Parallel Computing

Most modern computers possess more than one CPU, and several computers can be combined together in a cluster. Harnessing the power of these multiple CPUs allows many computations to be completed more quickly. There are two major factors that influence performance: the speed of the CPUs themselves, and the speed of their access to memory. In a cluster, it’s fairly obvious that a given CPU will have fastest access to the RAM within the same computer (node). Perhaps more surprisingly, similar issues are relevant on a typical multicore laptop, due to differences in the speed of main memory and the cache. Consequently, a good multiprocessing environment should allow control over the “ownership” of a chunk of memory by a particular CPU. Julia provides a multiprocessing environment based on message passing to allow programs to run on multiple processes in separate memory domains at once.

Julia’s implementation of message passing is different from other environments such as MPI [1]. Communication in Julia is generally “one-sided”, meaning that the programmer needs to explicitly manage only one process in a two-process operation. Furthermore, these operations typically do not look like “message send” and “message receive” but rather resemble higher-level operations like calls to user functions.

Parallel programming in Julia is built on two primitives: remote references and remote calls. A remote reference is an object that can be used from any process to refer to an object stored on a particular process. A remote call is a request by one process to call a certain function on certain arguments on another (possibly the same) process. A remote call returns a remote reference to its result. Remote calls return immediately; the process that made the call proceeds to its next operation while the remote call happens somewhere else. You can wait for a remote call to finish by calling wait on its remote reference, and you can obtain the full value of the result using fetch. You can store a value to a remote reference using put.

Let’s try this out. Starting with julia -p n provides n worker processes on the local machine. Generally it makes sense for n to equal the number of CPU cores on the machine.

$ ./julia -p 2

julia> r = remotecall(2, rand, 2, 2)
RemoteRef(2,1,5)

julia> fetch(r)
2x2 Float64 Array:
 0.60401   0.501111
 0.174572  0.157411

julia> s = @spawnat 2 1+fetch(r)
RemoteRef(2,1,7)

julia> fetch(s)
2x2 Float64 Array:
 1.60401  1.50111
 1.17457  1.15741

The first argument to remotecall is the index of the process that will do the work. Most parallel programming in Julia does not reference specific processes or the number of processes available, but remotecall is considered a low-level interface providing finer control. The second argument to remotecall is the function to call, and the remaining arguments will be passed to this function. As you can see, in the first line we asked process 2 to construct a 2-by-2 random matrix, and in the second line we asked it to add 1 to it. The result of both calculations is available in the two remote references, r and s. The @spawnat macro evaluates the expression in the second argument on the process specified by the first argument.

Occasionally you might want a remotely-computed value immediately. This typically happens when you read from a remote object to obtain data needed by the next local operation. The function remotecall_fetch exists for this purpose. It is equivalent to fetch(remotecall(...)) but is more efficient.

julia> remotecall_fetch(2, getindex, r, 1, 1)
0.10824216411304866

Remember that getindex(r,1,1) is equivalent to r[1,1], so this call fetches the first element of the remote reference r.

The syntax of remotecall is not especially convenient. The macro @spawn makes things easier. It operates on an expression rather than a function, and picks where to do the operation for you:

julia> r = @spawn rand(2,2)
RemoteRef(1,1,0)

julia> s = @spawn 1+fetch(r)
RemoteRef(1,1,1)

julia> fetch(s)
1.10824216411304866 1.13798233877923116
1.12376292706355074 1.18750497916607167

Note that we used 1+fetch(r) instead of 1+r. This is because we do not know where the code will run, so in general a fetch might be required to move r to the process doing the addition. In this case, @spawn is smart enough to perform the computation on the process that owns r, so the fetch will be a no-op.

(It is worth noting that @spawn is not built-in but defined in Julia as a macro. It is possible to define your own such constructs.)

One important point is that your code must be available on any process that runs it. For example, type the following into the julia prompt:

julia> function rand2(dims...)
         return 2*rand(dims...)
       end

julia> rand2(2,2)
2x2 Float64 Array:
 0.153756  0.368514
 1.15119   0.918912

julia> @spawn rand2(2,2)
RemoteRef(1,1,1)

julia> @spawn rand2(2,2)
RemoteRef(2,1,2)

julia> exception on 2: in anonymous: rand2 not defined

Process 1 knew about the function rand2, but process 2 did not. To make your code available to all processes, the require function will automatically load a source file on all currently available processes:

julia> require("myfile")

In a cluster, the contents of the file (and any files loaded recursively) will be sent over the network. It is also useful to execute a statement on all processes. This can be done with the @everywhere macro:

julia> @everywhere id = myid()

julia> remotecall_fetch(2, ()->id)
2

@everywhere include("defs.jl")

A file can also be preloaded on multiple processes at startup, and a driver script can be used to drive the computation:

julia -p <n> -L file1.jl -L file2.jl driver.jl

Each process has an associated identifier. The process providing the interactive julia prompt always has an id equal to 1, as would the julia process running the driver script in the example above. The processes used by default for parallel operations are referred to as workers. When there is only one process, process 1 is considered a worker. Otherwise, workers are considered to be all processes other than process 1.

The base Julia installation has in-built support for two types of clusters:

  • A local cluster specified with the -p option as shown above.
  • And a cluster spanning machines using the --machinefile option. This uses a passwordless ssh login to start julia worker processes (from the same path as the current host) on the specified machines.

Functions addprocs, rmprocs, workers and others, are available as a programmatic means of adding, removing and querying the processes in a cluster.

Other types of clusters can be supported by writing your own custom ClusterManager. See section on ClusterManagers.

Data Movement

Sending messages and moving data constitute most of the overhead in a parallel program. Reducing the number of messages and the amount of data sent is critical to achieving performance and scalability. To this end, it is important to understand the data movement performed by Julia’s various parallel programming constructs.

fetch can be considered an explicit data movement operation, since it directly asks that an object be moved to the local machine. @spawn (and a few related constructs) also moves data, but this is not as obvious, hence it can be called an implicit data movement operation. Consider these two approaches to constructing and squaring a random matrix:

# method 1
A = rand(1000,1000)
Bref = @spawn A^2
...
fetch(Bref)

# method 2
Bref = @spawn rand(1000,1000)^2
...
fetch(Bref)

The difference seems trivial, but in fact is quite significant due to the behavior of @spawn. In the first method, a random matrix is constructed locally, then sent to another process where it is squared. In the second method, a random matrix is both constructed and squared on another process. Therefore the second method sends much less data than the first.

In this toy example, the two methods are easy to distinguish and choose from. However, in a real program designing data movement might require more thought and likely some measurement. For example, if the first process needs matrix A then the first method might be better. Or, if computing A is expensive and only the current process has it, then moving it to another process might be unavoidable. Or, if the current process has very little to do between the @spawn and fetch(Bref) then it might be better to eliminate the parallelism altogether. Or imagine rand(1000,1000) is replaced with a more expensive operation. Then it might make sense to add another @spawn statement just for this step.

Parallel Map and Loops

Fortunately, many useful parallel computations do not require data movement. A common example is a Monte Carlo simulation, where multiple processes can handle independent simulation trials simultaneously. We can use @spawn to flip coins on two processes. First, write the following function in count_heads.jl:

function count_heads(n)
    c::Int = 0
    for i=1:n
        c += randbool()
    end
    c
end

The function count_heads simply adds together n random bits. Here is how we can perform some trials on two machines, and add together the results:

require("count_heads")

a = @spawn count_heads(100000000)
b = @spawn count_heads(100000000)
fetch(a)+fetch(b)

This example, as simple as it is, demonstrates a powerful and often-used parallel programming pattern. Many iterations run independently over several processes, and then their results are combined using some function. The combination process is called a reduction, since it is generally tensor-rank-reducing: a vector of numbers is reduced to a single number, or a matrix is reduced to a single row or column, etc. In code, this typically looks like the pattern x = f(x,v[i]), where x is the accumulator, f is the reduction function, and the v[i] are the elements being reduced. It is desirable for f to be associative, so that it does not matter what order the operations are performed in.

Notice that our use of this pattern with count_heads can be generalized. We used two explicit @spawn statements, which limits the parallelism to two processes. To run on any number of processes, we can use a parallel for loop, which can be written in Julia like this:

nheads = @parallel (+) for i=1:200000000
  int(randbool())
end

This construct implements the pattern of assigning iterations to multiple processes, and combining them with a specified reduction (in this case (+)). The result of each iteration is taken as the value of the last expression inside the loop. The whole parallel loop expression itself evaluates to the final answer.

Note that although parallel for loops look like serial for loops, their behavior is dramatically different. In particular, the iterations do not happen in a specified order, and writes to variables or arrays will not be globally visible since iterations run on different processes. Any variables used inside the parallel loop will be copied and broadcast to each process.

For example, the following code will not work as intended:

a = zeros(100000)
@parallel for i=1:100000
  a[i] = i
end

Notice that the reduction operator can be omitted if it is not needed. However, this code will not initialize all of a, since each process will have a separate copy if it. Parallel for loops like these must be avoided. Fortunately, distributed arrays can be used to get around this limitation, as we will see in the next section.

Using “outside” variables in parallel loops is perfectly reasonable if the variables are read-only:

a = randn(1000)
@parallel (+) for i=1:100000
  f(a[randi(end)])
end

Here each iteration applies f to a randomly-chosen sample from a vector a shared by all processes.

In some cases no reduction operator is needed, and we merely wish to apply a function to all integers in some range (or, more generally, to all elements in some collection). This is another useful operation called parallel map, implemented in Julia as the pmap function. For example, we could compute the singular values of several large random matrices in parallel as follows:

M = {rand(1000,1000) for i=1:10}
pmap(svd, M)

Julia’s pmap is designed for the case where each function call does a large amount of work. In contrast, @parallel for can handle situations where each iteration is tiny, perhaps merely summing two numbers. Only worker processes are used by both pmap and @parallel for for the parallel computation. In case of @parallel for, the final reduction is done on the calling process.

Synchronization With Remote References

Scheduling

Julia’s parallel programming platform uses Tasks (aka Coroutines) to switch among multiple computations. Whenever code performs a communication operation like fetch or wait, the current task is suspended and a scheduler picks another task to run. A task is restarted when the event it is waiting for completes.

For many problems, it is not necessary to think about tasks directly. However, they can be used to wait for multiple events at the same time, which provides for dynamic scheduling. In dynamic scheduling, a program decides what to compute or where to compute it based on when other jobs finish. This is needed for unpredictable or unbalanced workloads, where we want to assign more work to processes only when they finish their current tasks.

As an example, consider computing the singular values of matrices of different sizes:

M = {rand(800,800), rand(600,600), rand(800,800), rand(600,600)}
pmap(svd, M)

If one process handles both 800x800 matrices and another handles both 600x600 matrices, we will not get as much scalability as we could. The solution is to make a local task to “feed” work to each process when it completes its current task. This can be seen in the implementation of pmap:

function pmap(f, lst)
    np = nprocs()  # determine the number of processes available
    n = length(lst)
    results = cell(n)
    i = 1
    # function to produce the next work item from the queue.
    # in this case it's just an index.
    nextidx() = (idx=i; i+=1; idx)
    @sync begin
        for p=1:np
            if p != myid() || np == 1
                @async begin
                    while true
                        idx = nextidx()
                        if idx > n
                            break
                        end
                        results[idx] = remotecall_fetch(p, f, lst[idx])
                    end
                end
            end
        end
    end
    results
end

@async is similar to @spawn, but only runs tasks on the local process. We use it to create a “feeder” task for each process. Each task picks the next index that needs to be computed, then waits for its process to finish, then repeats until we run out of indexes. Note that the feeder tasks do not begin to execute until the main task reaches the end of the @sync block, at which point it surrenders control and waits for all the local tasks to complete before returning from the function. The feeder tasks are able to share state via nextidx() because they all run on the same process. No locking is required, since the threads are scheduled cooperatively and not preemptively. This means context switches only occur at well-defined points: in this case, when remotecall_fetch is called.

Distributed Arrays

Large computations are often organized around large arrays of data. In these cases, a particularly natural way to obtain parallelism is to distribute arrays among several processes. This combines the memory resources of multiple machines, allowing use of arrays too large to fit on one machine. Each process operates on the part of the array it owns, providing a ready answer to the question of how a program should be divided among machines.

Julia distributed arrays are implemented by the DArray type. A DArray has an element type and dimensions just like an Array. A DArray can also use arbitrary array-like types to represent the local chunks that store actual data. The data in a DArray is distributed by dividing the index space into some number of blocks in each dimension.

Common kinds of arrays can be constructed with functions beginning with d:

dzeros(100,100,10)
dones(100,100,10)
drand(100,100,10)
drandn(100,100,10)
dfill(x, 100,100,10)

In the last case, each element will be initialized to the specified value x. These functions automatically pick a distribution for you. For more control, you can specify which processes to use, and how the data should be distributed:

dzeros((100,100), workers()[1:4], [1,4])

The second argument specifies that the array should be created on the first four workers. When dividing data among a large number of processes, one often sees diminishing returns in performance. Placing DArrays on a subset of processes allows multiple DArray computations to happen at once, with a higher ratio of work to communication on each process.

The third argument specifies a distribution; the nth element of this array specifies how many pieces dimension n should be divided into. In this example the first dimension will not be divided, and the second dimension will be divided into 4 pieces. Therefore each local chunk will be of size (100,25). Note that the product of the distribution array must equal the number of processes.

distribute(a::Array) converts a local array to a distributed array.

localpart(a::DArray) obtains the locally-stored portion of a DArray.

localindexes(a::DArray) gives a tuple of the index ranges owned by the local process.

convert(Array, a::DArray) brings all the data to the local process.

Indexing a DArray (square brackets) with ranges of indexes always creates a SubArray, not copying any data.

Constructing Distributed Arrays

The primitive DArray constructor has the following somewhat elaborate signature:

DArray(init, dims[, procs, dist])

init is a function that accepts a tuple of index ranges. This function should allocate a local chunk of the distributed array and initialize it for the specified indices. dims is the overall size of the distributed array. procs optionally specifies a vector of process IDs to use. dist is an integer vector specifying how many chunks the distributed array should be divided into in each dimension.

The last two arguments are optional, and defaults will be used if they are omitted.

As an example, here is how to turn the local array constructor fill into a distributed array constructor:

dfill(v, args...) = DArray(I->fill(v, map(length,I)), args...)

In this case the init function only needs to call fill with the dimensions of the local piece it is creating.

Distributed Array Operations

At this time, distributed arrays do not have much functionality. Their major utility is allowing communication to be done via array indexing, which is convenient for many problems. As an example, consider implementing the “life” cellular automaton, where each cell in a grid is updated according to its neighboring cells. To compute a chunk of the result of one iteration, each process needs the immediate neighbor cells of its local chunk. The following code accomplishes this:

function life_step(d::DArray)
    DArray(size(d),procs(d)) do I
        top   = mod(first(I[1])-2,size(d,1))+1
        bot   = mod( last(I[1])  ,size(d,1))+1
        left  = mod(first(I[2])-2,size(d,2))+1
        right = mod( last(I[2])  ,size(d,2))+1

        old = Array(Bool, length(I[1])+2, length(I[2])+2)
        old[1      , 1      ] = d[top , left]   # left side
        old[2:end-1, 1      ] = d[I[1], left]
        old[end    , 1      ] = d[bot , left]
        old[1      , 2:end-1] = d[top , I[2]]
        old[2:end-1, 2:end-1] = d[I[1], I[2]]   # middle
        old[end    , 2:end-1] = d[bot , I[2]]
        old[1      , end    ] = d[top , right]  # right side
        old[2:end-1, end    ] = d[I[1], right]
        old[end    , end    ] = d[bot , right]

        life_rule(old)
    end
end

As you can see, we use a series of indexing expressions to fetch data into a local array old. Note that the do block syntax is convenient for passing init functions to the DArray constructor. Next, the serial function life_rule is called to apply the update rules to the data, yielding the needed DArray chunk. Nothing about life_rule is DArray-specific, but we list it here for completeness:

function life_rule(old)
    m, n = size(old)
    new = similar(old, m-2, n-2)
    for j = 2:n-1
        for i = 2:m-1
            nc = +(old[i-1,j-1], old[i-1,j], old[i-1,j+1],
                   old[i  ,j-1],             old[i  ,j+1],
                   old[i+1,j-1], old[i+1,j], old[i+1,j+1])
            new[i-1,j-1] = (nc == 3 ? 1 :
                            nc == 2 ? old[i,j] :
                            0)
        end
    end
    new
end

Shared Arrays (Experimental, UNIX-only feature)

Shared Arrays use system shared memory to map the same array across many processes. While there are some similarities to a DArray, the behavior of a SharedArray is quite different. In a DArray, each process has local access to just a chunk of the data, and no two processes share the same chunk; in contrast, in a SharedArray each “participating” process has access to the entire array. A SharedArray is a good choice when you want to have a large amount of data jointly accessible to two or more processes on the same machine.

SharedArray indexing (assignment and accessing values) works just as with regular arrays, and is efficient because the underlying memory is available to the local process. Therefore, most algorithms work naturally on SharedArrays, albeit in single-process mode. In cases where an algorithm insists on an Array input, the underlying array can be retrieved from a SharedArray by calling sdata(S). For other AbstractArray types, sdata just returns the object itself, so it’s safe to use sdata on any Array-type object.

The constructor for a shared array is of the form
SharedArray(T::Type, dims::NTuple; init=false, pids=Int[])

which creates a shared array of a bitstype T and size dims across the processes specified by pids. Unlike distributed arrays, a shared array is accessible only from those participating workers specified by the pids named argument (and the creating process too, if it is on the same host).

If an init function, of signature initfn(S::SharedArray), is specified, it is called on all the participating workers. You can arrange it so that each worker runs the init function on a distinct portion of the array, thereby parallelizing initialization.

Here’s a brief example:

julia> addprocs(3)
3-element Array{Any,1}:
 2
 3
 4

julia> S = SharedArray(Int, (3,4), init = S -> S[localindexes(S)] = myid())
3x4 SharedArray{Int64,2}:
 2  2  3  4
 2  3  3  4
 2  3  4  4

julia> S[3,2] = 7
7

julia> S
3x4 SharedArray{Int64,2}:
 2  2  3  4
 2  3  3  4
 2  7  4  4

localindexes provides disjoint one-dimensional ranges of indexes, and is sometimes convenient for splitting up tasks among processes. You can, of course, divide the work any way you wish:

julia> S = SharedArray(Int, (3,4), init = S -> S[myid()-1:nworkers():length(S)] = myid())
3x4 SharedArray{Int64,2}:
 2  2  2  2
 3  3  3  3
 4  4  4  4

Since all processes have access to the underlying data, you do have to be careful not to set up conflicts. For example:

@sync begin
    for p in workers()
        @async begin
            remotecall_wait(p, fill!, S, p)
        end
    end
end

would result in undefined behavior: because each process fills the entire array with its own pid, whichever process is the last to execute (for any particular element of S) will have its pid retained.

ClusterManagers

Julia worker processes can also be spawned on arbitrary machines, enabling Julia’s natural parallelism to function quite transparently in a cluster environment. The ClusterManager interface provides a way to specify a means to launch and manage worker processes. For example, ssh clusters are also implemented using a ClusterManager:

immutable SSHManager <: ClusterManager
    launch::Function
    manage::Function
    machines::AbstractVector

    SSHManager(; machines=[]) = new(launch_ssh_workers, manage_ssh_workers, machines)
end

function launch_ssh_workers(cman::SSHManager, np::Integer, config::Dict)
    ...
end

function manage_ssh_workers(id::Integer, config::Dict, op::Symbol)
    ...
end

where launch_ssh_workers is responsible for instantiating new Julia processes and manage_ssh_workers provides a means to manage those processes, e.g. for sending interrupt signals. New processes can then be added at runtime using addprocs:

addprocs(5, cman=LocalManager())

which specifies a number of processes to add and a ClusterManager to use for launching those processes.

Footnotes

[1]In this context, MPI refers to the MPI-1 standard. Beginning with MPI-2, the MPI standards committee introduced a new set of communication mechanisms, collectively referred to as Remote Memory Access (RMA). The motivation for adding RMA to the MPI standard was to facilitate one-sided communication patterns. For additional information on the latest MPI standard, see http://www.mpi-forum.org/docs.