Open Access

d2o: a distributed data object for parallel high-performance computing in Python

Journal of Big Data20163:17

https://doi.org/10.1186/s40537-016-0052-5

Received: 10 June 2016

Accepted: 30 August 2016

Published: 15 September 2016

Abstract

We introduce d2o, a Python module for cluster-distributed multi-dimensional numerical arrays. It acts as a layer of abstraction between the algorithm code and the data-distribution logic. The main goal is to achieve usability without losing numerical performance and scalability. d2o’s global interface is similar to the one of a numpy.ndarray, whereas the cluster node’s local data is directly accessible for use in customized high-performance modules. d2o is written in pure Python which makes it portable and easy to use and modify. Expensive operations are carried out by dedicated external libraries like numpy and mpi4py. The performance of d2o is on a par with numpy for serial applications and scales well when moving to an MPI cluster. d2o is open-source software available under the GNU General Public License v3 (GPL-3) at https://gitlab.mpcdf.mpg.de/ift/D2O.

Keywords

ParallelizationNumericsMPIPythonNumpyOpen source

Introduction

Background

Data sets in simulation and signal-reconstruction applications easily reach sizes too large for a single computer’s random access memory (RAM). A reasonable grid size for such tasks like galactic density reconstructions [1] or multi-frequency imaging in radio astronomy [2] is a cube with a side resolution of 2048. Such a cube contains \(2048^3 \approx 8.6 \cdot 10^9\) voxels. Storing a 64-bit double for every voxel therefore consumes \(64~\mathrm {GiB}\). In practice one has to handle several or even many instances of those arrays which ultimately prohibits the use of single shared memory machines. Apart from merely holding the arrays’ data in memory, parallelization is needed to process those huge arrays within reasonable time. This applies to basic arithmetics like addition and multiplication as well as to complex operations like Fourier transformation and advanced linear algebra, e.g. operator inversions or singular value decompositions. Thus parallelization is highly advisable for code projects that must be scaled to high resolutions.

To be specific, the initial purpose of d2o was to provide parallelization to the package for Numerical Information Field Theory (NIFTy)[3], which permits the abstract and efficient implementation of sophisticated signal processing methods. Typically, those methods are so complex on their own that a NIFTy user should not need to bother with parallelization details in addition to that. It turned out that providing a generic encapsulation for parallelization to NIFTy is not straightforward as the applications NIFTy is used for are highly diversified. The challenge hereby is that, despite all their peculiarities, for those applications numerical efficiency is absolutely crucial. Hence, for encapsulating the parallelization effort in NIFTy we needed an approach that is flexible enough to adapt to those different applications such that numerical efficiency can be preserved: d2o.

d2o is implemented in Python. As a high-level language with a low-entry barrier Python is widely used in computational science. It has a huge standard library and an active community for 3rd party packages. For computationally demanding applications Python is predominantly used as a steering language for external compiled modules because Python itself is slow for numerics.

This article is structured as follows. "Aim" section gives the aims of d2o, and "Alternative packages" section describes alternative data distribution packages. We dicuss the code architecture in "Code architecture" section, the basic usage of d2o in "Basic usage" section, and the numerical scaling behavior in section "Performance and scalability". "Summary and outlook" section contains our conclusion and "Appendix 1" describes the detailed usage of d2o.

Aim

As most scientists are not fully skilled software engineers, for them the hurdle for developing parallelized code is high. Our goal is to provide data scientists with a numpy array-like object (cf. numpy [4]) that distributes data among several nodes of a cluster in a user-controllable way. The user, however, shall not need to have profound knowledge about parallel programming with a system like MPI [5, 6] to achieve this. The transition to use distributed_data_objects instead of numpy arrays in existing code must be as straightforward as possible. Hence, d2o shall in principle run—at least in a non-parallelized manner—with standard-library dependencies available; the packages needed for parallel usage should be easily available. Whilst providing a global-minded interface, the node’s local data should be directly accessible in order to enable the usage in specialized high-performance modules. This approach matches with the theme of DistArray [7]: “Think globally, act locally”. Regarding d2o’s architecture we do not want to make any a-priori assumptions about the specific distribution strategy, but retain flexibility: it shall be possible to adapt to specific constraints induced from third-party libraries a user may incorporate. For example, a library for fast Fourier transformations like FFTW [8] may rely on a different data-distribution model than a package for linear algebra operations like ScaLAPACK [9].1 In the same manner it shall not matter whether a new distribution scheme stores data redundantly or not, e.g. when a node is storing not only a distinct piece of a global array, but also its neighboring (ghost) cells [10].

Our main focus is on rendering extremely costly computations possible in the first place; not on improving the speed of simple computations that can be done serially. Although primarily geared towards weak scaling, it turns out that d2o performs very well in strong-scaling scenarios, too; see "Performance and scalability" section for details.

Alternative packages

There are several alternatives to d2o. We discuss the differences to d2o and why the alternatives are not sufficient for our needs.

DistArray

DistArray [7] is very mature and powerful. Its approach is very similar to d2o: It mimics the interface of a multi dimensional numpy array while distributing the data among nodes in a cluster. However, DistArray involves a design decision that makes it inapt for our purposes: it has a strict client-worker architecture. DistArray either needs an ipython ipcluster [11] as back end or must be run with two or more MPI processes. The former must be started before an interactive ipython session is launched. This at least complicates the workflow in the prototyping phase and at most is not practical for batch system based computing on a cluster. The latter enforces tool-developers who build on top of DistArray to demand that their code always is run parallelized. Both scenarios conflict with our goal of minimal second order dependencies and maximal flexibility, cf. "Aim" section. Nevertheless, its theme also applies to d2o: “Think globally, act locally”.

Scalapy (ScaLAPACK)

scalapy is a Python wrapper around ScaLAPACK [9], which is “a library of high-performance linear algebra routines for parallel distributed memory machines” [12]. The scalapy.DistributedMatrix class essentially uses the routines from ScaLAPACK and therefore is limited to the functionality of that: two-dimensional arrays and very specific block-cyclic distribution strategies that optimize numerical efficiency in the context of linear algebra problems. In contrast, we are interested in n-dimensional arrays whose distribution scheme shall be arbitrary in the first place. Therefore scalapy is not extensive enough for us.

Petsc4py (PETSc)

petsc4py is a Python wrapper around PETSc, which “is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations” [13]. Regarding distributed arrays its scope is as focused as scalapy to its certain problem domain—here: solving partial differential equations. The class for distributed arrays petsc4py.PETSc.DMDA is limited to one, two and three dimensions as PETSc uses a highly problem-fitted distribution scheme. We in contrast need n-dimensional arrays with arbitrary distribution schemes. Hence, petsc4py is not suitable for us.

Code architecture

Choosing the right level of parallelization

d2o distributes numerical arrays over a cluster in order to parallelize and therefore to speed up operations on the arrays themselves. An application that is built on top of d2o can profit from its fast array operations that may be performed on a cluster. However, there are various approaches how to deploy an algorithm on a cluster and d2o implements only one of them. In order to understand the design decisions of d2o and its position respective to other packages, cf. "Alternative packages" section, we will now discuss the general problem setting of parallelization and possible approaches for that. Thereby we reenact the decision process which led to the characteristics d2o has today.

Vertical and horizontal scaling

Suppose we want to solve an expensive numerical problem which involves operations on data arrays. To reduce the computation time one can in principle do two things. Either use a faster machine—vertical scaling—or use more than one machine—horizontal scaling. Vertical scaling has the advantage that existing code does not need to be changed,2 but in many cases this is not appropriate. Maybe one already uses the fastest possible machine, scaling up is not affordable or even the fastest machine available is still too slow. Because of this, we choose horizontal scaling.

High- and low-level parallelization

With horizontal scaling we again face two choices: high- and low-level parallelization. With high-level parallelization, many replicas of the algorithm run simultaneously, potentially on multiple machines. Each instance then works independently if possible, solving an isolated part of the global problem. At the end, the individual results get collected and merged. The python framework pathos [14] provides functionality for this kind of procedure.

An example of high-level parallelization is a sample generator which draws from a probability distribution. Using high-level parallelization many instances of the generator produce their own samples, which involves very little communication overhead. The sample production process itself, however, is not sped up.

In low-level parallelization, several nodes work together on one basic task at a time. For the above sample generator, this means that all nodes work on the same sample at a time. Hence, the time needed for producing individual samples is reduced; they are serially generated by the cluster as a whole.

Downsides

Both of these approaches have their drawbacks. For high-level parallelization the algorithm itself must be parallelizable. Every finite algorithm has a maximum degree of intrinsic parallelization.3 If this degree is lower than the desired number of processes then high-level parallelization reaches its limits. This is particularly true for algorithms that cannot be parallelized by themselves, like iterative schemes. Furthermore, there can be an additional complication: if the numerical problem deals with extremely large objects it may be the case that it is not at all solvable by one machine alone.4

Now let us consider low-level parallelization. As stated above, we assume that the solution of the given numerical problem involves operations on data arrays. Examples for those are unary,5 binary6 or sorting operations, but also more advanced procedures like Fourier transformations or (other) linear algebra operations. Theoretically, the absolute maximum degree of intrinsic parallelization for an array operation is equal to the array’s number of elements. For comparison, the problems we want to tackle involve at least \(10^8\) elements but most of the TOP500 [15] supercomputers possess \(10^6\) cores or less. At first glance this seems promising. But with an increasing number of nodes that participate in one operation the computational efficiency may decrease considerably. This happens if the cost of the actual numerical operations becomes comparable to the generic program and inter-node communication overhead. The ratios highly depend on the specific cluster hardware and the array operations performed.

Problem sizes

Due to our background in signal reconstruction and grid-based simulations, we decide to use low-level parallelization for the following reasons. First, we have to speed up problems that one cannot parallelize algorithmically, like fixed-point iterations or step-wise simulations. Second, we want to scale our algorithms to higher resolutions while keeping the computing time at least constant. Thereby the involved data arrays become so big that a single computer would be oversubscribed. Because of this, the ratio of array size to desired degree of parallelization does not become such that the computational efficiency would decrease considerably. In practice we experience a good scaling behavior with up to \({\approx}10^3\) processes7 for problems of size \(8192^2\), cf. "Performance and scalability" section. Hence, for our applications the advantages of low-level parallelization clearly outweigh its drawbacks.

d2o as layer of abstraction

Compared to high-level parallelization, the low-level approach is more complicated to implement. In the best case, for the former one simply runs the serial code in parallel on the individual machines; when finished one collects and combines the results. For the latter, when doing the explicit coding one deals with local data portions of the global data array on the individual nodes of the cluster. Hence, one has to keep track of additional information: for example, given a distribution scheme, which portion of the global data is stored on which node of the cluster? Keeping the number of cluster nodes, the size and the dimensionality of the data arrays arbitrary implies a considerable complication for indexing purposes. By this, while implementing an application one has to take care of two non-trivial tasks. On the one hand, one must program the logic of distributing and collecting the data; i.e. the data handling. On the other hand, one must implement the application’s actual (abstract) algorithm. Those two tasks are conceptually completely different and therefore a mixture of implementations should be avoided. Otherwise there is the risk that features of an initial implementation—like the data distribution scheme—become hard-wired to the algorithm, inhibiting its further evolution. Thus it makes sense to insert a layer of abstraction between the algorithm code and the data distribution logic. Then the abstract algorithm can be written in a serial style from which all knowledge and methodology regarding the data distribution is encapsulated. This layer of abstraction is d2o.

Choosing a parallelization environment

To make the application spectrum of d2o as wide as possible we want to maximize its portability and reduce its dependencies. This implies that—despite its parallel architecture—d2o must just as well run within a single-process environment for cases when no elaborate parallelization back end is available. But nevertheless, d2o must be massively scalable. This relates to the question of which distribution environment should be used. There are several alternatives:
  • Threading and multiprocessing: These two options limit the application to a single machine which conflicts with the aim of massive scalability.

  • (py)Spark [16] and hadoop [17]: These modern frameworks are very powerful but regrettably too abstract for our purposes, as they prescind the location of individual portions of the full data. Building a numpy-like interface would be disproportionately hard or even unfeasible. In addition to that, implementing a low-level interface for highly optimized applications which interact with the node’s local data is not convenient within pySpark. Lastly, those frameworks are usually not installed as standard dependencies on scientific HPC clusters.

  • MPI [5, 6]: The Message Passing Interface is available on virtually every HPC cluster via well-tested implementations like OpenMPI [18], MPICH2 [19] or Intel MPI [20]. The open implementations are also available on commodity multicore hardware like desktops or laptops. A Python interface to MPI is given by the Python module mpi4py [21]. MPI furthermore offers the right level of abstraction for hands-on control of distribution strategies for the package developers.

Given these features we decide to use MPI as the parallelization environment for d2o. We stress that in order to fully utilize d2o on multiple cores, a user does not need to know how to program in MPI; it is only necessary to execute the program via MPI as shown in the example in "Distributed arrays" section.

Internal structure

Composed object

A main goal for the design of d2o was to make no a-priori assumptions about the specific distribution strategies that will be used in order to spread array data across the nodes of a cluster. Because of this, d2o’s distributed array—d2o.distributed_data_object—is a composed object; cf. Fig. 1.

The distributed_data_object itself provides a rich user interface, and makes sanity and consistency checks regarding the user input. In addition to that, the distributed_data_object possesses an attribute called data. Here the MPI processes’ local portion of the global array data is stored, even though the distributed_data_object itself will never make any assumptions about its specific content since the distribution strategy is arbitrary in the first place. The distributed_data_object is the only object of the d2o library that a casual user would interact with.

For all tasks that require knowledge about the certain distribution strategy every distributed_data_object possesses an instance of a d2o.distributor subclass. This object stores all the distribution-scheme and cluster related information it needs in order to scatter (gather) data to (from) the nodes and to serve for special methods, e.g. the array-cumulative sum. The distributed_data_object builds its rich user interface on top of those abstracted methods of its distributor.

The benefit of this strict separation is that the user interface becomes fully detached from the distribution strategy; may it be block-cyclic or slicing, or have neighbor ghost cells or not, et cetera. Currently there are two fundamental distributors available: a generic slicing-8 and a not-distributor. From the former, three special slicing distributors are derived: fftw 9, equal 10 and freeform.11 The latter, the not-distributor, does not do any data-distribution or -collection but stores the full data on every node redundantly.

Advantages of a global view interface

d2o’s global view interface makes it possible to build software that remains completely independent from the distribution strategy and the used number of cluster processes. This in turn enables the development of 3rd party libraries that are very end-use-case independent. An example for this may be a mathematical optimizer; an object which tries to find for a given scalar function f an input vector \(\vec{x}\) such that the output \(y = f(\vec{x})\) becomes minimal. It is interesting to note that many optimization algorithms solely use basic arithmetics like vector addition or scalar multiplication when acting on \(\vec{x}\). As such operations act locally on the elements of an array, there is no preference for one distribution scheme over another when distributing \(\vec{x}\) among nodes in a cluster. Two different distribution schemes will yield the same performance if their load-balancing is on a par with each other. Further assume that f is built on d2o, too. On this basis, one could now build an application that uses the minimizer but indeed has a preference for a certain distribution scheme. This may be the case if the load-balancing of the used operations is non-trivial and therefore only a certain distribution scheme guarantees high evaluation speeds. While the application’s developer therefore enforces this scheme, the minimizer remains completely unaffected by this as it is agnostic of the array’s distribution strategy.

Basic usage

In the subsequent sections we will illustrate the basic usage of d2o in order to explain its functionality and behavior. A more extended discussion is given in "Appendix 1". Our naming conventions are:
  • instances of the numpy.ndarray class are labeled a and b,

  • instances of d2o.distributed_data_object are labeled obj and p.

In addition to these examples, the interested reader is encouraged to have a look into the distributed_data_object method’s docstrings for further information; cf. the project’s web page https://gitlab.mpcdf.mpg.de/ift/D2O.

Initialization

Here we discuss how to initialize a distributed_data_object and compare some of its basic functionality to that of a numpy.ndarray. First we import the packages.
Now we set up some test data using numpy.
One way to initialize a distributed_data_object is to pass an existing numpy array.

The output of the obj call shows the local portion of the global data available in this process.

Arithmetics

Simple arithmetics and point-wise comparison work as expected from a numpy array.
Please note that the distributed_data_object tries to avoid inter-process communication whenever possible. Therefore the returned objects of those arithmetic operations are instances of distributed_data_object, too. However, the d2o user must be careful when combining distributed_data_objects with numpy arrays. If one combines two objects with a binary operator in Python (like +, −, \(\ast\), \, % or \(\ast\ast\), it will try to call the respective method (__add__, __sub__, etc.) of the first object. If this fails, i.e. if it throws an exception, Python will try to call the reverse methods of the second object (__radd__, __rsub__, etc.):
Depending on the conjunction’s ordering, the return type may vary when combining numpy arrays with distributed_data_objects. If the numpy array is in the first place, numpy will try to extract the second object’s array data using its __array__ method. This invokes the distributed_data_object’s get_full_data method that communicates the full data to every process. For large arrays this is extremely inefficient and should be avoided by all means. Hence, it is crucial for performance to assure that the distributed_data_object’s methods will be called by Python. In this case, the locally relevant parts of the array are extracted from the numpy array and then efficiently processed as a whole.

Array indexing

The distributed_data_object supports most of numpy’s indexing functionality, so it is possible to work with scalars, tuples, lists, numpy arrays and distributed_data_objects as input data. Every process will extract its locally relevant part of the given data-object and then store it; cf. "Array indexing" section.

By default it is assumed that all processes use the same key-object when accessing data. See "Local keys" section for more details regarding process-individual indexing.

Distribution strategies

In order to specify the distribution strategy explicitly one may use the “distribution_strategy” keyword:

See "Copy methods" section for more information on distribution strategies.

Distributed arrays

To use d2o in a distributed manner, one has to create an MPI job. This example shows how four MPI processes hold individual parts of the global data and how distributed read & write access works. The script is started via the command:
The distributed_data_object gets initialized in line 11 with the following array:
Here, the script is run in four MPI processes; cf. mpirun -n4 [...]. The data is split along the first axis; the print statement in line 14 yields the four pieces:
The second print statement (line 16) illustrates the behavior of data extraction; obj[0:3:2, 1:3] is slicing notation for the entries 1, 2, 9 and 10.12 This expression returns a distributed_data_object where the processes possess the individual portion of the requested data. This means that the distribution strategy of the new (sub-)array is determined by and aligned to that of the original array.
The result is a distributed_data_object where the processes 1 and 3 do not possess any data as they had no data to contribute to the slice in obj[0:3:2, 1:3]. In line 19 we store a small 2 × 2 block b in the lower middle of obj. The process’ local data reads:
Finally, in line 25 we use obj.get_full_data() in order to consolidate the distributed data; i.e. to communicate the individual pieces between the processes and merge them into a single numpy array.

Performance and scalability

In this section we examine the scaling behavior of a distributed_data_object that uses the equal distribution strategy. The timing measurements were performed on the C2PAP Computing Cluster [23].13 The software stack was built upon Intel MPI 5.1, mpi4py 2.0, numpy 1.11 and python 2.7.11. For measuring the individual timings we used the Python standard library module timeit with a fixed number of 100 repetitions.

Please note that d2o comes with an extensive suite of unit tests featuring a high code-coverage rate. By this we assure d2o’s continuous code development to be very robust and provide the users with a reliable interface definition.

Regarding d2o’s performance there are two important scaling directions: the size of the array data and the number of MPI processes. One may distinguish three different contributions to the overall computational cost. First, there is data management effort that every process has to cover itself. Second, there are the costs for inter-MPI-process communication. And third, there are the actual numerical costs.

d2o has size-independent management overhead compared to numpy. Hence, the larger the arrays are for which d2o is used, the more efficient the computations become. We will see below that there is a certain array size per node—roughly \(2^{16}\) elements—from which on d2o’s management overhead becomes negligible compared to the purely numerical costs. This size corresponds to a two-dimensional grid with a resolution of 256 × 256 or equivalently 0.5 MiB of 64-bit doubles. In "Scaling the array size" section we focus on this very ratio of management overhead to numerical costs.

d2o raises the claim to be able to operate well running with a single process as well as in a highly parallelized regime. In "Weak scaling: proportional number of processes and size of data" section, the scaling analysis regarding the MPI process count is done with a fixed local array size for which the process overhead is negligible compared to the numerical costs. For this weak scaling analysis we are interested in the costs arising from inter-process communication compared to those of actual numerics.

In the following three sections, we study the strong scaling of d2o where the performance is a result of the combination of all three cost contributions. "Strong scaling: varying number of processes with a fixed size of data" section covers the case in which the number of MPI processes is increased while the array size is left constant. In "Strong scaling: comparison with DistArray" section we compare d2o's to DistArray's [7] performance and finally, in section Strong scaling: real-world application speedup—the Wiener filter we benchmark d2o's strong-scaling behavior when applied to a real-world application: a Wiener filter signal reconstruction.

A discussion on d2o's efficient Python iterators can be found in "Appendix 2".

Scaling the array size

One may think of d2o as a layer of abstraction that is added to numpy arrays in order to take care of data distribution and collection among multiple MPI processes. This abstraction comes with inherent Python overhead, separately for each MPI process. Therefore, if one wants to analyze how the ratio of management overhead to actual numerical effort varies with the data size, only the individual process’ data size is important. Because of this, all timing tests for this subsection were carried out with one MPI process only.

A common task during almost all numerical operations is to create a new array object for storing its results.14 Hence, the speed of object creation can be crucial for overall performance there. Note that in contrast to a numpy array which basically just allocates RAM, several things must be done during the initialization of a distributed_data_object. The Python object instance itself must be created, a distributor must be initialized which involves parsing of user input, RAM must be allocated, the distributed_data_object must be registered with the d2o_librarian (cf. "The d2o librarian" section), and, if activated, inter-MPI-process communication must be done for performing sanity checks on the user input.

By default the initialization requires 60 μs to complete for a distributed_data_object with a shape of (1,) when run within one single MPI process. Using this trivial shape makes the costs for memory allocation negligible compared to the others tasks. Hence, those 60 μs represent d2o’s constant overhead compared to numpy, since a comparable numpy array requires ≈0.4 μs for initialization.

In order to speed up the initialization process one may disable all sanity checks on the user input that require MPI communication, e.g. if the same datatype was specified in all MPI processes. Even when run with one single process, skipping those checks reduces the costs by 27 μs from 60 to 33 μs .

Because of the high costs, it is important to avoid building distributed_data_objects from scratch over and over again. A simple measure against this is to use inplace operations like obj+= 1 instead of obj = obj + 1 whenever possible. This is generally a favorable thing to do—also for numpy arrays—as this saves the costs for repeated memory allocation. Nonetheless, also non-inplace operations can be improved in many cases, as often the produced and the initial distributed_data_object have all of their attributes in common, except for their data: they are of the same shape and datatype, and use the same distribution strategy and MPI communicator; cf. p = obj + 1. With obj.copy() and obj.copy_empty() there exist two cloning methods that we implemented to be as fast as allowed by pure Python. Those methods reuse as much already initialized components as possible and are therefore faster than a fresh initialization: for the distributed_data_object from above obj.copy() and obj.copy_empty() consume 7.9 and 4.3 \(\upmu\), respectively.

Table 3 shows the performance ratio in percent between serial d2o and numpy. The array sizes range from \(2^0=1\) to \(2^{25} \approx 3.3\cdot 10^7\) elements. In the table, \(100\,\%\) would mean that d2o is as fast as numpy.

The previous section already suggested that for tasks that primarily consist of initialization work—like array creation or copy_emptyd2o will clearly follow behind numpy. However, increasing the array size from \(2^{20}\) to \(2^{22}\) elements implies a considerable performance drop for numpy’s memory allocation. This in turn means that for arrays with more than \(2^{22}\) elements d2o’s relative overhead becomes less significant: e.g. np.copy_empty is then only a factor of four faster than obj.copy_empty().

Functions like max and sum return a scalar number; no expensive return-array must be created. Hence, d2o’s overhead is quite modest: even for size 1 arrays, d2o’s relative performance lies above 50 %. Once the size is greater than \(2^{18}\) elements the performance is higher than 95 %.

obj[::-2] is slicing syntax for “take every second element from the array in reverse order”. It illustrates the costs of the data-distribution and collection logic that even plays a significant role if there is no inter-process communication involved. Again, with a large-enough array size, d2o’s efficiency becomes comparable to that of numpy.

Similarly to obj[::-2], the remaining functions in the table return a distributed_data_object as their result and therefore suffer from its initialization costs. However, with an array size of \(2^{16}\) elements and larger d2o’s relative performance is at least greater than approximately 65 %.

An interesting phenomenon can be observed for obj + 0 and obj + obj: As for the other functions, their relative performance starts to increase significantly when an array size of \(2^{16}\) is reached. However, in contrast to obj+= obj which then immediately scales up above \(95\,\%\), the relative performance of the non-inplace additions temporarily decreases with increasing array size. This may be due to the fact that given our test scenario \(2^{18}\) elements almost take up half of the cache of C2PAP’s Intel E5-2680 CPUs. d2o’s memory overhead is now responsible for the fact, that its non-inplace operations—which need twice the initial memory—cannot profit that strongly from the cache anymore, whereas the numpy array still operates fast. Once the array size is above \(2^{22}\) elements numpy’s just as d2o’s array-object is too large for profiting from the cache and therefore become comparably fast again: the relative performance is then greater than \(98\,\%\).

Thus, when run within a single process, d2o is ideally used for arrays larger than \(2^{16}=65536\) elements which corresponds to \(512~\mathrm {KiB}\). From there the management overhead becomes less significant than the actual numerical costs.

Weak scaling: proportional number of processes and size of data

Now we analyze the scaling behavior of d2o when run with several MPI processes. Repeating the beginning of "Performance and scalability" section, there are three contributions to the execution time. First, the fixed management overhead that every process has to cover itself, second, the communication overhead and third, the actual numerical costs. In order to filter out the effect of a changing contribution of management overhead, in this section we fix the MPI processes’ local array size to a fixed value. Hence, now the global data size is proportional to the number of MPI processes.

Table 4 shows the performance of various array operations normalized to the time d2o needs when running with one process only. Assuming that d2o had no communication overhead and an operation scaled perfectly linearly with array size, the performance would be rated at 100 %.

In theory, operations that do not inherently require inter-process communication like point-wise array addition or subtraction ideally scale linearly. And in fact, d2o scales excellently with the number of processes involved for those functions: here we tested copy, copy_empty, sum(axis = 1), obj + 0, obj + obj, obj += obj and sqrt.

Comparing sum(axis = 0) with sum(axis = 1) illustrates the performance difference between those operations that involve inter-process communication and those that don’t: the equal distribution strategy slices the global array along its first axis in order to distribute the data among the individual MPI processes. Hence, sum(axis = 0)—which means to take the sum along the first axis—does intrinsically involve inter-process communication whereas sum(axis = 1) does not. Similarly to sum(axis = 0) also the remaining functions in Table 4 are affected by an increasing number of processes as they involve inter-process communication.

But still, even if—for example in case of sum(axis = 0)—the relative performance may drop to 28.2 % when using 256 processes, this means that the operation just took 3.5 times longer than the single-process run, whereat the array size has been increased by a factor of 256. This corresponds to a speedup factor of 72.2.

Strong scaling: varying number of processes with a fixed size of data

Similarly to the previous "Weak scaling: proportional number of processes and size of data" section, we vary the number of processes but now fix the data size globally instead of locally. This corresponds to the real-life scenario in which the problem size and resolution are already set—maybe by environmental conditions—and now one tries to reduce the run time by using more cores. Since the size of the local data varies with the number of processes, the overall scaling behavior is now a mixture of the varying ratio between management overhead and process-individual numerical costs, and the fact that an increasing amount of CPU cache becomes available at the expense of increased communication effort. Table 5 shows the benchmarking results for the same set of operations as used in the previous section on weak scaling and the results are reasonable. Those operations in the list that inherently cannot scale strongly as they consist of purely node-individual work, namely the initialization and copy_empty, show that their performance just does not increase with the number of processes. In contrast, operations without communication effort benefit strongly from the increasing total amount of CPU cache combined with smaller local arrays; above all copy which is about 3 times faster than what one would expect from linear scaling to 256 processes.15

In theory, the strong-scaling behavior is the combination of the size- and weak-scaling we discussed in sections "Scaling the array size" and "Weak scaling: proportional number of processes and size of data". In order to verify whether the strong-scaling behavior makes sense, we estimate the strong-scaling performance using the information from size- and weak-scaling.

We choose the sum() method as our test case. During the reduction phase, the n MPI-processes exchange their local results with each other. This corresponds to adding n times a fixed amount of communication time to the total computing time needed. Hence, we perform a linear fit to the weak-scaling data; cf. Table 4. Furthermore, we assume that the local computation time is roughly proportional to the local-array size. This is true for sufficiently large array sizes, since then numpy scales linearly and d2o is in a good efficiency regime, cf. Table 3. Again we performed a linear fit but now on the size-scaling timing data. Combining those two linear fits leads to the following run-time formula for applying sum() to an array with shape (4096, 4096):
$$\begin{aligned} t(n) = (0.0065n + 1.57/n)~\mathrm {s} \end{aligned}$$
(1)
In the case of linear scaling, t(n) is expected to be equal to t(1)/n. Hence, the relative performance p(n) is the ratio between the two:
$$\begin{aligned} p(n)_{\mathrm {estimated}} = \frac{\frac{t(1)}{n}}{t(n)} = \frac{241.8}{240.8 + n^2} \end{aligned}$$
(2)
Comparing the estimate with the actually measured relative performance—cf. Tables 1, 2, 3, 4, 5 shows that even under those rough assumptions the strong scaling behavior of sum() can be explained as the combination of size- and weak scaling within about 20 % accuracy.
Table 1

Strong scaling behavior: estimate vs. measurement

\(\#\mathrm {processes:~} n\)

1 (%)

4 (%)

8 (%)

16 (%)

32 (%)

64 (%)

128 (%)

256 (%)

\(p(n)_{\mathrm {estimated}}\)

100

94.2

79.3

48.7

19.1

5.58

1.45

0.37

\(p(n)_{\mathrm {measured}}\)

100

91.7

74.1

60.9

24.4

7.05

1.82

0.45

Table 2

Execution time scaling of a Wiener filter reconstruction on a grid of size \(8192 \times 8192\)

\(\#\mathrm {nodes}\)

1

1

1

1

2

4

8

16

32

\(\#\mathrm {processes:~} n\)

1

2

3

4

8

16

32

64

128

t [S]

1618

622.0

404.2

364.2

181.7

94.50

46.79

18.74

8.56

\(s_n = 2t_2/t_n\)

0.769

2.00

3.08

3.42

6.85

13.2

26.6

66.4

145

\(q_n = {1/(1+\log (\frac{n}{s_n}))}\)

0.900

1.00

1.01

0.94

0.94

0.92

0.93

1.02

1.06

Table 3

Overhead costs: d2o’s relative performance to numpy when scaling the array size and being run with one MPI-process

Array size

2\(^{\mathrm {0}}\) (%)

2\(^{\mathrm {2}}\) (%)

2\(^{\mathrm {4}}\) (%)

2\(^{\mathrm {6}}\) (%)

2\(^{\mathrm {8}}\) (2 KiB) (%)

2\(^{\mathrm {10}}\) (%)

2\(^{\mathrm {12}}\) (%)

2\(^{\mathrm {14}}\) (128 KiB) (%)

2\(^{\mathrm {16}}\) (%)

2\(^{\mathrm {18}}\) (%)

2\(^{\mathrm {20}}\) (8 MiB) (%)

\(2_{\mathrm {22}}\) (%)

2\(^{\mathrm {23}}\) (%)

2\(^{\mathrm {24}}\) (128 MiB) (%)

2\(^{\mathrm {25}}\) (%)

initialization

0.65

0.64

0.69

0.69

0.71

0.71

0.74

0.72

0.75

0.74

0.75

5.41

5.58

5.83

6.00

copy .empty

3.77

3.86

4.00

4.12

4.57

3.92

3.95

3.98

4.01

4.10

4.15

25.0

24.2

25.3

26.0

max

56.2

56.0

54.4

55.5

56.0

56.9

62.6

79.2

91.7

97.5

99.4

99.4

99.9

99.8

99.9

sum

59.7

59.0

57.5

60.1

58.6

59.8

62.5

74.8

88.4

95.9

99.0

99.3

99.7

99.7

100.0

obj[::-2]

1.18

1.17

1.20

1.24

1.34

1.50

2.14

4.77

15.0

22.2

28.7

24.1

45.8

47.3

47.7

copy

8.16

8.86

9.30

9.72

9. 66

10.4

14.8

26.7

65.8

95.5

98.7

99.1

99.9

99.4

98.4

obj \(+\) 0

6.58

6.48

6.67

7.08

7.37

9.51

16.2

35.0

65.2

34.7

44.7

98.7

99.7

96.5

100.0

obj \(+\) obj

3.07

3.10

3.24

3.59

3.98

5.70

13.3

31.9

64.0

34.6

45.0

98.7

99.9

99.5

99.8

obj \(+=\) obj

5.17

5.35

5.41

6.02

6.66

11.3

22.8

46.4

75.3

91.9

97.8

98.0

99.8

99.3

99.7

sqrt

3.25

3.17

3.26c

3.50

4.42

7.49

18.7

45.6

75.8

65.2

83.2

98.8

99.4

99.6

99.8

bincount

3.57

3.35

3.76

4.04

4.97

7.54

16.5

35.4

58.0

75.1

78.0

76.8

78.9

82.1

83.2

“100 %” corresponds to the case when d2o is as fast as numpy. In order to guide the eye, values <30 % are printed italic, values ≥90 % are printed bold-italics. Please see "Scaling the array size" section for discussion

Table 4

Weak scaling: d2o’s relative performance to the single-process case when increasing both, the number of processes and the global array size proportionally

Process count

1 (%)

2 (%)

3 (%)

4 (%)

S (%)

16 (%)

32 (%)

64 (%)

128 (%)

256 (%)

initialization

100.0

90.9

87.9

87.8

74.6

67.6

54.9

45.7

34.6

19.9

copy .empty

100.0

97.5

96.2

97.5

97.6

103.6

97.8

97.7

97.6

95.1

max

100.0

97.5

96.6

95.6

90.9

84.0

72.1

56.2

39.1

24.3

sum

100.0

98.0

95.3

93.5

87.3

79.2

65.1

48.3

32.2

19.2

sum(axis\(\,=\,\)0)

100.0

100.2

96.7

96.5

90.9

78.1

74.6

58.0

42.7

28.2

sum(axis\(\,=\,\)1)

100.0

105.2

103.2

102.2

100.6

100.0

98.3

95.8

93.2

88.6

obj[::-2]

100.0

70.4

65.9

64.0

46.2

46.6

42.8

33.6

31.1

25.3

copy

100.0

104.7

103.1

101.3

101.3

105.3

101.4

101.2

101.3

101.5

obj \(+\) 0

100.0

105.1

102.6

100.6

99.9

103.5

100.2

100.0

99.7

100.1

obj \(+\) obj

100.0

105.2

102.5

100.1

100.0

103.7

100.1

100.1

99.8

100.2

obj \(+\) = obj

100.0

102.3

99.3

98.6

98.2

101.8

98.2

98.2

98.2

98.4

sqrt

100.0

102.0

100.6

100.1

99.6

99.1

99.2

99.2

8.6

98.0

bincount

100.0

103.0

101.2

99.9

98.8

97.6

94.1

88.3

79.4

65.8

The arrays used for this tests had the global shape \((n*2048, 2048)\) with n being the number of processes. By this the local data size was fixed to \(2^{22}\) elements, which is equal to \(32~\mathrm {MiB}\). “100 %” in the table corresponds to the case were the speedup is equal to the number of processes. Example: the \(95.1\,\%\) for copy_empty on 256 processes correspond to a speedup-factor of 243.5. In order to guide the eye, values \(<30\,\%\) are printed italic, values \(\ge 90\,\%\) are printed bold-italics. Please see "Weak scaling: proportional number of processes and size of data" section for discussion

Table 5

Strong scaling: d2o’s relative performance to a single process when increasing the number of processes while fixing the global array size to \((4096, 4096)=128~\mathrm {MiB}\)

Processes (local size)

1 (128 MiB) (%)

2 (64 MiB) (%)

3 (42.7 MiB) (%)

4 (32MiB) (%)

8 (16 MiB) (%)

16 (8 MiB) (%)

32 (4 MiB) (%)

64 (2 MiB) (%)

128 (1 MiB) (%)

256 (512 KiB) (%)

initialization

100.0

40.3

23.7

17.1o

6.80

2.55

0.89

0.29

0.10,

0.03

copy .empty

100.0

47.8

33.5

26.3

23.2

12.1

6.16

3.09

1.55

0.79

max

100.0

99.0

96.7

94.0

80.1

64.3

29.6

9.10

2. 41

0.60

sum

100.0

100.4

95.4

91.7

74.1

60.9

24-4

7.05

1.82

0.45

sum(axis\(\,=\,\)0)

100.0

98.0

94.2

89.9

62.2

45.3

19.8

6.34

1.78:

0.45

sum(axis\(\,=\,\)1)

100.0

100.5

92.3

92.6

79.0

77.7

47.1

20.6

4. 27

1.25

obj[::-2]

100.0

65.4

62.4

58.5

40.7

33.1

26.6

18.3

8. 78

3.25

copy

100.0

103.6

105.9

98.0

145.1

156.4

155.3

152.3

157.5

306.7

obj \(+\) 0

100.0

105.9

109.1

100.4

59.0

97.3

75.7

79.2

79.4

149.8

obj \(+\) obj

100.0

106.2

109.2

100.3

59.0

97.6

74.9

79.0

80.1

150.2

obj \(+=\) obj

100.0

103.1

101.3

98.0

97.8

124.0

122.8

117.9

108.1

94.4

sqrt

100.0

101.8

99.4

98.7

95.7

88.8

76.0

56.0

37.1

16.4

bincount

100.0

102.3

99.5

98.0

111.1

107.3

84.2

40.5

13.2

3.57

“100 %” corresponds to the case were the speedup is equal to the number of processes. Example: the 94.4 % for obj+=obj on 256 processes correspond to a speedup-factor of 241.7. In order to guide the eye, values <30 % are printed italic, values ≥90 % are printed bold-italics. Please see "Strong scaling: varying number of processes with a fixed size of data" section for discussion

Strong scaling: comparison with distArray

In "Alternative packages" section we discussed several competitors to d2o Because of their similarities, we conducted the strong scaling tests—as far as possible16—also with DistArray and compare the performance. In Table 6 the results are shown for the subset of all operations from the previous sections that where available for DistArray, too.
Table 6

Strong scaling comparison: d2o’s relative performance compared to DistArray[7] when increasing the number of processes while fixing the global array size to \((4096, 4096)=128~\mathrm {MiB}\)

Processes (local size)

1 (128 MiB)

2 (64 MiB)

3 (42.7 MiB)

4 (32 MiB)

8 (16 MiB)

16 (8 MiB)

32 (4 MiB)

64 (2 MiB)

128 (1MiB)

256 (512 KiB)

copy_empty

23.49

27.70

33.99

40.03

1.11 × 10\(^{\mathrm {2}}\)

1.12 × 10\(^{\mathrm {3}}\)

1.06 × 10\(^{\mathrm {3}}\)

1.91 × 10\(^{\mathrm {3}}\)

5.57 × 10\(^{\mathrm {3}}\)

1.90 × 10\(^{\mathrm {4}}\)

Max

1.05

1.11

1.16

1.20

1.34

5.00

4.74

3.77

3.25

4.17

Sum

1.07

1.20

1.25

1.33

1.48

6.57

5.57

4.20

3.55

4.61

sum(axis\(\,=\,\)0)

1.02

1.09

1.12

1.22

1.03

3.61

3.35

2.75

2.42

3.06

sum(axis\(\,=\,\)1)

1.03

1.15

1.15

1.28

1.63

11.35

12.56

19.29

22.55

42.48

obj \(+\) 0

1.02

1.09

1.20

1.15

0.44

2.88

4.24

7.81

33.94

3.78 × 10\(^{\mathrm {2}}\)

obj \(+\) obj

1.02

1.09

1.20

1.16

0.44

2.90

4.23

8.08

34.40

3.81 × 10\(^{\mathrm {2}}\)

obj \(+=\) obj

2.27

2.38

2.51

2.53

1.60

8.23

14.89

26.24

1.04 × 10\(^{\mathrm {2}}\)

5.35 × 10\(^{\mathrm {2}}\)

Sqrt

1.00

1.03

1.03

1.04

0.81

1.57

2.06

2.59

6.66

16.77

“2” corresponds to the case were d2o is twice as fast as DistArray. Please see "Strong scaling: comparison with DistArray" section for discussion

While being at least on a par for numerical operations when being run single-threaded, d2o outperforms DistArray more and more with an increasing number of processes. Furthermore, it is conspicuous that DistArray does not seem to support inplace array operations. Because of this, the inplace addition obj+=obj is way slower with DistArray than with d2o which is on a par with numpy in most cases, cf. Tables 3, 4 and 5.

The fact that d2o is way more efficient when doing numerics on very small arrays—like obj+0 using 256 processes—indicates that d2o’s organizatorial overhead is much smaller than that of DistArray. Supporting evidence for this is that the initialization of an empty DistArray (copy_empty) becomes disproportionately costly when increasing the number of processes used.

Strong scaling: real-world application speedup—the Wiener filter

d2o was initially developed for NIFTy [3], a library for building signal inference algorithms in the framework of information field theory (IFT) [24]. Within NIFTy v2 all of the parallelization is done via d2o; the code of NIFTy itself is almost completely agnostic of the parallelization and completely agnostic of MPI.
Fig. 1

Object structure. Here the main object composition structure of d2o is shown. distributed data objects are composed objects, cf. "Composed object" section. All tasks that need information related to the distribution strategy are outsourced to a distributor. In this figure, three distributed data objects are shown where obj1 and obj2 share the same distributor. This is possible because they are essentially identical: they have the same global shape, datatype, and distribution strategy. Since it is expensive to instantiate new distributors, the distributed data objects get their instance from the distributor_factory that takes care of caching those distributors that have already been created. Furthermore, we illustrate the d2o_librarian that keeps weak references to the individual distributed data objects and assigns a unique cluster-wide identifier to them: the index

A standard computational operation in IFT-based signal reconstruction is the Wiener filter [25]. For this performance test, we use a Wiener filter implemented in NIFTy to reconstruct the realization of a Gaussian random field—called the signal. Assume we performed a hypothetical measurement which produced some data. The data model is
$$\begin{aligned} \mathrm {data} = R(\mathrm {signal}) + \mathrm {noise} \end{aligned}$$
(3)
where R is a smoothing operator and noise is additive Gaussian noise. In Fig. 2 one sees the three steps:
  • the true signal we try to reconstruct,

  • the data one gets from the hypothetical measurement, and

  • the reconstructed signal field that according to the Wiener filter has most likely produced the data.

Table 2 shows the scaling behavior of the reconstruction code, run with a resolution of \(8192 \times 8192\). Here, n is the number of used processes and \(t_n\) the respective execution time. The relative speedup \(s_n = 2t_2/t_n\) 17 is the ratio of execution times: parallel versus serial. In the case of exactly linear scaling \(s_n\) is equal to n. Furthermore we define the scaling quality \(q = 1/(1+\log (n/s_n))\), which compares \(s_n\) with linear scaling in terms of orders of magnitude. A value \(q=1\) represents linear scaling and \(q \ge 1\) super-linear scaling.
Fig. 2

Wiener filter reconstruction. Top left true signal to be reconstructed. Top right data which is the result of smoothing the signal and adding Gaussian noise. Bottom reconstructed maximum of the posterior (conditional) probability density function for the signal given the data P(signaljdata)

This benchmark illustrates that even in a real-life application super-linear scaling is possible to achieve for a sufficiently large number of processes. This is due to the operations that are needed in order to perform the Wiener filtering: basic point-wise arithmetics that do not involve any inter-process communication and Fourier transformations that are handled by the high-performance library FFTW [26]. While the problem size remains constant, the amount of available CPU cache increases with the number of processes, which explains the super-linear scaling, cf. "Strong scaling: varying number of processes with a fixed size of data" section.

Summary and outlook

We introduced d2o, a Python module for cluster-distributed multi-dimensional numerical arrays. It can be understood as a layer of abstraction between abstract algorithm code and actual data-distribution logic. We argued why we developed d2o as a package following a low-level parallelization ansatz and why we built it on MPI. Compared to other packages available for data parallelization, d2o has the advantage of being ready for action on one as well as several hundreds of CPUs, of being highly portable and customizable as it is built with Python, that it is faster in many circumstances, and that it is able to treat arrays of arbitrary dimension.

For the future, we plan to cover more of numpy’s interface such that working with d2o becomes even more convenient. Furthermore we evaluate the option to build a d2o distributor in order to support scalapy’s block-cyclic distribution strategy directly. This will open up a whole new class of applications d2o then can be used for.

d2o is open source software licensed under the GNU General Public License v3 (GPL-3) and is available by https://gitlab.mpcdf.mpg.de/ift/D2O.

Footnotes
1

FFTW distributes slices of data, while ScaLAPACK uses a block-cyclic distribution pattern.

 
2

This is true if scaling up does not involve a change of the processor architecture.

 
3

For the exemplary sample generator the maximum degree of parallelization is the total number of requested samples.

 
4

In case of the sample generator this would be the case if even one sample would be too large for an individual machine’s RAM.

 
5

E.g. the positive, negative or absolute values of the array’s individual elements or the maximum, minimum, median or mean of all its elements.

 
6

E.g. the sum, difference or product of two data arrays.

 
7

This was the maximum number of processes available for testing.

 
8

The slicing is done along the first array axis.

 
9

The fftw-distributor uses routines from the pyFFTW [8, 22] package [8] for the data partitioning.

 
10

The equal-distributor tries to split the data in preferably equal-sized parts.

 
11

The local data array’s first axis is of arbitrary length for the freeform-distributor.

 
12

This notation can be decoded as follows. The numbers in a slice correspond to start:stop:step with stop being exclusive. obj[0:3:2,1:3] means to take every second line from the lines 0, 1 and 2, and to then take from this the elements in collumns 1 and 2.

 
13

The C2PAP computing cluster consists of 128 nodes, each possessing two Intel Xeon CPU E5-2680 (8 cores 2.70 GHz + hyper-threading, 64 KiB L1 per core, 256 KiB L2 cache per core, 20 MiB L3 cache shared for all 8 cores) and 64 GiB RAM each. The nodes are connected via Mellanox Infiniband 40 Gbits/s.

 
14

Exceptions to this are inplace operations which reuse the input array for the output data.

 
15

This very strong scaling is indeed realistic: when analyzing pure numpy arrays one gets speedups in the order of magnitude of even 800 %.

 
16

DistArray does, for example, not support negative step-sizes for slicing ([::-2]) and also the special method bincount is not available.

 
17

Since the combination of NIFTy and pyfftw exhibits an unexpected speed malus for one process, we chose the two-process timing as the benchmark’s baseline.

 
18

Depending on whether pyfftw is available or not, the equal- or the fftw-distribution strategy is used, respectively; cf. "Distribution strategies" section.

 
19

This mimics numpys behavior.

 
20

These are the methods getting called through Python’s obj[...]=... notation.

 
21

This has the downside, that if the iteration was stopped prematurely, data has been communicated in vain.

 

Declarations

Authors' contributions

TS is the principal researcher for the work proposed in this article. His contributions include the primal idea, the implementation of presented software package, the conduction of the performance tests and the writing of the article. MG, FB and TE helped working out the conceptual structure of the software package and drafting the manuscript. FB also played a pivotal role for executing the performance tests. Furthermore, TE also fulfilled the role of the principal investigator. All authors read and approved the final manuscript.

Acknowledgements

We want to thank Jait Dixit, Philipp Franck, Reimar Leike, Fotis Megas, Martin Reinecke and Csongor Varady for useful discussions and support. We acknowledge the support by the DFG Cluster of Excellence “Origin and Structure of the Universe” and the Studienstiftung des deutschen Volkes. The performance tests have been carried out on the computing facilities of the Computational Center for Particle and Astrophysics (C2PAP). We are grateful for the support by Dr. Alexey Krukau through the Computational Center for Particle and Astrophysics (C2PAP).

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Max Planck Institut für Astrophysik
(2)
Ludwig-Maximilians-Universität München
(3)
Exzellenzcluster Universe

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