code: def _run_dependency_balance_test(
      dependencies: Mapping[str, int],
      dependency_placement: list[list[str]],
      task_placement: list[list[list[str]]],
      correct_placement_fn: Callable[[list[list[list[str]]]], bool],
      config: dict | None = None,
  ) -> None:
      """Run a test for balancing with task dependencies according to the provided
      specifications.
  
      This method executes the test logic for all permutations of worker placements
      and generates a new cluster for each one.
  
      Parameters
      ----------
      dependencies
          Mapping of task dependencies to their weight.
      dependency_placement
          List of list of dependencies to be placed on the worker corresponding
          to the index of the outer list.
      task_placement
          List of list of tasks to be placed on the worker corresponding to the
          index of the outer list. Each task is a list of names of dependencies.
      correct_placement_fn
          Callable used to determine if stealing placed the tasks as expected.
      config
          Optional configuration to apply to the test.
      See Also
      --------
      _dependency_balance_test_permutation
      """
      nworkers = len(task_placement)
      for permutation in itertools.permutations(range(nworkers)):
  
          async def _run(
              *args,
              permutation=permutation,
              **kwargs,
          ):
              await _dependency_balance_test_permutation(
                  dependencies,
                  dependency_placement,
                  task_placement,
                  correct_placement_fn,
                  permutation,
                  *args,
                  **kwargs,
              )
  
          default_task_durations = {
              compose_task_prefix(deps): "1s"
              for tasks in task_placement
              for deps in tasks
          }
  
          gen_cluster(
              client=True,
              nthreads=[("", 1)] * len(task_placement),
              config=merge(
                  NO_AMM,
                  config or {},
                  {
                      "distributed.scheduler.default-task-durations": default_task_durations,
                  },
              ),
              # Avoid heartbeats since comm costs are sensitive to bandwidth updates
              worker_kwargs={"heartbeat_interval": "100s"},
          )(_run)()
  
imports: from __future__ import annotations
  import asyncio
  import contextlib
  import itertools
  import logging
  import math
  import random
  import weakref
  from collections import defaultdict
  from collections.abc import Callable, Coroutine, Iterable, Mapping, Sequence
  from operator import mul
  from time import sleep
  import pytest
  from tlz import merge, sliding_window
  import dask
  from dask.utils import key_split, parse_bytes
  from distributed import Client, Event, Lock, Nanny, Scheduler, Worker, profile, wait, worker_client
  from distributed.client import Future
  from distributed.compatibility import LINUX
  from distributed.core import Status
  from distributed.metrics import time
  from distributed.system import MEMORY_LIMIT
  from distributed.utils import wait_for
  from distributed.utils_test import NO_AMM, BlockedGetData, captured_logger, freeze_batched_send, gen_cluster, gen_nbytes, inc, nodebug_setup_module, nodebug_teardown_module, slowadd, slowidentity, slowinc, wait_for_state
  from distributed.worker_state_machine import ExecuteSuccessEvent, FreeKeysEvent, StealRequestEvent
label_desc: Use image node_id 11 for calling a global function with example usage: _run_dependency_balance_test(dependencies, dependency_placement, task_placement, correct_placement_fn, config) without return types