>
a
>
Par
()
get
::
IVar
a
>
Par
a
Think of an IVar
as a box that starts empty. The put
operation stores a value in the box, and get
reads the value. If
the get
operation finds the box empty, then it waits until the box is
filled by a put
. So an IVar
lets you communicate values between
parallel Par
computations, because you can put
a value in the box
in one place and get
it in another.
Once filled, the box stays full; the get
operation doesn’t remove
the value from the box. It is an error to call put
more than once
on the same IVar
.
The IVar
type is a relative of the MVar
type that we shall see
later in the context of Concurrent Haskell (Communication: MVars), the main
difference being that an IVar
can be written only once. An IVar
is also like a future or promise, concepts that may be familiar to
you from other parallel or concurrent languages.
There is nothing in the types to stop you from returning an IVar
from
runPar
and passing it to another call of runPar
. This is a Very
Bad Idea; don’t do it. The implementation of the Par
monad assumes
that IVar
s are created and used within the same runPar
, and
breaking this assumption could lead to a runtime error, deadlock,
or worse.
The library could prevent you from doing this using qualified types
in the same way that the ST
monad prevents you from returning an
STRef
from runST
. This is planned for a future version.
Together, fork
and IVar
s allow the construction of dataflow
networks. Let’s see how that works with a few simple examples.
We’ll start in the same way we did in Chapter 2: write some
code to perform two independent computations in parallel. As before,
I’m going to use the fib
function to simulate some work we want
to do:
runPar
$
do
i
<
new

j
<
new

fork
(
put
i
(
fib
n
))

fork
(
put
j
(
fib
m
))

a
<
get
i

b
<
get
j

return
(
a
+
b
)

Creates two new IVar
s to hold the results, i
and j
.
fork
two independent Par
computations. The first puts the
value of fib n
into the IVar
i
, and the second puts the value of
fib m
into the IVar
j
.
The parent of the two fork
s calls get
to wait for the
results from i
and j
.
Finally, add the results and return.
When run, this program evaluates fib n
and fib m
in parallel.
To try it yourself, compile parmonad.hs and run it passing two
values for n
and m
, for example:
$ ./parmonad 34 35 +RTS N2
The pattern in this program is represented graphically in Figure 41.
The diagram makes it clear that what we are creating is a dataflow
graph: that is, a graph in which the nodes (fib n
, etc.) contain the computation
and data flows down the edges (i
and j
). To be concrete, each fork
in the
program creates a node, each new
creates an edge, and get
and
put
connect the edges to the nodes.
From the diagram, we can see that the two nodes containing fib n
and
fib m
are independent of each other, and that is why they can be
computed in parallel, which is exactly what the monadpar
library
will do. However, the dataflow graph doesn’t exist in any explicit
form at runtime; the library works by keeping track of all the
computations that can currently be performed (a work pool), and
dividing those amongst the available processors using an appropriate
scheduling strategy. The dataflow graph is just a way to visualize
and understand the structure of the parallelism. Unfortunately, right
now there’s no way to generate a visual representation of the dataflow graph from some Par
monad
code, but hopefully in the future someone will write a tool to do
that.
Using dataflow to express parallelism is quite an old idea; there were people experimenting with custom hardware architectures designed around dataflow back in the 1970s and 1980s. In contrast to those designs that were focused on exploiting very finegrained parallelism automatically, here we are using dataflow as an explicit parallel programming model. But we are using dataflow here for the same reasons that it was attractive back then: instead of saying what is to be done in parallel, we only describe the data dependencies, thereby exposing all the implicit parallelism to be exploited.
While the Par
monad is particularly suited to expressing dataflow
networks, it can also express other common patterns. For example,
we can build an equivalent of the parMap
combinator that we saw
earlier in Chapter 2. To make it easier to define parMap
,
let’s first build a simple abstraction for a parallel computation that
returns a result:
spawn
::
NFData
a
=>
Par
a
>
Par
(
IVar
a
)
spawn
p
=
do
i
<
new
fork
(
do
x
<
p
;
put
i
x
)
return
i
The spawn
function forks a computation in parallel and returns an IVar
that can be used to wait for the result. For convenience,
spawn
is already provided by Control
.Monad.Par
.
Parallel map consists of calling spawn
to apply the function to
each element of the list and then waiting for all the results:
parMapM
::
NFData
b
=>
(
a
>
Par
b
)
>
[
a
]
>
Par
[
b
]
parMapM
f
as
=
do
ibs
<
mapM
(
spawn
.
f
)
as
mapM
get
ibs
(parMapM
is also provided by Control.Monad.Par
, albeit in a more
generalized form than the version shown here.)
Note that the function argument, f
, returns its result in the Par
monad; this means that f
itself can create further parallelism using
fork
and the other Par
operations. When the function argument of
a map is monadic, convention is to add the M
suffix to the function
name, hence parMapM
.
It is straightforward to define a variant of parMapM
that takes a
nonmonadic function instead, by inserting a return
:
parMap
::
NFData
b
=>
(
a
>
b
)
>
[
a
]
>
Par
[
b
]
parMap
f
as
=
do
ibs
<
mapM
(
spawn
.
return
.
f
)
as
mapM
get
ibs
One other thing to note here is that, unlike the parMap
we saw in
Chapter 2, parMapM
and parMap
wait for all the results
before returning. Depending on the context, this may or may not be the
most useful behavior. If you don’t want to wait for the results, then you could always just use mapM (spawn . f)
, which
returns a list of IVar
s.
The FloydWarshall algorithm finds the lengths of the shortest paths
between all pairs of nodes in a weighted directed graph. The
algorithm is quite simple and can be expressed as a function over
three vertices. Assuming vertices are numbered from one, and we have
a function weight g i j
that gives the weight of the edge from i
to j
in graph g
, the algorithm is described by this pseudocode:
shortestPath :: Graph > Vertex > Vertex > Vertex > Weight shortestPath g i j 0 = weight g i j shortestPath g i j k = min (shortestPath g i j (k1)) (shortestPath g i k (k1) + shortestPath g k j (k1))
You can think of the algorithm intuitively this way: shortestPath g i j k
gives the length of the shortest path from i
to j
, passing through vertices up to k
only. At k == 0
, the paths between each pair of vertices consists
of the direct edges only. For a nonzero k
, there are two cases:
either the shortest path from i
to j
passes through k
, or it
does not. The shortest path passing through k
is given by the sum
of the shortest path from i
to k
and from k
to j
. Then the
shortest path from i
to j
is the minimum of the two choices,
either passing through k
or not.
We wouldn’t want to implement the algorithm like this directly,
because it requires an exponential number of recursive calls. This is a
classic example of a dynamic programming problem: rather than
recursing topdown, we can build the solution bottomup, so that
earlier results can be used when computing later ones. In this case,
we want to start by computing the shortest paths between all pairs of
nodes for k == 0
, then for k == 1
, and so on until k
reaches the
maximum vertex. Each step is O(n^{2}) in the vertices, so the
whole algorithm is O(n^{3}).
The algorithm is often run over an adjacency matrix, which is a very efficient representation of a dense graph. But here we’re going to assume a sparse graph (most pairs of vertices do not have an edge between them), and use a representation more suitable for this case:
type
Vertex
=
Int
type
Weight
=
Int
type
Graph
=
IntMap
(
IntMap
Weight
)
weight
::
Graph
>
Vertex
>
Vertex
>
Maybe
Weight
weight
g
i
j
=
do
jmap
<
Map
.
lookup
i
g
Map
.
lookup
j
jmap
The graph is essentially a mapping from pairs of nodes to weights, but
it is represented more efficiently as a twolayer map. For example, to find
the edge between i
and j
, we look up i
in the outer map,
yielding another map in which we look up j
to find the weight. The
function weight
embodies this pair of lookups using the Maybe
monad. If there is no edge between the two vertices, then weight
returns Nothing
.
Here is the sequential implementation of the shortest path algorithm:
shortestPaths
::
[
Vertex
]
>
Graph
>
Graph
shortestPaths
vs
g
=
foldl'
update
g
vs

where
update
g
k
=
Map
.
mapWithKey
shortmap
g

where
shortmap
::
Vertex
>
IntMap
Weight
>
IntMap
Weight
shortmap
i
jmap
=
foldr
shortest
Map
.
empty
vs

where
shortest
j
m
=
case
(
old
,
new
)
of

(
Nothing
,
Nothing
)
>
m
(
Nothing
,
Just
w
)
>
Map
.
insert
j
w
m
(
Just
w
,
Nothing
)
>
Map
.
insert
j
w
m
(
Just
w1
,
Just
w2
)
>
Map
.
insert
j
(
min
w1
w2
)
m
where
old
=
Map
.
lookup
j
jmap

new
=
do
w1
<
weight
g
i
k

w2
<
weight
g
k
j
return
(
w1
+
w2
)
shortestPaths
takes a list of vertices in addition to the graph; we
could have derived this from the graph, but it’s slightly more
convenient to pass it in. The result is also a Graph
, but instead
of containing the weights of the edges between vertices, it contains
the lengths of the shortest paths between vertices. For simplicity,
we’re not returning the shortest paths themselves, although this can
be added without affecting the asymptotic time or space complexity.
The algorithm as a whole is a leftfold over the list of vertices;
this corresponds to iterating over values of k
in the pseudocode
description shown earlier. At each stage we add a new vertex to the set of
vertices that can be used to construct paths, until at the end we have
paths that can use all the vertices. Note that we use the strict
leftfold, foldl'
, to ensure that we’re evaluating the graph at
every step and not building up a chain of thunks (we’re also using a
strict IntMap
to avoid thunks building up inside the Graph
; this
turns out to be vital for avoiding a space leak).
The update
function computes each step by mapping the function
shortmap
over the outer IntMap
in the graph. There’s no need to
map over the whole list of vertices because we know that any vertex
that does not have an entry in the outer map cannot have a path to any
other vertex (although it might have incoming paths).
shortmap
takes i
, the current vertex, and jmap
, the mapping
of shortest paths from i
. This function does need to consider
every vertex in the graph as a possible destination because there may
be vertices that we can reach from i
via k
, but which do
not currently have an entry in jmap
. So here we’re building up a
new jmap
by folding over the list of vertices, vs
.
For a given j
, look up the current shortest path from i
to
j
, and call it old
.
Look up the shortest path from i
to j
via k
(if one
exists), and call it new
.
Find the minimum of old
and new
, and insert it into the new
mapping. Naturally, one path is the winner if the other path does not
exist.
The algorithm is a nest of three loops. The outer loop is a leftfold with a data dependency between iterations, so it cannot be parallelized (as a side note, folds can be parallelized only when the operation being folded is associative, and then the linear fold can be turned into a tree). The next loop, however, is a map:
update
g
k
=
Map
.
mapWithKey
shortmap
g
As we know, maps parallelize nicely. Will this give the right
granularity? The map is over the outer IntMap
of the Graph
, so
there will be as many tasks as there are vertices without edges.
There will typically be at least hundreds of edges in the graph, so
there are clearly enough separate work items to keep even tens of
cores busy. Furthermore, each task is an O(n) loop over the list
of vertices, so we are unlikely to have problems with the granularity
being too fine.
Let’s consider how to add parallelism here. It’s not an ordinary
map
—we’re using the mapWithKey
function provided by Data.IntMap
to map directly over the IntMap
. We could turn the IntMap
into a
list, run a standard parMap
over that, and then turn it back into an
IntMap
, but the conversion to and from a list would add some
overhead. Fortunately, the IntMap
library provides a way to
traverse an IntMap
in a monad:
traverseWithKey
::
Applicative
t
=>
(
Key
>
a
>
t
b
)
>
IntMap
a
>
t
(
IntMap
b
)
Don’t worry if you’re not familiar with the Applicative
type class; most of the time, you can read Applicative
as Monad
and you’ll be fine. All the standard Monad
types are also Applicative
, and in general any Monad
can be made into an Applicative
easily.^{[12]}
So traverseWithKey
essentially maps a monadic function over the
IntMap
, for any monad t
. The monadic function is passed not only
the element a
, but also the Key
, which is just what we need here:
shortmap
needs both the key (the source vertex) and the element (the
map from destination vertices to weights).
So we want to behave like parMap
, except that we’ll use
traverseWithKey
to map over the IntMap
. Here is the parallel code
for update
:
update
g
k
=
runPar
$
do
m
<
Map
.
traverseWithKey
(
\
i
jmap
>
spawn
(
return
(
shortmap
i
jmap
)))
g
traverse
get
m
We’ve put runPar
inside update
; the rest of the shortestPaths
function will remain as before, and all the parallelism is confined to
update
. We’re calling traverseWithKey
to spawn
a call to
shortmap
for each of the elements of the IntMap
. The result of
this call will be an IntMap (IVar (IntMap Weight))
; that is, there’s an IVar
in place of each element. To get the new Graph
, we need
to call get
on each of these IVar
s and produce a new Graph
with all the elements, which is what the final call to traverse
does. The traverse
function is from the Traversable
class; for
our purposes here, it is like traverseWithKey
but doesn’t pass the
Key
to the function.
Let’s take a look at the speedup we get from this code. Running the original program on a random graph with 800 edges over 1,000 vertices:
$ ./fwsparse 1000 800 +RTS s ... Total time 4.16s ( 4.17s elapsed)
And our parallel version, first on one core:
$ ./fwsparse1 1000 800 +RTS s ... Total time 4.54s ( 4.57s elapsed)
Adding the parallel traversal has cost us about 10% overhead; this is
quite a lot, and if we were optimizing this program for real, we would
want to look into whether that overhead can be reduced. Perhaps it is
caused by doing one runPar
per iteration (a runPar
is
quite expensive) or perhaps traverseWithKey
is expensive.
The speedup on four cores is fairly respectable:
$ ./fwsparse1 1000 800 +RTS s N4 ... Total time 5.27s ( 1.38s elapsed)
This gives us a speedup of 3.02 over the sequential version. To improve this speedup further, the first target would be to reduce the overhead of the parallel version.
Next, we’re going to look at a different way to expose parallelism: pipeline parallelism. Back in Parallelizing Lazy Streams with parBuffer, we saw how to use parallelism in a program that consumed and produced input lazily, although in that case we used data parallelism, which is parallelism between the stream elements. Here, we’re going to show how to make use of parallelism between the stages of a pipeline. For example, we might have a pipeline that looks like this:
Each stage of the pipeline is doing some computation on the
stream elements and maintaining state as it does so. When a
pipeline stage maintains some state, we can’t exploit parallelism between
the stream elements as we did in Parallelizing Lazy Streams with parBuffer. Instead, we would
like each of the pipeline stages to run on a separate core, with the
data streaming between them. The Par
monad, together with the
techniques in this section, allows us to do that.
The basic idea is as follows: instead of representing the stream as a lazy list, use an explicit representation of a stream:
data
IList
a
=
Nil

Cons
a
(
IVar
(
IList
a
))
type
Stream
a
=
IVar
(
IList
a
)
An IList
is a list with an IVar
as the tail. This allows the
producer to generate the list incrementally, while a consumer runs in
parallel, grabbing elements as they are produced. A Stream
is an
IVar
containing an IList
.
We’ll need a few functions for working with Stream
s. First, we need a
generic producer that turns a lazy list into a Stream
:
streamFromList
::
NFData
a
=>
[
a
]
>
Par
(
Stream
a
)
streamFromList
xs
=
do
var
<
new

fork
$
loop
xs
var

return
var

where
loop
[]
var
=
put
var
Nil

loop
(
x
:
xs
)
var
=
do

tail
<
new

put
var
(
Cons
x
tail
)

loop
xs
tail

Creates the IVar
that will be the Stream
itself.
fork
s the loop that will create the Stream
contents.
Returns the Stream
to the caller. The Stream
is now being created in parallel.
This loop
traverses the input list, producing the IList
as it
goes. The first argument is the list, and the second argument is the
IVar
into which to store the IList
. In the case of an empty list, we
simply store an empty IList
into the IVar
.
In the case of a nonempty list,
we create a new IVar
for the
tail,
and store a Cons
cell representing this element into the current
IVar
. Note that this fully evaluates the list element x
, because
put
is strict.
Recurse to create the rest of the stream.
Next, we’ll write a consumer of Stream
s, streamFold
:
streamFold
::
(
a
>
b
>
a
)
>
a
>
Stream
b
>
Par
a
streamFold
fn
!
acc
instrm
=
do
ilst
<
get
instrm
case
ilst
of
Nil
>
return
acc
Cons
h
t
>
streamFold
fn
(
fn
acc
h
)
t
This is a leftfold over the Stream
and is defined exactly as you
would expect: recursing through the IList
and accumulating the result
until the end of the Stream
is reached. If the streamFold
consumes all the available stream elements and catches up with the
producer, it will block in the get
call waiting for the next
element.
The final operation we’ll need is a map over Stream
s. This is
both a producer and a consumer:
streamMap
::
NFData
b
=>
(
a
>
b
)
>
Stream
a
>
Par
(
Stream
b
)
streamMap
fn
instrm
=
do
outstrm
<
new
fork
$
loop
instrm
outstrm
return
outstrm
where
loop
instrm
outstrm
=
do
ilst
<
get
instrm
case
ilst
of
Nil
>
put
outstrm
Nil
Cons
h
t
>
do
newtl
<
new
put
outstrm
(
Cons
(
fn
h
)
newtl
)
loop
t
newtl
There’s nothing particularly surprising here—the pattern is a
combination of the producer we saw in streamFromList
and the consumer
in streamFold
.
To demonstrate that this works, I’ll construct an example using the RSA encryption code that we saw earlier in Parallelizing Lazy Streams with parBuffer. However, this time, in order to construct a nontrivial pipeline, I’ll compose together encryption and decryption; encryption will produce a stream that decryption consumes (admittedly this isn’t a realistic use case, but it does demonstrate pipeline parallelism).
Previously, encrypt
and decrypt
consumed and produced lazy
ByteStrings
. Now they work over Stream ByteString
in the Par
monad, and are expressed as a streamMap
:
encrypt
::
Integer
>
Integer
>
Stream
ByteString
>
Par
(
Stream
ByteString
)
encrypt
n
e
s
=
streamMap
(
B
.
pack
.
show
.
power
e
n
.
code
)
s
decrypt
::
Integer
>
Integer
>
Stream
ByteString
>
Par
(
Stream
ByteString
)
decrypt
n
d
s
=
streamMap
(
B
.
pack
.
decode
.
power
d
n
.
integer
)
s
The following function composes these together and also adds a
streamFromList
to create the input Stream
and a streamFold
to
consume the result:
pipeline
::
Integer
>
Integer
>
Integer
>
ByteString
>
ByteString
pipeline
n
e
d
b
=
runPar
$
do
s0
<
streamFromList
(
chunk
(
size
n
)
b
)
s1
<
encrypt
n
e
s0
s2
<
decrypt
n
d
s1
xs
<
streamFold
(
\
x
y
>
(
y
:
x
))
[]
s2
return
(
B
.
unlines
(
reverse
xs
))
Note that the streamFold
produces a list of ByteString
s at the end,
to which we apply unlines
, and then the caller prints out the
result.
This works rather nicely: I see a speedup of 1.45 running the program
over a large text file. What’s the maximum speedup we can achieve
here? Well, there are four independent pipeline stages:
streamFromList
, two streamMap
s, and a streamFold
, although
only the two maps really involve any significant
computation.^{[13]}
So the best we can hope for is to reduce the running time to the
longer of the two maps. We can verify, by timing the rsa.hs program,
that encryption takes approximately twice as long as decryption, which
means that we can expect a speedup of about 1.5, which is close to
the sample run here.
The ThreadScope profile of this program is quite revealing. Figure 42 is a typical section from it.
HEC 0 appears to be doing the encryption, and HEC 1 the decryption. Decryption is faster than encryption, so HEC 1 repeatedly gets stuck waiting for the next element of the encrypted stream; this accounts for the gaps in execution we see on HEC 1.
One interesting thing to note about this profile is that the pipeline
stages tend to stay on the same core. This is because each pipeline
stage is a single fork
(a single node in the dataflow
graph) and the Par
scheduler will run a task to completion on the
current core before starting on the next task. Keeping each pipeline
stage running on a single core is good for locality.
In our previous example, the consumer was faster than the producer. If,
instead, the producer had been faster than the consumer, then there would be
nothing to stop the producer from getting a long way ahead of the consumer
and building up a long IList
chain in memory. This is undesirable,
because large heap data structures incur overhead due to garbage
collection, so we might want to ratelimit the producer to avoid it
getting too far ahead. There’s a trick that adds some automatic
ratelimiting to the stream API. It entails adding another
constructor to the IList
type:
data
IList
a
=
Nil

Cons
a
(
IVar
(
IList
a
))

Fork
(
Par
()
)
(
IList
a
)

The idea is that the creator of the IList
produces a fixed
amount of the list and inserts a Fork
constructor containing
another Par
computation that will produce more of the list. The
consumer, upon finding a Fork
, calls fork
to start production of
the next chunk of the list. The Fork
doesn’t have to be at the end;
for example, the list might be produced in chunks of 200 elements,
with the first Fork
being at the 100 element mark, and every 200
elements thereafter. This would mean that at any time there would be
at least 100 and up to 300 elements waiting to be consumed.
I’ll leave the rest of the implementation of this idea as an exercise for you to try on your own.
See if you can modify streamFromList
, streamFold
, and streamMap
to incorporate the Fork
constructor. The chunk size and fork
distance should be parameters to the producers (streamFromList
and
streamMap
).
Pipeline parallelism is limited in that we can expose only as much parallelism as we have pipeline stages. It therefore tends to be less effective than data parallelism, which can expose a lot more parallelism. Still, pipeline parallelism is a useful tool to have in your toolbox.
The earlier example also exposes a limitation of the Par
monad; we cannot
produce a lazy stream from runPar
itself. The call to streamFold
accumulates the entire list before it returns. You can’t return an
IList
from runPar
and consume it in another runPar
, because
returning an IVar
from runPar
is illegal and will
probably result in an error. Besides, runPar
always runs all the
forked Par
computations to completion before returning, because this
is necessary to ensure deterministic results. There is an IO
version of the Par
monad that we’ll encounter in
The ParIO monad, and you could use that for lazy streaming,
although unlike the pure Par
monad, determinism is not guaranteed
when using the IO
version.
In this section, we’ll look at a program that finds a valid timetable for a conference.^{[14]} The outline of the problem is this:
Here’s a small example. Suppose we have two tracks and two slots, and four talks named A, B, C, and D. There are four attendees—P, Q, R, and S—and each wants to go to two talks:
One solution is:
Track  Slot 1  Slot 2 
1  B  C 
2  D  A 
There are other solutions, but they are symmetrical with this one (interchange either the tracks or the slots or both).
Timetabling is an instance of a constraint satisfaction problem: we’re finding assignments for variables (talk slots) that satisfy the constraints (attendees’ preferences). The problem requires an exhaustive search, but we can be more clever than just generating all the possible assignments and testing each one. We can fill in the timetable incrementally: assign a talk to the first slot of the first track, then find a talk for the first slot of the second track that doesn’t introduce a conflict, and so on until we’ve filled up the first slot of all the tracks. Then we proceed to the second slot and so on until we’ve filled the whole timetable. This incremental approach prunes a lot of the search space because we avoid searching for solutions when the partial grid already contains a conflict.^{[15]}
If at any point we cannot fill a slot without causing a conflict, we have to backtrack to the previous slot and choose a different talk instead. If we exhaust all the possibilities at the previous slot, then we have to backtrack further. So, in general, the search pattern is a tree. A fragment of the search tree for the example above is shown in Figure 43.
If we are interested in all the solutions (perhaps because we want to pick the best one according to some criteria), we have to explore the whole tree.
Algorithms that have this treeshaped structure are often called divide and conquer. A divideandconquer algorithm is one in which the problem is recursively split into smaller subproblems that are solved separately, then combined to form the whole solution. In this case, starting from the empty timetable, we’re dividing the solution space according to which talk goes in the first slot, and then by which talk goes in the second slot, and so on recursively until we fill the timetable. Divideandconquer algorithms have some nice properties, not least of which is that they parallelize well because the branches are independent of one another.
So let’s look at how to code up a solution in Haskell. First, we need a type to represent talks; for simplicity, I’ll just number them:
newtype
Talk
=
Talk
Int
deriving
(
Eq
,
Ord
)
instance
NFData
Talk
instance
Show
Talk
where
show
(
Talk
t
)
=
show
t
An attendee is represented by her name and the talks she wants to attend:
data
Person
=
Person
{
name
::
String
,
talks
::
[
Talk
]
}
deriving
(
Show
)
And the complete timetable is represented as a list of lists of Talk
. Each list represents a
single slot. So if there are four tracks and three slots, for example,
the timetable will be three lists of four elements each.
type
TimeTable
=
[[
Talk
]]
Here’s the toplevel function: it takes a list of Person
, a list of
Talk
, the number of tracks and slots, and returns a list of
TimeTable
:
timetable
::
[
Person
]
>
[
Talk
]
>
Int
>
Int
>
[
TimeTable
]
timetable
people
allTalks
maxTrack
maxSlot
=
First, I’m going to cache some information about which talks clash with
each other. That is, for each talk, we want to know which other talks
cannot be scheduled in the same slot, because one or more attendees
want to see both of them. This information is collected in a Map
called clashes
, which is built from the [Person]
passed to
timetable
:
clashes
::
Map
Talk
[
Talk
]
clashes
=
Map
.
fromListWith
union
[
(
t
,
ts
)

s
<
people
,
(
t
,
ts
)
<
selects
(
talks
s
)
]
The auxiliary function selects
takes a list and returns a list of
pairs, one pair for each item in the input list. The first element of
each pair is an element, and the second is the original list with that
element removed. For efficient implementation, selects
does not
preserve the order of the elements. Example output:
*Main> selects [1..3] [(1,[2,3]),(2,[1,3]),(3,[2,1])]
Now we can write the algorithm itself. Remember that the algorithm
is recursive: at each stage, we start with a partially filledin
timetable, and we want to determine all the possible ways of filling
in the next slot and recursively generate all the solutions from
those. The recursive function is called generate
. Here is its type:
generate
::
Int
 current slot number
>
Int
 current track number
>
[[
Talk
]]
 slots allocated so far
>
[
Talk
]
 talks in this slot
>
[
Talk
]
 talks that can be allocated in this slot
>
[
Talk
]
 all talks remaining to be allocated
>
[
TimeTable
]
 all possible solutions
The first two arguments tell us where in the timetable we are, and the second two arguments are the partially complete timetable. In fact, we’re filling in the slots in reverse order, but the slots are independent of one another so it makes no difference. The last two arguments keep track of which talks remain to be assigned: the first is the list of talks that we can put in the current slot (taking into account clashes with other talks already in this slot), and the second is the complete list of talks still left to assign.
The implementation of generate
looks a little dense, but I’ll walk
through it step by step:
generate
slotNo
trackNo
slots
slot
slotTalks
talks

slotNo
==
maxSlot
=
[
slots
]


trackNo
==
maxTrack
=
generate
(
slotNo
+
1
)
0
(
slot
:
slots
)
[]
talks
talks


otherwise
=
concat

[
generate
slotNo
(
trackNo
+
1
)
slots
(
t
:
slot
)
slotTalks'
talks'


(
t
,
ts
)
<
selects
slotTalks

,
let
clashesWithT
=
Map
.
findWithDefault
[]
t
clashes

,
let
slotTalks'
=
filter
(`
notElem
`
clashesWithT
)
ts

,
let
talks'
=
filter
(
/=
t
)
talks

]
If we’ve filled in all the slots, we’re done; the current list of slots is a solution.
If we’ve filled in all the tracks for the current slot, move on to the next slot.
Otherwise, we’re going to fill in the next talk in this slot. The result is the concatenation of all the solutions arising from the possibilities for filling in that talk.
Here we select all the possibilities for the next talk from
slotTalks
, binding the next talk to t
.
Decide which other talks clash with t
.
Remove from slotTalks
the talks that clash with t
.
Remove t
from talks
.
For each t
, recursively call generate
with the new partial solution.
Finally, we need to call generate
with the empty timetable to start
things off:
generate
0
0
[]
[]
allTalks
allTalks
The program is equipped with some machinery to generate test data, so we can see how long it takes with a variety of inputs. Unfortunately, it turns out to be hard to find some parameters that don’t either take forever or complete instantaneously, but here’s one set:
$ ./timetable 4 3 11 10 3 +RTS s
The commandline arguments set the parameters for the search: 4 slots, 3 tracks, 11 total talks, and 10 participants who each want to go to 3 talks. This takes about 1 second to calculate the number of possible timetables (about 31,000).
This code is already quite involved, and if we try to parallelize it directly, it is likely to get more complicated. We’d prefer to separate the parallelism as far as possible from the algorithm code. With Strategies (Chapter 3), we did this by generating a lazy data structure. But this application is an example where generating a lazy data structure doesn’t work very well, because we would have to return the entire search tree as a data structure.
Instead, I want to demonstrate another technique for separating the
parallelism from the algorithm: building a parallel skeleton. A
parallel skeleton is nothing more than a higherorder function that
abstracts a pattern of computation. We’ve already seen one parallel
skeleton: parMap
, the function that describes data parallelism,
abstracted over the function to apply in parallel. Here we need a
different skeleton, which I’ll call the search skeleton (although
it’s a variant of a more general divideandconquer skeleton).
I’ll start by refactoring the algorithm into a skeleton and its
instantiation, and then add parallelism to the skeleton. The type of
the search
skeleton is as follows:
search
::
(
partial
>
Maybe
solution
)

>
(
partial
>
[
partial
]
)

>
partial

>
[
solution
]

The search
function is polymorphic in two types: partial
is the type of partial
solutions, and solution
is the type of complete solutions. We’ll
see how these are instantiated in our example shortly.
The first argument to search
is a function that tells whether a
particular partial
solution corresponds to a complete solution
,
and if so, what the solution is.
The second argument takes a partial
solution and refines it to a
list of further partial
solutions. It is expected that this process
doesn’t continue forever!
To get things started, we need an initial, empty value of type
partial
.
The result is a list of solution
s.
The definition of search
is quite straightforward. It’s one of
those functions that is almost impossible to get wrong, because the
type describes exactly what it does:
search
finished
refine
emptysoln
=
generate
emptysoln
where
generate
partial

Just
soln
<
finished
partial
=
[
soln
]

otherwise
=
concat
(
map
generate
(
refine
partial
))
Now to refactor timetable
to use search
. The basic idea is that
the arguments to generate
constitute the partial solution, so we’ll
just package them up:
type
Partial
=
(
Int
,
Int
,
[[
Talk
]],
[
Talk
],
[
Talk
],
[
Talk
])
The rest of the refactoring is mechanical, so I won’t describe it in detail. The result is:
timetable
::
[
Person
]
>
[
Talk
]
>
Int
>
Int
>
[
TimeTable
]
timetable
people
allTalks
maxTrack
maxSlot
=
search
finished
refine
emptysoln
where
emptysoln
=
(
0
,
0
,
[]
,
[]
,
allTalks
,
allTalks
)
finished
(
slotNo
,
trackNo
,
slots
,
slot
,
slotTalks
,
talks
)

slotNo
==
maxSlot
=
Just
slots

otherwise
=
Nothing
clashes
::
Map
Talk
[
Talk
]
clashes
=
Map
.
fromListWith
union
[
(
t
,
ts
)

s
<
people
,
(
t
,
ts
)
<
selects
(
talks
s
)
]
refine
(
slotNo
,
trackNo
,
slots
,
slot
,
slotTalks
,
talks
)

trackNo
==
maxTrack
=
[(
slotNo
+
1
,
0
,
slot
:
slots
,
[]
,
talks
,
talks
)]

otherwise
=
[
(
slotNo
,
trackNo
+
1
,
slots
,
t
:
slot
,
slotTalks'
,
talks'
)

(
t
,
ts
)
<
selects
slotTalks
,
let
clashesWithT
=
Map
.
findWithDefault
[]
t
clashes
,
let
slotTalks'
=
filter
(`
notElem
`
clashesWithT
)
ts
,
let
talks'
=
filter
(
/=
t
)
talks
]
The algorithm works exactly as before. All we did was pull out the search pattern as a higherorder function and call it.
Now to parallelize the search
skeleton. As you might expect, the basic idea is that at
each stage, we’ll spawn off the recursive calls in parallel and then collect the results. Here’s how to express that using the Par
monad:
parsearch
::
NFData
solution
=>
(
partial
>
Maybe
solution
)
>
(
partial
>
[
partial
]
)
>
partial
>
[
solution
]
parsearch
finished
refine
emptysoln
=
runPar
$
generate
emptysoln
where
generate
partial

Just
soln
<
finished
partial
=
return
[
soln
]

otherwise
=
do
solnss
<
parMapM
generate
(
refine
partial
)
return
(
concat
solnss
)
We’re using parMapM
to call generate
in parallel on the
list of partial solutions returned by refine
, and then concatenating
the results. However, this doesn’t work out too well; on the
parameter set we used before, it adds a factor of five overhead. The
problem is that as we get near the leaves of the search tree, the
granularity is too fine in relation to the overhead of spawning the
calls in parallel.
So we need a way to make the granularity coarser. We can’t use chunking, because we don’t have a flat list here; we have a tree. For treeshaped parallelism we need to use a different technique: a depth threshold. The basic idea is quite simple: spawn recursive calls in parallel down to a certain depth, and below that depth use the original sequential algorithm.
Our parsearch
function needs an extra parameter, namely the depth
to parallelize to:
parsearch
::
NFData
solution
=>
Int
>
(
partial
>
Maybe
solution
)
 finished?
>
(
partial
>
[
partial
]
)
 refine a solution
>
partial
 initial solution
>
[
solution
]
parsearch
maxdepth
finished
refine
emptysoln
=
runPar
$
generate
0
emptysoln
where
generate
d
partial

d
>=
maxdepth

=
return
(
search
finished
refine
partial
)
generate
d
partial

Just
soln
<
finished
partial
=
return
[
soln
]

otherwise
=
do
solnss
<
parMapM
(
generate
(
d
+
1
))
(
refine
partial
)
return
(
concat
solnss
)
Using a depth of three in this case works reasonably well and gets us a speedup of about three on four cores relative to the original sequential version. Adding the skeleton unfortunately incurs some overhead, but in return it gains us some worthwhile modularity: it would have been difficult to add the depth threshold without first abstracting the skeleton.
Here are the main points to take away from this example:
In this section, we will parallelize a type inference engine, such as you might find in a compiler for a functional language. The purpose of this example is to demonstrate two things: one, that parallelism can be readily applied to program analysis problems, and two, that the dataflow model works well even when the structure of the parallelism is entirely dependent on the input and cannot be predicted beforehand.
The outline of the problem is as follows: given a list of bindings of
the form x = e
for a variable x
and expression e
, infer the
types for each of the variables. Expressions consist of integers,
variables, application, lambda expressions, let
expressions, and
arithmetic operators (+
, 
, *
, /
).
We can test the type inference engine on a few simple examples. Load it in GHCi (from the directory parinfer in the sample code):
$ ghci parinfer.hs
The function test
typechecks an expression. Simple arithmetic
expressions have type Int
:
*Main> test "1 + 2" Int
We can use lambda expressions, let
expressions, and higherorder
functions, just like in Haskell:
*Main> test "\\x > x" a0 > a0 *Main> test "\\x > x + 1" Int > Int *Main> test "\\g > \\h > g (h 3)" (a2 > a3) > (Int > a2) > a3
(Note that in order to get the backslash character in a Haskell String
, we
need to use "\\"
.)
When the type inferencer is run as a standalone program, it typechecks a file of bindings, and infers a type for each one. For simplicity, we assume the list of bindings is ordered and nonrecursive; any variable used in an expression has to be defined earlier in the list. Later bindings may also shadow earlier ones.
For example, consider the following set of bindings for which we want to infer types:
f = ... g = ... f ... h = ... f ... j = ... g ... h ...
I’m using the notation "... f ...
" to stand for an expression
involving f
. The specific expression isn’t important here, only
that it mentions f
.
We could proceed in a linear fashion through the list of bindings:
first inferring the type for f
, then the type for g
, then the type for
h
, and so on. However, if we look at the dataflow graph for this
set of bindings (Figure 44), we can see that there is some parallelism.
Clearly we can infer the types of g
and h
in parallel, once the type of f
is known. When viewed this way, we can see that type inference is a natural fit for the dataflow model; we can consider each binding to be a node in the
graph, and the edges of the graph carry inferred types from bindings
to usage sites in the program.
Building a dataflow graph for the type inference problem allows parallelism to be automatically extracted from the type inference process. The actual amount of parallelism present depends entirely on the structure of the input program, however. An input program in which every binding depends on the previous one in the list would have no parallelism to extract. Fortunately, most programs aren’t like that—usually there is a decent amount of parallelism implicit in the dependency structure.
Note that we’re not necessarily exploiting all the available parallelism here. There might be parallelism available within the inference of individual bindings. However, to try to parallelize too deeply might cause granularity problems, and parallelizing the outer level is likely to gain the most reward.
The type inference engine that I’m using for this example is a rather ancient piece of code that I modified to add parallelism.^{[16]} The changes to add parallelism were quite modest.^{[17]}
The types from the inference engine that we will need to work with are as follows:
type
VarId
=
String
 Variables
data
Term
 Terms in the input program
data
Env
 Environment, mapping VarId to PolyType
data
PolyType
 Polymorphic types
In programming language terminology, an environment is a mapping that
assigns some meaning to the variables of an expression. A type
inference engine uses an environment to assign types to variables;
this is the purpose of the Env
type. When we typecheck an
expression, we must supply an Env
that gives the types of the
variables that the expression mentions. An Env
is created using
makeEnv
:
makeEnv
::
[(
VarId
,
PolyType
)]
>
Env
To determine which variables we need to populate the Env
with, we
need a way to extract the free (unbound) variables of an expression;
this is what the freeVars
function does:
freeVars
::
Term
>
[
VarId
]
The underlying type inference engine for expressions takes a Term
and an Env
that supplies the types for the free variables of the
Term
and delivers a PolyType
:
inferTopRhs
::
Env
>
Term
>
PolyType
While the sequential part of the inference
engine uses an Env
that maps VarId
s to PolyType
s, the parallel
part of the inference engine will use an environment that maps
VarId
s to IVar PolyType
, so that we can fork
the inference
engine for a given binding, and then wait for its result
later.^{[18]} The
environment for the parallel type inferencer is called TopEnv
:
type
TopEnv
=
Map
VarId
(
IVar
PolyType
)
All that remains is to write the toplevel loop. We’ll do this in two stages. First, a function to infer the type of a single binding:
inferBind
::
TopEnv
>
(
VarId
,
Term
)
>
Par
TopEnv
inferBind
topenv
(
x
,
u
)
=
do
vu
<
new

fork
$
do

let
fu
=
Set
.
toList
(
freeVars
u
)

tfu
<
mapM
(
get
.
fromJust
.
flip
Map
.
lookup
topenv
)
fu

let
aa
=
makeEnv
(
zip
fu
tfu
)

put
vu
(
inferTopRhs
aa
u
)

return
(
Map
.
insert
x
vu
topenv
)

Create an IVar
, vu
, to hold the type of this binding.
Fork the computation that does the type inference.
The inputs to this type inference are the types of the variables
mentioned in the expression u
. Hence we call freeVars
to get
those variables.
For each of the free variables, look up its IVar
in the
topenv
, and then call get
on it. Hence this step will wait until
the types of all the free variables are available before proceeding.
Build an Env
from the free variables and their types.
Infer the type of the expression u
, and put the result in the
IVar
we created at the beginning.
Back in the parent, return topenv
extended with x
mapped to
the new IVar
vu
.
Next we use inferBind
to define inferTop
, which infers types for a
list of bindings:
inferTop
::
TopEnv
>
[(
VarId
,
Term
)]
>
Par
[(
VarId
,
PolyType
)]
inferTop
topenv0
binds
=
do
topenv1
<
foldM
inferBind
topenv0
binds

mapM
(
\
(
v
,
i
)
>
do
t
<
get
i
;
return
(
v
,
t
))
(
Map
.
toList
topenv1
)

This parallel implementation works quite nicely. To demonstrate it, I’ve constructed a synthetic input for the type checker, a fragment of which is given below (the full version is in the file parinfer/benchmark.in).
id = \x>x ; a = \f > f id id ; a = \f > f a a ; a = \f > f a a ; ... a = let f = a in \x > x ; b = \f > f id id ; b = \f > f b b ; b = \f > f b b ; ... b = let f = b in \x > x ; c = \f > f id id ; c = \f > f c c ; c = \f > f c c ; ... c = let f = c in \x > x ; d = \f > f id id ; d = \f > f d d ; d = \f > f d d ; ... d = let f = d in \x > x ;
There are four sequences of bindings that can be inferred in
parallel. The first sequence is the set of bindings for a
(each
successive binding for a
shadows the previous one), then
identical sequences named b
, c
, and d
. Each binding in a
sequence depends on the previous one, but the sequences are
independent of one another. This means that our parallel typechecking
algorithm should automatically infer types for the a
, b
, c
, and
d
bindings in parallel, giving a maximum speedup of 4.
With one processor, the result should be something like this:
$ ./parinfer <benchmark.in +RTS s ... Total time 4.71s ( 4.72s elapsed)
The result with two processors represents a speedup of 1.96:
$ ./parinfer <benchmark.in +RTS s N2 ... Total time 4.79s ( 2.41s elapsed)
With three processors, the result is:
$ ./parinfer <benchmark.in +RTS s N3 ... Total time 4.92s ( 2.42s elapsed)
This is almost exactly the same as with two processors! But this is to be expected: there are four independent problems, so the best we can do is to overlap the first three and then run the final one. Thus the program will take the same amount of time as with two processors, where we could overlap two problems at a time. Adding the fourth processor allows all four problems to be overlapped, resulting in a speedup of 3.66:
$ ./parinfer <benchmark.in +RTS s N4 ... Total time 5.10s ( 1.29s elapsed)
The Par
monad is implemented as a library in Haskell, so
aspects of its behavior can be changed without changing GHC or its
runtime system. One way in which this is useful is in changing the
scheduling strategy; certain scheduling strategies are better suited
to certain patterns of execution.
The monadpar
library comes with two schedulers: the “Trace”
scheduler and the “Direct” scheduler, where the latter is the default. In general the Trace scheduler performs slightly worse than the Direct scheduler, but not always; it’s worth trying both with your code to see which gives the better results.
To choose one or the other, just import the appropriate module. For example, to use the Trace scheduler instead of the Direct scheduler:
import
Control.Monad.Par.Scheds.Trace
 instead of Control.Monad.Par
Remember that you need to make this change in all the modules of
your program that import Control.Monad.Par
.
I’ve presented two different parallel programming models, each with advantages and disadvantages. In reality, though, both approaches are
suitable for a wide range of tasks; most Parallel Haskell benchmarks
achieve broadly similar results when coded with either Strategies or
the Par
monad. So which to choose is to some extent a matter of
personal preference. However, there are a number of tradeoffs that
are worth bearing in mind, as these might tip the balance one way
or the other for your code:
Strategy
to evaluate it in parallel will probably work well. If not, then it can be more straightforward to use the Par
monad to express the parallelism.
runPar
function itself is relatively expensive, whereas
runEval
is free. So when using the Par
monad, you should usually
try to thread the Par
monad around to all the places that need
parallelism to avoid needing multiple runPar
calls. If this is
inconvenient, then Eval
or Strategies might be a better choice.
In particular, nested calls to runPar
(where a runPar
is
evaluated during the course of executing another Par
computation)
usually give poor results.
Par
monad has more overhead than the Eval
monad. At the
present time, Eval
tends to perform better at finer granularities,
due to the direct runtime system support for sparks. At larger
granularities, Par
and Eval
perform approximately the same.
Par
monad is implemented entirely in a Haskell library (the
monadpar
package), and is thus easily modified. There is a choice of scheduling strategies (see Using Different Schedulers).
Eval
monad has more diagnostics in ThreadScope. There are
graphs that show different aspects of sparks: creation rate,
conversion rate, and so on. The Par
monad is not currently
integrated with ThreadScope.
Par
monad does not support speculative parallelism in the
sense that rpar
does (GC’d Sparks and Speculative Parallelism); parallelism in the Par
monad is always executed.
^{[11] }IVar
has this name because it is an implementation of IStructures, a concept from an early Parallel
Haskell variant called pH.
^{[12] }For more details, see the documentation for Control.Monad.Applicative
.
^{[13] }You can verify this for yourself by profiling the
rsa.hs program. Most of the execution time is spent in power
.
^{[14] }I’m avoiding the term “schedule” here because we already use it a lot in concurrent programming.
^{[15] }I should mention that even with some pruning, an exhaustive search will be impractical beyond a small number of slots. Realworld solutions to this kind of problem use heuristics.
^{[16] }This code was authored by Philip Wadler and found in the nofib
benchmark suite of Haskell programs.
^{[17] }I did, however, take the liberty of modernizing the code in various ways, although that wasn’t strictly necessary.
^{[18] }We are ignoring the possibility of type errors here; in a real implementation, the IVar
would probably contain an Either
type representing either the inferred type or an error.
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