# mpi4py.futures

Added in version 3.0.0.

This package provides a high-level interface for asynchronously executing callables on a pool of worker processes using MPI for inter-process communication.

The `mpi4py.futures` package is based on `concurrent.futures` from the Python standard library. More precisely, `mpi4py.futures` provides the `MPIPoolExecutor` class as a concrete implementation of the abstract class `Executor`. The `submit()` interface schedules a callable to be executed asynchronously and returns a `Future` object representing the execution of the callable. `Future` instances can be queried for the call result or exception. Sets of `Future` instances can be passed to the `wait()` and `as_completed()` functions.

See also

Module `concurrent.futures`

  Documentation of the `concurrent.futures` standard module.

## MPIPoolExecutor

The `MPIPoolExecutor` class uses a pool of MPI processes to execute calls asynchronously. By performing computations in separate processes, it allows to side-step the global interpreter lock but also means that only picklable objects can be executed and returned. The `__main__` module must be importable by worker processes, thus `MPIPoolExecutor` instances may not work in the interactive interpreter.

`MPIPoolExecutor` takes advantage of the dynamic process management features introduced in the MPI-2 standard. In particular, the `MPI.Intracomm.Spawn` method of `MPI.COMM_SELF` is used in the master (or parent) process to spawn new worker (or child) processes running a Python interpreter. The master process uses a separate thread (one for each `MPIPoolExecutor` instance) to communicate back and forth with the workers. The worker processes serve the execution of tasks in the main (and only) thread until they are signaled for completion.

Note

The worker processes must import the main script in order to unpickle any callable defined in the `__main__` module and submitted from the master process. Furthermore, the callables may need access to other global variables. At the worker processes, `mpi4py.futures` executes the main script code (using the `runpy` module) under the `__worker__` namespace to define the `__main__` module. The `__main__` and `__worker__` modules are added to `sys.modules` (both at the master and worker processes) to ensure proper pickling and unpickling.

Warning

During the initial import phase at the workers, the main script cannot create and use new `MPIPoolExecutor` instances. Otherwise, each worker would attempt to spawn a new pool of workers, leading to infinite recursion. `mpi4py.futures` detects such recursive attempts to spawn new workers and aborts the MPI execution environment. As the main script code is run under the `__worker__` namespace, the easiest way to avoid spawn recursion is using the idiom `if __name__ == '__main__': ...` in the main script.

class mpi4py.futures.MPIPoolExecutor(max_workers=None, initializer=None, initargs=(), **kwargs)

  An `Executor` subclass that executes calls asynchronously using a pool of at most max_workers processes. If max_workers is `None` or not given, its value is determined from the `MPI4PY_FUTURES_MAX_WORKERS` environment variable if set, or the MPI universe size if set, otherwise a single worker process is spawned. If max_workers is lower than or equal to `0`, then a `ValueError` will be raised.

  initializer is an optional callable that is called at the start of each worker process before executing any tasks; initargs is a tuple of arguments passed to the initializer. If initializer raises an exception, all pending tasks and any attempt to submit new tasks to the pool will raise a `BrokenExecutor` exception.

  Other parameters:

  - python_exe: Path to the Python interpreter executable used to spawn worker processes, otherwise `sys.executable` is used.

  - python_args: `list` or iterable with additional command line flags to pass to the Python executable. Command line flags determined from inspection of `sys.flags`, `sys.warnoptions` and `sys._xoptions` in are passed unconditionally.

  - mpi_info: `dict` or iterable yielding `(key, value)` pairs. These `(key, value)` pairs are passed (through an `MPI.Info` object) to the `MPI.Intracomm.Spawn` call used to spawn worker processes. This mechanism allows telling the MPI runtime system where and how to start the processes. Check the documentation of the backend MPI implementation about the set of keys it interprets and the corresponding format for values.

  - globals: `dict` or iterable yielding `(name, value)` pairs to initialize the main module namespace in worker processes.

  - main: If set to `False`, do not import the `__main__` module in worker processes. Setting main to `False` prevents worker processes from accessing definitions in the parent `__main__` namespace.

  - path: `list` or iterable with paths to append to `sys.path` in worker processes to extend the module search path.

  - wdir: Path to set the current working directory in worker processes using `os.chdir()`. The initial working directory is set by the MPI implementation. Quality MPI implementations should honor a `wdir` info key passed through mpi_info, although such feature is not mandatory.

  - env: `dict` or iterable yielding `(name, value)` pairs with environment variables to update `os.environ` in worker processes. The initial environment is set by the MPI implementation. MPI implementations may allow setting the initial environment through mpi_info, however such feature is not required nor recommended by the MPI standard.

  - use_pkl5: If set to `True`, use `pickle` with out-of-band buffers for interprocess communication. If use_pkl5 is set to `None` or not given, its value is determined from the `MPI4PY_FUTURES_USE_PKL5` environment variable. Using `pickle` with out-of-band buffers may benefit applications dealing with large buffer-like objects like NumPy arrays. See `mpi4py.util.pkl5` for additional information.

  - backoff: `float` value specifying the maximum number of seconds a worker thread or process suspends execution with `time.sleep()` while idle-waiting. If not set, its value is determined from the `MPI4PY_FUTURES_BACKOFF` environment variable if set, otherwise the default value of 0.001 seconds is used. Lower values will reduce latency and increase execution throughput for very short-lived tasks, albeit at the expense of spinning CPU cores and increased energy consumption.

  submit(fn, /, *args, **kwargs)

    Schedule the callable fn to be executed as `fn(*args, **kwargs)` and returns a `Future` object representing the execution of the callable.

        executor = MPIPoolExecutor(max_workers=1)
        future = executor.submit(pow, 321, 1234)
        print(future.result())

  map(fn, *iterables, timeout=None, chunksize=1, buffersize=None, **kwargs)

    Similar to `map(fn, *iterables)` except:

    - The iterables are consumed immediately rather than lazily, unless buffersize is specified to limit the number of submitted tasks whose results have not yet been yielded. If the task buffer is full, the caller blocks and iteration over the iterables pauses until a result is yielded from the buffer.

    - fn is executed asynchronously and several calls to fn may be made concurrently, out-of-order, in separate processes.

    The returned iterator raises a `TimeoutError` if `__next__()` is called and the result isn’t available after timeout seconds from the original call to `map()`. timeout can be an int or a float. If timeout is not specified or `None`, there is no limit to the wait time.

    If fn raises an exception, then that exception will be raised when its value is retrieved from the iterator.

    This method chops iterables into a number of chunks which it submits to the pool as separate tasks. The (approximate) size of these chunks can be specified by setting chunksize to a positive integer. For very long iterables, using a large value for chunksize can significantly improve performance compared to the default size of one.

    By default, the returned iterator yields results in-order, waiting for successive tasks to complete . This behavior can be changed by passing the keyword argument unordered as `True`, then the result iterator will yield a result as soon as any of the tasks complete.

        executor = MPIPoolExecutor(max_workers=3)
        for result in executor.map(pow, [2] * 32, range(32)):
            print(result)

    Changed in version 4.1.0: Added the buffersize parameter.

  starmap(fn, iterable, timeout=None, chunksize=1, buffersize=None, **kwargs)

    Similar to `itertools.starmap(fn, iterable)`. Used instead of `map()` when argument parameters are already grouped in tuples from a single iterable (the data has been “pre-zipped”). `map(fn, *iterable)` is equivalent to `starmap(fn, zip(*iterable))`.

        executor = MPIPoolExecutor(max_workers=3)
        iterable = ((2, n) for n in range(32))
        for result in executor.starmap(pow, iterable):
            print(result)

    Changed in version 4.1.0: Added the buffersize parameter.

  shutdown(wait=True, cancel_futures=False)

    Signal the executor that it should free any resources that it is using when the currently pending futures are done executing. Calls to `submit()` and `map()` made after `shutdown()` will raise `RuntimeError`.

    If wait is `True` then this method will not return until all the pending futures are done executing and the resources associated with the executor have been freed. If wait is `False` then this method will return immediately and the resources associated with the executor will be freed when all pending futures are done executing. Regardless of the value of wait, the entire Python program will not exit until all pending futures are done executing.

    If cancel_futures is `True`, this method will cancel all pending futures that the executor has not started running. Any futures that are completed or running won’t be cancelled, regardless of the value of cancel_futures.

    You can avoid having to call this method explicitly if you use the `with` statement, which will shutdown the executor instance (waiting as if `shutdown()` were called with wait set to `True`).

        import time
        with MPIPoolExecutor(max_workers=1) as executor:
            future = executor.submit(time.sleep, 2)
        assert future.done()

  bootup(wait=True)

    Signal the executor that it should allocate eagerly any required resources (in particular, MPI worker processes). If wait is `True`, then `bootup()` will not return until the executor resources are ready to process submissions. Resources are automatically allocated in the first call to `submit()`, thus calling `bootup()` explicitly is seldom needed.

  num_workers

    Number or worker processes in the pool.

MPI4PY_FUTURES_MAX_WORKERS

  If the max_workers parameter to `MPIPoolExecutor` is `None` or not given, the `MPI4PY_FUTURES_MAX_WORKERS` environment variable provides a fallback value for the maximum number of MPI worker processes to spawn.

  Added in version 3.1.0.

MPI4PY_FUTURES_USE_PKL5

  If the use_pkl5 keyword argument to `MPIPoolExecutor` is `None` or not given, the `MPI4PY_FUTURES_USE_PKL5` environment variable provides a fallback value for whether the executor should use `pickle` with out-of-band buffers for interprocess communication. Accepted values are `0` and `1` (interpreted as `False` and `True`, respectively), and strings specifying a YAML boolean value (case-insensitive). Using `pickle` with out-of-band buffers may benefit applications dealing with large buffer-like objects like NumPy arrays. See `mpi4py.util.pkl5` for additional information.

  Added in version 4.0.0.

MPI4PY_FUTURES_BACKOFF

  If the backoff keyword argument to `MPIPoolExecutor` is not given, the `MPI4PY_FUTURES_BACKOFF` environment variable can be set to a `float` value specifying the maximum number of seconds a worker thread or process suspends execution with `time.sleep()` while idle-waiting. If not set, the default backoff value is 0.001 seconds. Lower values will reduce latency and increase execution throughput for very short-lived tasks, albeit at the expense of spinning CPU cores and increased energy consumption.

  Added in version 4.0.0.

Note

As the master process uses a separate thread to perform MPI communication with the workers, the backend MPI implementation should provide support for `MPI.THREAD_MULTIPLE`. However, some popular MPI implementations do not support yet concurrent MPI calls from multiple threads. Additionally, users may decide to initialize MPI with a lower level of thread support. If the level of thread support in the backend MPI is less than `MPI.THREAD_MULTIPLE`, `mpi4py.futures` will use a global lock to serialize MPI calls. If the level of thread support is less than `MPI.THREAD_SERIALIZED`, `mpi4py.futures` will emit a `RuntimeWarning`.

Warning

If the level of thread support in the backend MPI is less than `MPI.THREAD_SERIALIZED` (i.e, it is either `MPI.THREAD_SINGLE` or `MPI.THREAD_FUNNELED`), in theory `mpi4py.futures` cannot be used. Rather than raising an exception, `mpi4py.futures` emits a warning and takes a “cross-fingers” attitude to continue execution in the hope that serializing MPI calls with a global lock will actually work.

## MPICommExecutor

Legacy MPI-1 implementations (as well as some vendor MPI-2 implementations) do not support the dynamic process management features introduced in the MPI-2 standard. Additionally, job schedulers and batch systems in supercomputing facilities may pose additional complications to applications using the `MPI_Comm_spawn()` routine.

With these issues in mind, `mpi4py.futures` supports an additional, more traditional, SPMD-like usage pattern requiring MPI-1 calls only. Python applications are started the usual way, e.g., using the mpiexec command. Python code should make a collective call to the `MPICommExecutor` context manager to partition the set of MPI processes within a MPI communicator in one master processes and many workers processes. The master process gets access to an `MPIPoolExecutor` instance to submit tasks. Meanwhile, the worker process follow a different execution path and team-up to execute the tasks submitted from the master.

Besides alleviating the lack of dynamic process management features in legacy MPI-1 or partial MPI-2 implementations, the `MPICommExecutor` context manager may be useful in classic MPI-based Python applications willing to take advantage of the simple, task-based, master/worker approach available in the `mpi4py.futures` package.

class mpi4py.futures.MPICommExecutor(comm=None, root=0)

  Context manager for `MPIPoolExecutor`. This context manager splits a MPI (intra)communicator comm (defaults to `MPI.COMM_WORLD` if not provided or `None`) in two disjoint sets: a single master process (with rank root in comm) and the remaining worker processes. These sets are then connected through an intercommunicator. The target of the `with` statement is assigned either an `MPIPoolExecutor` instance (at the master) or `None` (at the workers).

      from mpi4py import MPI
      from mpi4py.futures import MPICommExecutor

      with MPICommExecutor(MPI.COMM_WORLD, root=0) as executor:
          if executor is not None:
             future = executor.submit(abs, -42)
             assert future.result() == 42
             answer = set(executor.map(abs, [-42, 42]))
             assert answer == {42}

Warning

If `MPICommExecutor` is passed a communicator of size one (e.g., `MPI.COMM_SELF`), then the executor instance assigned to the target of the `with` statement will execute all submitted tasks in a single worker thread, thus ensuring that task execution still progress asynchronously. However, the GIL will prevent the main and worker threads from running concurrently in multicore processors. Moreover, the thread context switching may harm noticeably the performance of CPU-bound tasks. In case of I/O-bound tasks, the GIL is not usually an issue, however, as a single worker thread is used, it progress one task at a time. We advice against using `MPICommExecutor` with communicators of size one and suggest refactoring your code to use instead a `ThreadPoolExecutor`.

## Command line

Recalling the issues related to the lack of support for dynamic process management features in MPI implementations, `mpi4py.futures` supports an alternative usage pattern where Python code (either from scripts, modules, or zip files) is run under command line control of the `mpi4py.futures` package by passing `-m mpi4py.futures` to the python executable. The `mpi4py.futures` invocation should be passed a pyfile path to a script (or a zipfile/directory containing a `__main__.py` file). Additionally, `mpi4py.futures` accepts `-m mod` to execute a module named mod, `-c cmd` to execute a command string cmd, or even `-` to read commands from standard input (`sys.stdin`). Summarizing, `mpi4py.futures` can be invoked in the following ways:

- `$ mpiexec -n numprocs python -m mpi4py.futures pyfile [arg] ...`

- `$ mpiexec -n numprocs python -m mpi4py.futures -m mod [arg] ...`

- `$ mpiexec -n numprocs python -m mpi4py.futures -c cmd [arg] ...`

- `$ mpiexec -n numprocs python -m mpi4py.futures - [arg] ...`

Before starting the main script execution, `mpi4py.futures` splits `MPI.COMM_WORLD` in one master (the process with rank 0 in `MPI.COMM_WORLD`) and numprocs - 1 workers and connects them through an MPI intercommunicator. Afterwards, the master process proceeds with the execution of the user script code, which eventually creates `MPIPoolExecutor` instances to submit tasks. Meanwhile, the worker processes follow a different execution path to serve the master. Upon successful termination of the main script at the master, the entire MPI execution environment exists gracefully. In case of any unhandled exception in the main script, the master process calls `MPI.COMM_WORLD.Abort(1)` to prevent deadlocks and force termination of entire MPI execution environment.

Warning

Running scripts under command line control of `mpi4py.futures` is quite similar to executing a single-process application that spawn additional workers as required. However, there is a very important difference users should be aware of. All `MPIPoolExecutor` instances created at the master will share the pool of workers. Tasks submitted at the master from many different executors will be scheduled for execution in random order as soon as a worker is idle. Any executor can easily starve all the workers (e.g., by calling `MPIPoolExecutor.map()` with long iterables). If that ever happens, submissions from other executors will not be serviced until free workers are available.

See also

Command line

  Documentation on Python command line interface.

## Parallel tasks

The `mpi4py.futures` package favors an embarrassingly parallel execution model involving a series of sequential tasks independent of each other and executed asynchronously. Albeit unnatural, `MPIPoolExecutor` can still be used for handling workloads involving parallel tasks, where worker processes communicate and coordinate each other via MPI.

mpi4py.futures.get_comm_workers()

  Access an intracommunicator grouping MPI worker processes.

Executing parallel tasks with `mpi4py.futures` requires following some rules, cf. highlighted lines in example cpi.py :

- Use `MPIPoolExecutor.num_workers` to determine the number of worker processes in the executor and submit exactly one callable per worker process using the `MPIPoolExecutor.submit()` method.

- The submitted callable must use `get_comm_workers()` to access an intracommunicator grouping MPI worker processes. Afterwards, it is highly recommended calling the `Barrier()` method on the communicator. The barrier synchronization ensures that every worker process is executing the submitted callable exactly once. Afterwards, the parallel task can safely perform any kind of point-to-point or collective operation using the returned communicator.

- The `Future` instances returned by `MPIPoolExecutor.submit()` should be collected in a sequence. Use `wait()` with the sequence of `Future` instances to ensure logical completion of the parallel task.

## Utilities

The `mpi4py.futures` package provides additional utilities for handling `Future` instances.

mpi4py.futures.collect(fs)

  Gather a collection of futures in a new future.

  Parameters:

    fs – Collection of futures.

  Returns:

    New future producing as result a list with results from fs.

mpi4py.futures.compose(future, resulthook=None, excepthook=None)

  Compose the completion of a future with result and exception handlers.

  Parameters:

    - future – Input future instance.

    - resulthook – Function to be called once the input future completes with success. Once the input future finish running with success, its result value is the input argument for resulthook. The result of resulthook is set as the result of the output future. If resulthook is `None`, the output future is completed directly with the result of the input future.

    - excepthook – Function to be called once the input future completes with failure. Once the input future finish running with failure, its exception value is the input argument for excepthook. If excepthook returns an `Exception` instance, it is set as the exception of the output future. Otherwise, the result of excepthook is set as the result of the output future. If excepthook is `None`, the output future is set as failed with the exception from the input future.

  Returns:

    Output future instance to be completed once the input future is completed and either resulthook or excepthook finish executing.

## Examples

### Computing the Julia set

The following julia.py script computes the Julia set and dumps an image to disk in binary PGM format. The code starts by importing `MPIPoolExecutor` from the `mpi4py.futures` package. Next, some global constants and functions implement the computation of the Julia set. The computations are protected with the standard `if __name__ == '__main__': ...` idiom. The image is computed by whole scanlines submitting all these tasks at once using the `map` method. The result iterator yields scanlines in-order as the tasks complete. Finally, each scanline is dumped to disk.

`julia.py`

     1from mpi4py.futures import MPIPoolExecutor
     2
     3x0, x1, w = -2.0, +2.0, 640*2
     4y0, y1, h = -1.5, +1.5, 480*2
     5dx = (x1 - x0) / w
     6dy = (y1 - y0) / h
     7
     8c = complex(0, 0.65)
     9
    10def julia(x, y):
    11    z = complex(x, y)
    12    n = 255
    13    while abs(z) < 3 and n > 1:
    14        z = z**2 + c
    15        n -= 1
    16    return n
    17
    18def julia_line(k):
    19    line = bytearray(w)
    20    y = y1 - k * dy
    21    for j in range(w):
    22        x = x0 + j * dx
    23        line[j] = julia(x, y)
    24    return line
    25
    26if __name__ == '__main__':
    27
    28    with MPIPoolExecutor() as executor:
    29        image = executor.map(julia_line, range(h))
    30        with open('julia.pgm', 'wb') as f:
    31            f.write(b'P5 %d %d %d\n' % (w, h, 255))
    32            for line in image:
    33                f.write(line)

The recommended way to execute the script is by using the mpiexec command specifying one MPI process (master) and (optional but recommended) the desired MPI universe size, which determines the number of additional dynamically spawned processes (workers). The MPI universe size is provided either by a batch system or set by the user via command-line arguments to mpiexec or environment variables. Below we provide examples for MPICH and Open MPI implementations [1]. In all of these examples, the mpiexec command launches a single master process running the Python interpreter and executing the main script. When required, `mpi4py.futures` spawns the pool of 16 worker processes. The master submits tasks to the workers and waits for the results. The workers receive incoming tasks, execute them, and send back the results to the master.

When using MPICH implementation or its derivatives based on the Hydra process manager, users can set the MPI universe size via the `-usize` argument to mpiexec:

    $ mpiexec -n 1 -usize 17 python julia.py

or, alternatively, by setting the `MPIEXEC_UNIVERSE_SIZE` environment variable:

    $ env MPIEXEC_UNIVERSE_SIZE=17 mpiexec -n 1 python julia.py

In the Open MPI implementation, the MPI universe size can be set via the `-host` argument to mpiexec:

    $ mpiexec -n 1 -host localhost:17 python julia.py

Another way to specify the number of workers is to use the `mpi4py.futures`-specific environment variable `MPI4PY_FUTURES_MAX_WORKERS`:

    $ env MPI4PY_FUTURES_MAX_WORKERS=16 mpiexec -n 1 python julia.py

Note that in this case, the MPI universe size is ignored.

Alternatively, users may decide to execute the script in a more traditional way, that is, all the MPI processes are started at once. The user script is run under command-line control of `mpi4py.futures` passing the -m flag to the python executable:

    $ mpiexec -n 17 python -m mpi4py.futures julia.py

As explained previously, the 17 processes are partitioned in one master and 16 workers. The master process executes the main script while the workers execute the tasks submitted by the master.

[1]

When using an MPI implementation other than MPICH or Open MPI, please check the documentation of the implementation and/or batch system for the ways to specify the desired MPI universe size.

### Computing Pi (parallel task)

The number \(\pi\) can be approximated via numerical integration with the simple midpoint rule, that is:

\[\pi = \int_{0}^{1} \frac{4}{1+x^2} \,dx \approx \frac{1}{n} \sum_{i=1}^{n} \frac{4}{1 + \left[\frac{1}{n} \left(i-\frac{1}{2}\right) \right]^2} .\]

The following cpi.py script computes such approximations using `mpi4py.futures` with a parallel task involving a collective reduction operation. Highlighted lines correspond to the rules discussed in Parallel tasks.

`cpi.py`

     1import math
     2import sys
     3from mpi4py.futures import MPIPoolExecutor, wait
     4from mpi4py.futures import get_comm_workers
     5
     6
     7def compute_pi(n):
     8    # Access intracommunicator and synchronize
     9    comm = get_comm_workers()
    10    comm.Barrier()
    11
    12    rank = comm.Get_rank()
    13    size = comm.Get_size()
    14
    15    # Local computation
    16    h = 1.0 / n
    17    s = 0.0
    18    for i in range(rank + 1, n + 1, size):
    19        x = h * (i - 0.5)
    20        s += 4.0 / (1.0 + x**2)
    21    pi_partial = s * h
    22
    23    # Parallel reduce-to-all
    24    pi = comm.allreduce(pi_partial)
    25
    26    # All workers return the same value
    27    return pi
    28
    29
    30if __name__ == '__main__':
    31    n = int(sys.argv[1]) if len(sys.argv) > 1 else 256
    32
    33    with MPIPoolExecutor() as executor:
    34        # Submit exactly one callable per worker
    35        P = executor.num_workers
    36        fs = [executor.submit(compute_pi, n) for _ in range(P)]
    37
    38        # Wait for all workers to finish
    39        wait(fs)
    40
    41        # Get result from the first future object.
    42        # In this particular example, due to using reduce-to-all,
    43        # all the other future objects hold the same result value.
    44        pi = fs[0].result()
    45        print(
    46            f"pi: {pi:.16f}, error: {abs(pi - math.pi):.3e}",
    47            f"({n:d} intervals, {P:d} workers)",
    48        )

To run in modern MPI-2 mode:

    $ env MPI4PY_FUTURES_MAX_WORKERS=4 mpiexec -n 1 python cpi.py 128
    pi: 3.1415977398528137, error: 5.086e-06 (128 intervals, 4 workers)

    $ env MPI4PY_FUTURES_MAX_WORKERS=8 mpiexec -n 1 python cpi.py 512
    pi: 3.1415929714812316, error: 3.179e-07 (512 intervals, 8 workers)

To run in legacy MPI-1 mode:

    $ mpiexec -n 5 python -m mpi4py.futures cpi.py 128
    pi: 3.1415977398528137, error: 5.086e-06 (128 intervals, 4 workers)

    $ mpiexec -n 9 python -m mpi4py.futures cpi.py 512
    pi: 3.1415929714812316, error: 3.179e-07 (512 intervals, 8 workers)

## Citation

If `mpi4py.futures` been significant to a project that leads to an academic publication, please acknowledge our work by citing the following article [mpi4py-futures]:

[mpi4py-futures]

M. Rogowski, S. Aseeri, D. Keyes, and L. Dalcin, mpi4py.futures: MPI-Based Asynchronous Task Execution for Python, IEEE Transactions on Parallel and Distributed Systems, 34(2):611-622, 2023. https://doi.org/10.1109/TPDS.2022.3225481