code: def auto_logical_to_physical_tpu(
      logical_shape,
      physical_shape,
      return_coordinates=False,
      device_assignment=None,
  ):
      """Set up a mapping from logical to physical cores for TPU.
  
      We will try to set up a mapping so that allreduce operations are relatively
      fast, prioritizing the later dimensions in the mesh_shape.
  
      Example:
  
      auto_logical_to_physical_tpu(
        logical_shape=[16, 8], physical_shape=[8, 8, 1, 2])
  
      Heuristics in this function subject to change.
  
      Args:
        logical_shape: a list of integers
        physical_shape: a list of integers - typically [X, Y, 1, cores]
        return_coordinates: a boolean - return a list of integer lists (coordinates)
           instead of a list of processor indices
        device_assignment:  the tpu logical to physical device assignment
  
      Returns:
        logical_to_physical: a permutation of range(product(physical_shape)))
      """
      # Convert physical shape to 3d
      if len(physical_shape) == 4:
          physical_shape = physical_shape_3d_from_topology_proto_4d(
              physical_shape
          )
  
      tf.logging.info(
          "auto_logical_to_physical_tpu "
          "logical_shape=%s physical_shape=%s"
          % (logical_shape, physical_shape)
      )
      if mtf.list_product(logical_shape) != mtf.list_product(
          physical_shape
      ):
          raise ValueError(
              "physical and logical shapes must have the same product "
              "physical_shape=%s logical_shape=%s"
              % (physical_shape, logical_shape)
          )
      # drop logical dimensions of size 1
      logical_shape = [i for i in logical_shape if i != 1]
      num_cores = mtf.list_product(logical_shape)
  
      # For physical shapes different from what we are used to [2^a, 2^b, 2],
      #   return a simple default value (a lexicographic ordering)
      def _default_value():
          default = list(range(num_cores))
          if return_coordinates:
              default = [
                  mtf.pnum_to_processor_coordinates(i) for i in default
              ]
          return default
  
      if len(physical_shape) != 3:
          tf.logging.warning(
              "Unrecognized format for tpu physical shape"
          )
          return _default_value()
      # physical_shape is a triple of rows, cols, cores
      p0, p1, p2 = physical_shape
      if p2 != FLAGS.logical_cores_per_chip:
          return _default_value
      for dimsize in [p0, p1]:
          # if dimsize not a power of 2, give up
          if dimsize & (dimsize - 1):
              return _default_value()
      # At this point, the physical shape has at least 1x1x2=2 cores, so there
      #   must be at least one logical dimension.
      assert logical_shape
      if len(logical_shape) == 1:
          # ring of p0 x p1 chips
          ring = _ring_2d(p0, p1)
          logical_to_physical = []
          for logical_pnum in range(num_cores):
              core_on_chip = logical_pnum % FLAGS.logical_cores_per_chip
              chip_num = logical_pnum // FLAGS.logical_cores_per_chip
              i, j = ring[chip_num]
              logical_to_physical.append((i, j, core_on_chip))
      else:
          # We have a p0 x p1 rectangle of chips, which we will tile with rectangular
          #   tiles.  The first logical dimension correspond to the number of tiles,
          #   and the other logical dimensions will correspond to position within a
          #   tile.
          num_tiles = logical_shape[0]
          tile_chips = num_cores // num_tiles // p2
          # If we can, we make each tile occupy exactly one row or column of chips.
          # Otherwise, we make each tile approximately square.
          if len(logical_shape) == 2 and tile_chips == p0:
              t0, t1 = [tile_chips, 1]
          elif len(logical_shape) == 2 and tile_chips == p1:
              t0, t1 = [1, tile_chips]
          else:
              # try to make the tile approximately square
              lg_tile_chips = int(math.log(tile_chips, 2))
              t0 = 2 ** (lg_tile_chips // 2)
              # make sure that the tile fits in the mesh - i.e.
              #   t0 <= p0
              #   t1 == tile_chips // t0 <= p1
              t0 = min(t0, p0)
              t0 = max(t0, tile_chips // p1)
              t1 = tile_chips // t0
          # recursive call to find mapping for one tile
          tile_logical_to_physical = auto_logical_to_physical_tpu(
              logical_shape[1:], [t0, t1, p2], return_coordinates=True
          )
          tiles_ring = _ring_2d(p0 // t0, p1 // t1)
          logical_to_physical = []
          for logical_pnum in range(num_cores):
              logical_tile_num = logical_pnum // (t0 * t1 * p2)
              logical_pos_in_tile = logical_pnum % (t0 * t1 * p2)
              logical_to_physical.append(
                  (
                      tiles_ring[logical_tile_num][0] * t0
                      + tile_logical_to_physical[logical_pos_in_tile][
                          0
                      ],
                      tiles_ring[logical_tile_num][1] * t1
                      + tile_logical_to_physical[logical_pos_in_tile][
                          1
                      ],
                      tile_logical_to_physical[logical_pos_in_tile][2],
                  )
              )
      tf.logging.info(
          "auto_logical_to_physical_tpu logical_to_physical = %s"
          % logical_to_physical
      )
      if return_coordinates:
          return logical_to_physical
      else:
          if FLAGS.logical_cores_per_chip > 1:
              return [
                  mtf.processor_coordinates_to_pnum(
                      physical_shape, coord
                  )
                  for coord in logical_to_physical
              ]
          else:
              # check device_assignment
              if device_assignment is None:
                  raise ValueError(
                      "physical_shape or device_assignment unset"
                  )
              return mtf.processor_coordinates_to_pnum_map_nd(
                  physical_shape, logical_to_physical, device_assignment
              )
  
imports: from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  import math
  import os
  import gin
  from mesh_tensorflow import ops_with_redefined_builtins
  from mesh_tensorflow import tpu_variables
  from mesh_tensorflow import utils
  from six.moves import xrange
  import tensorflow.compat.v1
  from tensorflow.python.tpu.ops import tpu_ops
label_desc: Use image node_id 5 for calling a global function with example usage: auto_logical_to_physical_tpu(logical_shape, physical_shape, return_coordinates, device_assignment) and returns: default, _default_value, _default_value, logical_to_physical, _default_value, unknown, mtf