code: def cat_rule(op_schema: OpSchema) -> OutputSharding:
      # torch.cat requires all tensors must either have the same shape (except
      # in the concatenating dimension) or be "empty". "Empty" here strictly means
      # tensor.shape is torch.Size([0]). When tensor.ndim > 1, it will be treated
      # as a non-empty tensor and the shape must match on non-cat dimensions.
      def is_empty(spec: DTensorSpec) -> bool:
          return list(spec.shape) == [0]
  
      # the first arg is a list of input tensor specs
      tensor_list_specs = cast(
          List[DTensorSpec], op_schema.args_schema[0]
      )
      assert (
          len(tensor_list_specs) > 0
      ), "torch.cat expects a non-empty list of tensors"
      non_empty_specs = [
          spec for spec in tensor_list_specs if not is_empty(spec)
      ]
  
      if len(non_empty_specs) == 0:
          # all tensors are empty, we can return any output sharding
          return OutputSharding(
              output_spec=DTensorSpec(
                  mesh=tensor_list_specs[0].mesh,
                  placements=tensor_list_specs[0].placements,
              )
          )
  
      assert all(
          spec.ndim == non_empty_specs[0].ndim
          for spec in non_empty_specs
      ), f"Expect all tensors to have same shape or empty, but got {tensor_list_specs}"
      assert all(
          spec.mesh == tensor_list_specs[0].mesh
          for spec in tensor_list_specs
      ), f"Expect all tensors to have same mesh, but got {tensor_list_specs}"
  
      # ndim will also be the result's ndim
      ndim = 1
      for spec in tensor_list_specs:
          ndim = max(ndim, spec.ndim)
  
      dim = 0  # default dim = 0
      if len(op_schema.args_schema) > 1:
          dim = cast(int, op_schema.args_schema[1])
      dim = normalize_dim(dim, ndim)
  
      # Make sure all tensors are replciated on cat dimension
      need_reshard = False
      tensor_list_specs_after: List[DTensorSpec] = []
      for spec in tensor_list_specs:
          if not is_empty(spec) and (
              is_tensor_dim_sharded(spec, dim=dim)
              or is_tensor_partial(spec)
          ):
              need_reshard = True
              tensor_list_specs_after.append(
                  DTensorSpec(
                      mesh=spec.mesh,
                      placements=replicate_tensor_dim(
                          spec.placements, dim=dim
                      ),
                      tensor_meta=spec.tensor_meta,
                  )
              )
          else:
              tensor_list_specs_after.append(spec)
  
      tensor_list_specs = tensor_list_specs_after
  
      # align non-cat dimensions placements based on reshard cost
      non_empty_specs = [
          spec for spec in tensor_list_specs if not is_empty(spec)
      ]
      mesh = non_empty_specs[0].mesh
      ndim = non_empty_specs[0].ndim
      new_placements: List[Placement] = []
      for mesh_dim in range(mesh.ndim):
          # compute the minimum cost of resharding on this mesh_dim
          if any(
              spec.placements[mesh_dim]
              != non_empty_specs[0].placements[mesh_dim]
              for spec in non_empty_specs
          ):
              # only reshard if there is a mismatch
              need_reshard = True
              reshard_cost = []
              for shard_dim in range(ndim):
                  # compute the cost of resharding on this shard_dim
                  cost: float = 0.0
                  for spec in non_empty_specs:
                      global_shape = spec.shape
                      if global_shape[shard_dim] < mesh.size(mesh_dim):
                          # found one tensor where the shard_dim is smaller than
                          # mesh_dim. In this case, we cannot shard on this shard_dim,
                          # and hence set cost to infinity.
                          cost = +float("inf")
                      elif (
                          is_tensor_dim_sharded(spec, dim=shard_dim)
                          or prod(global_shape) == 0
                      ):
                          continue
                      else:
                          local_shape = compute_local_shape(
                              global_shape, spec.mesh, spec.placements
                          )
                          cost += prod(local_shape) * spec.mesh.size(
                              mesh_dim
                          )
                  reshard_cost.append(cost)
              best_dim = reshard_cost.index(min(reshard_cost))
              new_placements.append(Shard(best_dim))
          else:
              # no mismatch, keep the original placement
              new_placements.append(
                  non_empty_specs[0].placements[mesh_dim]
              )
  
      if need_reshard:
          tensor_list_specs_after = []
          for spec in tensor_list_specs:
              if is_empty(spec):
                  tensor_list_specs_after.append(spec)
              else:
                  tensor_list_specs_after.append(
                      DTensorSpec(
                          mesh=spec.mesh,
                          placements=tuple(new_placements),
                          tensor_meta=spec.tensor_meta,
                      )
                  )
  
          return OutputSharding(
              output_spec=None,
              schema_suggestions=[
                  OpSchema(
                      op=op_schema.op,
                      args_schema=(
                          tuple(tensor_list_specs_after),
                          *op_schema.args_schema[1:],
                      ),
                      kwargs_schema=op_schema.kwargs_schema,
                  ),
              ],
          )
      else:
          # at this point, the cat dim is not sharded,
          return OutputSharding(
              output_spec=DTensorSpec(
                  mesh=non_empty_specs[0].mesh,
                  placements=non_empty_specs[0].placements,
              ),
          )
  
imports: from typing import cast, List, Optional, Sequence, Tuple
  import torch
  from torch.distributed._tensor._utils import compute_local_shape
  from torch.distributed._tensor.op_schema import OpSchema, OpStrategy, OutputSharding, PlacementStrategy, RuntimeSchemaInfo, StrategyType
  from torch.distributed._tensor.ops.common_rules import pointwise_rule
  from torch.distributed._tensor.ops.utils import is_tensor_dim_sharded, is_tensor_partial, normalize_dim, prod, register_op_strategy, register_prop_rule
  from torch.distributed._tensor.placement_types import _Partial, DTensorSpec, Placement, Replicate, Shard
  from torch.distributed.device_mesh import DeviceMesh
label_desc: Use image node_id 13 for calling a global function with example usage: cat_rule(op_schema) and returns: unknown, OutputSharding, OutputSharding, OutputSharding