code: def merge(
      df: Union[DataFrame, Series],
      right: Union[DataFrame, Series],
      how: str = "inner",
      on: str = None,
      left_on: str = None,
      right_on: str = None,
      left_index: bool = False,
      right_index: bool = False,
      sort: bool = False,
      suffixes: Tuple[Optional[str], Optional[str]] = ("_x", "_y"),
      copy: bool = True,
      indicator: bool = False,
      validate: str = None,
      method: str = "auto",
      auto_merge: str = "both",
      auto_merge_threshold: int = 8,
      bloom_filter: Union[bool, str] = "auto",
      bloom_filter_options: Dict[str, Any] = None,
  ) -> DataFrame:
      """
      Merge DataFrame or named Series objects with a database-style join.
  
      A named Series object is treated as a DataFrame with a single named column.
  
      The join is done on columns or indexes. If joining columns on
      columns, the DataFrame indexes *will be ignored*. Otherwise if joining indexes
      on indexes or indexes on a column or columns, the index will be passed on.
      When performing a cross merge, no column specifications to merge on are
      allowed.
  
      Parameters
      ----------
      right : DataFrame or named Series
          Object to merge with.
      how : {'left', 'right', 'outer', 'inner'}, default 'inner'
          Type of merge to be performed.
  
          * left: use only keys from left frame, similar to a SQL left outer join;
            preserve key order.
          * right: use only keys from right frame, similar to a SQL right outer join;
            preserve key order.
          * outer: use union of keys from both frames, similar to a SQL full outer
            join; sort keys lexicographically.
          * inner: use intersection of keys from both frames, similar to a SQL inner
            join; preserve the order of the left keys.
  
      on : label or list
          Column or index level names to join on. These must be found in both
          DataFrames. If `on` is None and not merging on indexes then this defaults
          to the intersection of the columns in both DataFrames.
      left_on : label or list, or array-like
          Column or index level names to join on in the left DataFrame. Can also
          be an array or list of arrays of the length of the left DataFrame.
          These arrays are treated as if they are columns.
      right_on : label or list, or array-like
          Column or index level names to join on in the right DataFrame. Can also
          be an array or list of arrays of the length of the right DataFrame.
          These arrays are treated as if they are columns.
      left_index : bool, default False
          Use the index from the left DataFrame as the join key(s). If it is a
          MultiIndex, the number of keys in the other DataFrame (either the index
          or a number of columns) must match the number of levels.
      right_index : bool, default False
          Use the index from the right DataFrame as the join key. Same caveats as
          left_index.
      sort : bool, default False
          Sort the join keys lexicographically in the result DataFrame. If False,
          the order of the join keys depends on the join type (how keyword).
      suffixes : list-like, default is ("_x", "_y")
          A length-2 sequence where each element is optionally a string
          indicating the suffix to add to overlapping column names in
          `left` and `right` respectively. Pass a value of `None` instead
          of a string to indicate that the column name from `left` or
          `right` should be left as-is, with no suffix. At least one of the
          values must not be None.
      copy : bool, default True
          If False, avoid copy if possible.
      indicator : bool or str, default False
          If True, adds a column to the output DataFrame called "_merge" with
          information on the source of each row. The column can be given a different
          name by providing a string argument. The column will have a Categorical
          type with the value of "left_only" for observations whose merge key only
          appears in the left DataFrame, "right_only" for observations
          whose merge key only appears in the right DataFrame, and "both"
          if the observation's merge key is found in both DataFrames.
      validate : str, optional
          If specified, checks if merge is of specified type.
  
          * "one_to_one" or "1:1": check if merge keys are unique in both
            left and right datasets.
          * "one_to_many" or "1:m": check if merge keys are unique in left
            dataset.
          * "many_to_one" or "m:1": check if merge keys are unique in right
            dataset.
          * "many_to_many" or "m:m": allowed, but does not result in checks.
      method : {"auto", "shuffle", "broadcast"}, default auto
          "broadcast" is recommended when one DataFrame is much smaller than the other,
          otherwise, "shuffle" will be a better choice. By default, we choose method
          according to actual data size.
      auto_merge : {"both", "none", "before", "after"}, default both
          Auto merge small chunks before or after merge
  
          * "both": auto merge small chunks before and after,
          * "none": do not merge small chunks
          * "before": only merge small chunks before merge
          * "after": only merge small chunks after merge
      auto_merge_threshold : int, default 8
          When how is "inner", merged result could be much smaller than original DataFrame,
          if the number of chunks is greater than the threshold,
          it will merge small chunks automatically.
      bloom_filter: bool, str, default "auto"
          Use bloom filter to optimize merge
      bloom_filter_options: dict
          * "max_elements": max elements in bloom filter,
            default value is the max size of all input chunks
          * "error_rate": error raite, default 0.1.
          * "apply_chunk_size_threshold": min chunk size of input chunks to apply bloom filter, default 10
            when chunk size of left and right is greater than this threshold, apply bloom filter
          * "filter": "large", "small", "both", default "large"
            decides to filter on large, small or both DataFrames.
  
      Returns
      -------
      DataFrame
          A DataFrame of the two merged objects.
  
      Examples
      --------
      >>> import mars.dataframe as md
      >>> df1 = md.DataFrame({'lkey': ['foo', 'bar', 'baz', 'foo'],
      ...                     'value': [1, 2, 3, 5]})
      >>> df2 = md.DataFrame({'rkey': ['foo', 'bar', 'baz', 'foo'],
      ...                     'value': [5, 6, 7, 8]})
      >>> df1.execute()
          lkey value
      0   foo      1
      1   bar      2
      2   baz      3
      3   foo      5
      >>> df2.execute()
          rkey value
      0   foo      5
      1   bar      6
      2   baz      7
      3   foo      8
  
      Merge df1 and df2 on the lkey and rkey columns. The value columns have
      the default suffixes, _x and _y, appended.
  
      >>> df1.merge(df2, left_on='lkey', right_on='rkey').execute()
        lkey  value_x rkey  value_y
      0  foo        1  foo        5
      1  foo        1  foo        8
      2  foo        5  foo        5
      3  foo        5  foo        8
      4  bar        2  bar        6
      5  baz        3  baz        7
  
      Merge DataFrames df1 and df2 with specified left and right suffixes
      appended to any overlapping columns.
  
      >>> df1.merge(df2, left_on='lkey', right_on='rkey',
      ...           suffixes=('_left', '_right')).execute()
        lkey  value_left rkey  value_right
      0  foo           1  foo            5
      1  foo           1  foo            8
      2  foo           5  foo            5
      3  foo           5  foo            8
      4  bar           2  bar            6
      5  baz           3  baz            7
  
      Merge DataFrames df1 and df2, but raise an exception if the DataFrames have
      any overlapping columns.
  
      >>> df1.merge(df2, left_on='lkey', right_on='rkey', suffixes=(False, False)).execute()
      Traceback (most recent call last):
      ...
      ValueError: columns overlap but no suffix specified:
          Index(['value'], dtype='object')
  
      >>> df1 = md.DataFrame({'a': ['foo', 'bar'], 'b': [1, 2]})
      >>> df2 = md.DataFrame({'a': ['foo', 'baz'], 'c': [3, 4]})
      >>> df1.execute()
            a  b
      0   foo  1
      1   bar  2
      >>> df2.execute()
            a  c
      0   foo  3
      1   baz  4
  
      >>> df1.merge(df2, how='inner', on='a').execute()
            a  b  c
      0   foo  1  3
  
      >>> df1.merge(df2, how='left', on='a').execute()
            a  b  c
      0   foo  1  3.0
      1   bar  2  NaN
      """
      if method is None:
          method = "auto"
      if method not in [
          "auto",
          "shuffle",
          "broadcast",
      ]:  # pragma: no cover
          raise NotImplementedError(f"{method} merge is not supported")
      if auto_merge not in [
          "both",
          "none",
          "before",
          "after",
      ]:  # pragma: no cover
          raise ValueError(
              f"auto_merge can only be `both`, `none`, `before` or `after`, got {auto_merge}"
          )
      if bloom_filter not in [True, False, "auto"]:
          raise ValueError(
              f'bloom_filter can only be True, False, or "auto", got {bloom_filter}'
          )
      if bloom_filter_options:
          if not isinstance(bloom_filter_options, dict):
              raise TypeError(
                  f"bloom_filter_options must be a dict, got {type(bloom_filter_options)}"
              )
          for k, v in bloom_filter_options.items():
              if k not in BLOOM_FILTER_OPTIONS:
                  raise ValueError(
                      f"Invalid bloom filter option {k}, available: {BLOOM_FILTER_OPTIONS}"
                  )
              if k == "filter" and v not in BLOOM_FILTER_ON_OPTIONS:
                  raise ValueError(
                      f"Invalid filter {k}, available: {BLOOM_FILTER_ON_OPTIONS}"
                  )
      op = DataFrameMerge(
          how=how,
          on=on,
          left_on=left_on,
          right_on=right_on,
          left_index=left_index,
          right_index=right_index,
          sort=sort,
          suffixes=suffixes,
          copy=copy,
          indicator=indicator,
          validate=validate,
          method=method,
          auto_merge=auto_merge,
          auto_merge_threshold=auto_merge_threshold,
          bloom_filter=bloom_filter,
          bloom_filter_options=bloom_filter_options,
          output_types=[OutputType.dataframe],
      )
      return op(df, right)
  
imports: import itertools
  from collections import namedtuple
  from enum import Enum
  from typing import Any, Dict, List, Optional, Union, Tuple
  import numpy
  import pandas
  from ...None import opcodes
  from ...core import OutputType, recursive_tile, TileStatus
  from ...core.context import get_context
  from ...core.operand import OperandStage, MapReduceOperand
  from ...serialization.serializables import AnyField, BoolField, DictField, StringField, TupleField, KeyField, Int32Field, NamedTupleField
  from ...typing import TileableType
  from ...utils import has_unknown_shape, lazy_import
  from ..base.bloom_filter import filter_by_bloom_filter
  from ..core import DataFrame, Series, DataFrameChunk
  from ..operands import DataFrameOperand, DataFrameOperandMixin, DataFrameShuffleProxy
  from ..utils import auto_merge_chunks, build_concatenated_rows_frame, build_df, parse_index, hash_dataframe_on, infer_index_value, is_cudf
  import logging
label_desc: Use image node_id 2 for calling a global function with example usage: merge(df, right, how, on, left_on, right_on, left_index, right_index, sort, suffixes, copy, indicator, validate, method, auto_merge, auto_merge_threshold, bloom_filter, bloom_filter_options) and returns: op