Repository: pandas-dev/pandas
Issue category: Feature Request

New engine for MultiIndex?
Currently, ``MultiIndex.get_loc()`` and ``MultiIndex.get_indexer()`` both rely on an ``_engine`` which is either a ``MultiIndexObjectEngine`` or a ``MultiIndexHashEngine``: but both of these are thin layers over the flat ``ObjectEngine``. This means that the actual structure of labels and levels is completely discarded (except e.g. for partial indexing, see ``_get_level_indexer()``).

In principle, a completely different scheme could be used:
- first look for the key elements in ``levels``, and find the corresponding code
- then look for the code in the levels

In most cases, the second part should be the computationally expensive one. It would consist in running ``nlevels`` searches in arrays of ``dtype=int`` (the ``.labels``) rather than (as it is now) one search in an ``object`` array in which each element is actually a ``tuple`` of ``nlevels`` elements. My guess is that thanks to vectorization the former should be much faster than the latter.

Moreover (and maybe more importantly), with the current engine fixing a bug such as #18485 is a nightmare. And the same applies to
```
In [2]: (4, True) in pd.MultiIndex.from_tuples([(4, 1)])
Out[2]: True
```
and probably others. This is because even though levels are not mixed, the elements are compared as ``objects``.

One caveat is that the single levels would be very often non-unique, and I'm not sure what is the impact of this with the current implementation of hash tables.
No automatic type casting in complete indexing of a large MultiIndex
#### Code Sample, a copy-pastable example if possible

```python
In [2]: pd.MultiIndex.from_product([[1., 2.], range(5000)]).get_loc((1, 0))
Out[2]: 0

In [3]: pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc((1., 0))
Out[3]: 0

In [4]: pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc(1)
Out[4]: slice(0, 5001, None)

In [5]: pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc((1, 0))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-5-a8f6fcc9f5d9> in <module>()
----> 1 pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc((1, 0))

/home/nobackup/repo/pandas/pandas/core/indexes/multi.py in get_loc(self, key, method)
   2141             key = _values_from_object(key)
   2142             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 2143             return self._engine.get_loc(key)
   2144 
   2145         # -- partial selection or non-unique index

/home/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15856)()
    645         return algos.pad_object(values, other, limit=limit)
    646 
--> 647     cpdef get_loc(self, object val):
    648         if is_definitely_invalid_key(val):
    649             raise TypeError("'{val}' is an invalid key".format(val=val))

/home/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15703)()
    654 
    655         try:
--> 656             return self.mapping.get_item(val)
    657         except TypeError:
    658             raise KeyError(val)

/home/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24621)()
   1446             return False
   1447 
-> 1448     cpdef get_item(self, object key):
   1449         cdef:
   1450             khiter_t k

/home/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24576)()
   1460             return loc
   1461         else:
-> 1462             raise KeyError(key)
   1463 
   1464     cpdef set_item(self, object key, Py_ssize_t val):

KeyError: (1, 0)
```

#### Problem description

Related to #18519 - this is an inherent limit of the current design of the engine for large ``MultiIndex``es, and an [improved](https://github.com/pandas-dev/pandas/issues/18519#issuecomment-347323053) ``MultiIndexEngine`` should solve this too.

#### Expected Output

```python
Out[5]: 0
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: b5f1e716d89487423bb12d6cc4e6da2b39184531
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-4-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.22.0.dev0+371.gb5f1e716d
pytest: 3.2.3
pip: 9.0.1
setuptools: 36.7.0
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.4.1
dateutil: 2.6.1
pytz: 2017.2
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.3.0
xlsxwriter: 0.9.6
lxml: 4.1.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1


</details>

MultiIndexHashEngine can't manage mixed type tuples
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex.from_product([[True, False], range(1, 10)])

In [3]: mi.get_loc((False, 1))
Out[3]: 9

In [4]: mi = pd.MultiIndex.from_product([[True, False], range(1, 10000)])

In [5]: mi.get_loc((False, 1))
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-5-68f8698878ec> in <module>()
----> 1 mi.get_loc((False, 1))

/home/pietro/nobackup/repo/pandas/pandas/core/indexes/multi.py in get_loc(self, key, method)
   2134             key = _values_from_object(key)
   2135             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 2136             return self._engine.get_loc(key)
   2137 
   2138         # -- partial selection or non-unique index

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15854)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15701)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24621)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24468)()

/home/pietro/nobackup/repo/pandas/pandas/core/indexes/multi.py in _hashed_indexing_key(self, key)
    819         key = tuple([f(k, stringify)
    820                      for k, stringify in zip(key, self._have_mixed_levels)])
--> 821         return hash_tuple(key)
    822 
    823     @Appender(base._shared_docs['duplicated'] % _index_doc_kwargs)

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in hash_tuple(val, encoding, hash_key)
    186               for v in val)
    187 
--> 188     h = _combine_hash_arrays(hashes, len(val))[0]
    189 
    190     return h

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in _combine_hash_arrays(arrays, num_items)
     31     """
     32     try:
---> 33         first = next(arrays)
     34     except StopIteration:
     35         return np.array([], dtype=np.uint64)

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in <genexpr>(.0)
    184     """
    185     hashes = (_hash_scalar(v, encoding=encoding, hash_key=hash_key)
--> 186               for v in val)
    187 
    188     h = _combine_hash_arrays(hashes, len(val))[0]

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in _hash_scalar(val, encoding, hash_key)
    330 
    331     return hash_array(vals, hash_key=hash_key, encoding=encoding,
--> 332                       categorize=False)

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in hash_array(vals, encoding, hash_key, categorize)
    290 
    291         try:
--> 292             vals = hashing.hash_object_array(vals, hash_key, encoding)
    293         except TypeError:
    294             # we have mixed types

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashing.pyx in pandas._libs.hashing.hash_object_array (pandas/_libs/hashing.c:1764)()

ValueError: Does not understand character buffer dtype format string ('?')

In [6]: mi = pd.MultiIndex.from_product([[1, 0], range(1, 10000)])

In [7]: mi.get_loc((1, 1))
Out[7]: 0

```
#### Problem description

The two engines should give the same result.

#### Expected Output

``Out[3]:``

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: f745e52e168790bff06a55928c3491cdea389508
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-3-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.22.0.dev0+241.gf745e52e1.dirty
pytest: 3.2.3
pip: 9.0.1
setuptools: 36.7.0
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.4.1
dateutil: 2.6.1
pytz: 2017.2
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: None
xlrd: 1.0.0
xlwt: 1.1.2
xlsxwriter: 0.9.6
lxml: None
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1


</details>

MultiIndex.get_loc misbehaves on NaNs
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex(levels=[[1, 2, 3, 5], [4, 6]], labels=[[3, 1, 2, 0], [1, -1, 0, -1]])

In [3]: mi.get_loc((2, np.nan))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-3-d77b20d4d7a2> in <module>()
----> 1 mi.get_loc((2, np.nan))

/home/pietro/nobackup/repo/pandas/pandas/core/indexes/multi.py in get_loc(self, key, method)
   2119             key = _values_from_object(key)
   2120             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 2121             return self._engine.get_loc(key)
   2122 
   2123         # -- partial selection or non-unique index

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexObjectEngine.get_loc (pandas/_libs/index.c:14965)()
    616         return super(MultiIndexObjectEngine, self).get_indexer(values)
    617 
--> 618     cpdef get_loc(self, object val):
    619 
    620         # convert a MI to an ndarray

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexObjectEngine.get_loc (pandas/_libs/index.c:14886)()
    621         if hasattr(val, 'values'):
    622             val = val.values
--> 623         return super(MultiIndexObjectEngine, self).get_loc(val)
    624 
    625 

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5832)()
    137             util.set_value_at(arr, loc, value)
    138 
--> 139     cpdef get_loc(self, object val):
    140         if is_definitely_invalid_key(val):
    141             raise TypeError("'{val}' is an invalid key".format(val=val))

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5678)()
    159 
    160         try:
--> 161             return self.mapping.get_item(val)
    162         except (TypeError, ValueError):
    163             raise KeyError(val)

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.PyObjectHashTable.get_item (pandas/_libs/hashtable.c:21018)()
   1263                                        sizeof(uint32_t)) # flags
   1264 
-> 1265     cpdef get_item(self, object val):
   1266         cdef khiter_t k
   1267         if val != val or val is None:

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.PyObjectHashTable.get_item (pandas/_libs/hashtable.c:20972)()
   1271             return self.table.vals[k]
   1272         else:
-> 1273             raise KeyError(val)
   1274 
   1275     cpdef set_item(self, object key, Py_ssize_t val):

KeyError: (2, nan)

In [4]: mi.get_indexer(mi.copy())
Out[4]: array([ 0, -1,  2, -1])

In [5]: mi == mi.copy()
Out[5]: array([ True, False,  True, False], dtype=bool)

```
#### Problem description

I think _this_ is actually the cause for [this example](https://github.com/pandas-dev/pandas/issues/18455#issuecomment-346700652), which is different from the one reported at the top of #18455 .

#### Expected Output

``array([ 0, 1,  2, 3])``

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: b45325e283b16ec8869aaea407de8256fc234f33
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-4-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: en_GB.UTF-8
LOCALE: en_GB.UTF-8

pandas: 0.22.0.dev0+201.gb45325e28
pytest: 3.0.6
pip: 9.0.1
setuptools: 33.1.1
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.18.1
pyarrow: None
xarray: None
IPython: 5.2.2
sphinx: None
patsy: 0.4.1+dev
dateutil: 2.6.0
pytz: 2016.10
blosc: None
bottleneck: 1.2.0
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.2.0
xlsxwriter: None
lxml: 3.7.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.8
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: None


</details>

No implicit casting of int -> float on full indexing of multiindex
#### Code Sample, a copy-pastable example if possible

```python

In [2]: mi = pd.MultiIndex.from_product([[0., 1.], ['begin', 'end']])

In [3]: mi.get_loc(1)
Out[3]: slice(2, 4, None)

In [4]: mi.get_indexer_for([(1, 'end')])
Out[4]: array([3])

In [5]: mi.get_loc((1, 'end'))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-5-eb387fc9b362> in <module>()
----> 1 mi.get_loc((1, 'end'))

/home/pietro/nobackup/repo/pandas/pandas/indexes/multi.py in get_loc(self, key, method)
   1962             key = _values_from_object(key)
   1963             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 1964             return self._engine.get_loc(key)
   1965 
   1966         # -- partial selection or non-unique index

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexEngine.get_loc (pandas/_libs/index.c:13171)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexEngine.get_loc (pandas/_libs/index.c:13018)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:23625)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:23578)()

KeyError: (1, 'end')
```

#### Problem description

We implicitly cast if indexing only one level, and if asking a list of (complete) keys: we should do the same for a single complete key.

#### Expected Output

``` python
In [6]: mi.get_loc((1., 'end'))
Out[6]: 3
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: None
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.7.0-1-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.utf8
LOCALE: it_IT.UTF-8

pandas: 0.19.0+783.gcd35d22a0.dirty
pytest: 3.0.6
pip: 9.0.1
setuptools: 33.1.1
Cython: 0.25.2
numpy: 1.12.0
scipy: 0.18.1
xarray: 0.9.1
IPython: 5.1.0.dev
sphinx: 1.4.9
patsy: 0.3.0-dev
dateutil: 2.5.3
pytz: 2016.7
blosc: None
bottleneck: 1.2.0
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.1.2
xlsxwriter: 0.9.6
lxml: 3.7.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.8
s3fs: None
pandas_gbq: None
pandas_datareader: 0.2.1

</details>

MultiIndex (and not flat Index) casts bool to int
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex.from_product([['a', 'b'], [0, 1]])

In [3]: mi.get_loc(('a', True))
Out[3]: 1

In [4]: mi.get_level_values(1).get_loc(True)
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
~/nobackup/repo/pandas/pandas/core/indexes/base.py in get_loc(self, key, method, tolerance)
   2513             try:
-> 2514                 return self._engine.get_loc(key)
   2515             except KeyError:

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5834)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5617)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine._get_loc_duplicates (pandas/_libs/index.c:6064)()

~/nobackup/repo/pandas/pandas/_libs/index_class_helper.pxi in pandas._libs.index.Int64Engine._maybe_get_bool_indexer (pandas/_libs/index.c:19249)()

KeyError: True

During handling of the above exception, another exception occurred:

KeyError                                  Traceback (most recent call last)
<ipython-input-4-0c9f78315be2> in <module>()
----> 1 mi.get_level_values(1).get_loc(True)

~/nobackup/repo/pandas/pandas/core/indexes/base.py in get_loc(self, key, method, tolerance)
   2514                 return self._engine.get_loc(key)
   2515             except KeyError:
-> 2516                 return self._engine.get_loc(self._maybe_cast_indexer(key))
   2517 
   2518         indexer = self.get_indexer([key], method=method, tolerance=tolerance)

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5834)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5617)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine._get_loc_duplicates (pandas/_libs/index.c:6064)()

~/nobackup/repo/pandas/pandas/_libs/index_class_helper.pxi in pandas._libs.index.Int64Engine._maybe_get_bool_indexer (pandas/_libs/index.c:19249)()

KeyError: True

In [5]: mi.get_indexer([('a', True), ('b', False)])
Out[5]: array([1, 2])

In [6]: mi.get_level_values(1).get_indexer([True, False])
Out[6]: array([-1, -1])

```

#### Problem description

From https://github.com/pandas-dev/pandas/issues/18519#issue-277042995 , just to keep track of it (and add a test in #19074 , which fixes it).

#### Expected Output

``KeyError: True`` in ``Out[3]: 1`` too, ``Out[5]:`` like ``Out[6]:``.

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: None
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-4-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.23.0.dev0+42.g93033151a
pytest: 3.2.3
pip: 9.0.1
setuptools: 36.7.0
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.4.1
dateutil: 2.6.1
pytz: 2017.2
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.3.0
xlsxwriter: 0.9.6
lxml: 4.1.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1

</details>

New engine for MultiIndex?
Currently, ``MultiIndex.get_loc()`` and ``MultiIndex.get_indexer()`` both rely on an ``_engine`` which is either a ``MultiIndexObjectEngine`` or a ``MultiIndexHashEngine``: but both of these are thin layers over the flat ``ObjectEngine``. This means that the actual structure of labels and levels is completely discarded (except e.g. for partial indexing, see ``_get_level_indexer()``).

In principle, a completely different scheme could be used:
- first look for the key elements in ``levels``, and find the corresponding code
- then look for the code in the levels

In most cases, the second part should be the computationally expensive one. It would consist in running ``nlevels`` searches in arrays of ``dtype=int`` (the ``.labels``) rather than (as it is now) one search in an ``object`` array in which each element is actually a ``tuple`` of ``nlevels`` elements. My guess is that thanks to vectorization the former should be much faster than the latter.

Moreover (and maybe more importantly), with the current engine fixing a bug such as #18485 is a nightmare. And the same applies to
```
In [2]: (4, True) in pd.MultiIndex.from_tuples([(4, 1)])
Out[2]: True
```
and probably others. This is because even though levels are not mixed, the elements are compared as ``objects``.

One caveat is that the single levels would be very often non-unique, and I'm not sure what is the impact of this with the current implementation of hash tables.
No automatic type casting in complete indexing of a large MultiIndex
#### Code Sample, a copy-pastable example if possible

```python
In [2]: pd.MultiIndex.from_product([[1., 2.], range(5000)]).get_loc((1, 0))
Out[2]: 0

In [3]: pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc((1., 0))
Out[3]: 0

In [4]: pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc(1)
Out[4]: slice(0, 5001, None)

In [5]: pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc((1, 0))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-5-a8f6fcc9f5d9> in <module>()
----> 1 pd.MultiIndex.from_product([[1., 2.], range(5001)]).get_loc((1, 0))

/home/nobackup/repo/pandas/pandas/core/indexes/multi.py in get_loc(self, key, method)
   2141             key = _values_from_object(key)
   2142             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 2143             return self._engine.get_loc(key)
   2144 
   2145         # -- partial selection or non-unique index

/home/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15856)()
    645         return algos.pad_object(values, other, limit=limit)
    646 
--> 647     cpdef get_loc(self, object val):
    648         if is_definitely_invalid_key(val):
    649             raise TypeError("'{val}' is an invalid key".format(val=val))

/home/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15703)()
    654 
    655         try:
--> 656             return self.mapping.get_item(val)
    657         except TypeError:
    658             raise KeyError(val)

/home/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24621)()
   1446             return False
   1447 
-> 1448     cpdef get_item(self, object key):
   1449         cdef:
   1450             khiter_t k

/home/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24576)()
   1460             return loc
   1461         else:
-> 1462             raise KeyError(key)
   1463 
   1464     cpdef set_item(self, object key, Py_ssize_t val):

KeyError: (1, 0)
```

#### Problem description

Related to #18519 - this is an inherent limit of the current design of the engine for large ``MultiIndex``es, and an [improved](https://github.com/pandas-dev/pandas/issues/18519#issuecomment-347323053) ``MultiIndexEngine`` should solve this too.

#### Expected Output

```python
Out[5]: 0
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: b5f1e716d89487423bb12d6cc4e6da2b39184531
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-4-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.22.0.dev0+371.gb5f1e716d
pytest: 3.2.3
pip: 9.0.1
setuptools: 36.7.0
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.4.1
dateutil: 2.6.1
pytz: 2017.2
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.3.0
xlsxwriter: 0.9.6
lxml: 4.1.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1


</details>

MultiIndexHashEngine can't manage mixed type tuples
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex.from_product([[True, False], range(1, 10)])

In [3]: mi.get_loc((False, 1))
Out[3]: 9

In [4]: mi = pd.MultiIndex.from_product([[True, False], range(1, 10000)])

In [5]: mi.get_loc((False, 1))
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-5-68f8698878ec> in <module>()
----> 1 mi.get_loc((False, 1))

/home/pietro/nobackup/repo/pandas/pandas/core/indexes/multi.py in get_loc(self, key, method)
   2134             key = _values_from_object(key)
   2135             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 2136             return self._engine.get_loc(key)
   2137 
   2138         # -- partial selection or non-unique index

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15854)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexHashEngine.get_loc (pandas/_libs/index.c:15701)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24621)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:24468)()

/home/pietro/nobackup/repo/pandas/pandas/core/indexes/multi.py in _hashed_indexing_key(self, key)
    819         key = tuple([f(k, stringify)
    820                      for k, stringify in zip(key, self._have_mixed_levels)])
--> 821         return hash_tuple(key)
    822 
    823     @Appender(base._shared_docs['duplicated'] % _index_doc_kwargs)

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in hash_tuple(val, encoding, hash_key)
    186               for v in val)
    187 
--> 188     h = _combine_hash_arrays(hashes, len(val))[0]
    189 
    190     return h

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in _combine_hash_arrays(arrays, num_items)
     31     """
     32     try:
---> 33         first = next(arrays)
     34     except StopIteration:
     35         return np.array([], dtype=np.uint64)

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in <genexpr>(.0)
    184     """
    185     hashes = (_hash_scalar(v, encoding=encoding, hash_key=hash_key)
--> 186               for v in val)
    187 
    188     h = _combine_hash_arrays(hashes, len(val))[0]

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in _hash_scalar(val, encoding, hash_key)
    330 
    331     return hash_array(vals, hash_key=hash_key, encoding=encoding,
--> 332                       categorize=False)

/home/pietro/nobackup/repo/pandas/pandas/core/util/hashing.py in hash_array(vals, encoding, hash_key, categorize)
    290 
    291         try:
--> 292             vals = hashing.hash_object_array(vals, hash_key, encoding)
    293         except TypeError:
    294             # we have mixed types

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashing.pyx in pandas._libs.hashing.hash_object_array (pandas/_libs/hashing.c:1764)()

ValueError: Does not understand character buffer dtype format string ('?')

In [6]: mi = pd.MultiIndex.from_product([[1, 0], range(1, 10000)])

In [7]: mi.get_loc((1, 1))
Out[7]: 0

```
#### Problem description

The two engines should give the same result.

#### Expected Output

``Out[3]:``

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: f745e52e168790bff06a55928c3491cdea389508
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-3-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.22.0.dev0+241.gf745e52e1.dirty
pytest: 3.2.3
pip: 9.0.1
setuptools: 36.7.0
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.4.1
dateutil: 2.6.1
pytz: 2017.2
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: None
xlrd: 1.0.0
xlwt: 1.1.2
xlsxwriter: 0.9.6
lxml: None
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1


</details>

MultiIndex.get_loc misbehaves on NaNs
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex(levels=[[1, 2, 3, 5], [4, 6]], labels=[[3, 1, 2, 0], [1, -1, 0, -1]])

In [3]: mi.get_loc((2, np.nan))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-3-d77b20d4d7a2> in <module>()
----> 1 mi.get_loc((2, np.nan))

/home/pietro/nobackup/repo/pandas/pandas/core/indexes/multi.py in get_loc(self, key, method)
   2119             key = _values_from_object(key)
   2120             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 2121             return self._engine.get_loc(key)
   2122 
   2123         # -- partial selection or non-unique index

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexObjectEngine.get_loc (pandas/_libs/index.c:14965)()
    616         return super(MultiIndexObjectEngine, self).get_indexer(values)
    617 
--> 618     cpdef get_loc(self, object val):
    619 
    620         # convert a MI to an ndarray

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexObjectEngine.get_loc (pandas/_libs/index.c:14886)()
    621         if hasattr(val, 'values'):
    622             val = val.values
--> 623         return super(MultiIndexObjectEngine, self).get_loc(val)
    624 
    625 

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5832)()
    137             util.set_value_at(arr, loc, value)
    138 
--> 139     cpdef get_loc(self, object val):
    140         if is_definitely_invalid_key(val):
    141             raise TypeError("'{val}' is an invalid key".format(val=val))

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5678)()
    159 
    160         try:
--> 161             return self.mapping.get_item(val)
    162         except (TypeError, ValueError):
    163             raise KeyError(val)

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.PyObjectHashTable.get_item (pandas/_libs/hashtable.c:21018)()
   1263                                        sizeof(uint32_t)) # flags
   1264 
-> 1265     cpdef get_item(self, object val):
   1266         cdef khiter_t k
   1267         if val != val or val is None:

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.PyObjectHashTable.get_item (pandas/_libs/hashtable.c:20972)()
   1271             return self.table.vals[k]
   1272         else:
-> 1273             raise KeyError(val)
   1274 
   1275     cpdef set_item(self, object key, Py_ssize_t val):

KeyError: (2, nan)

In [4]: mi.get_indexer(mi.copy())
Out[4]: array([ 0, -1,  2, -1])

In [5]: mi == mi.copy()
Out[5]: array([ True, False,  True, False], dtype=bool)

```
#### Problem description

I think _this_ is actually the cause for [this example](https://github.com/pandas-dev/pandas/issues/18455#issuecomment-346700652), which is different from the one reported at the top of #18455 .

#### Expected Output

``array([ 0, 1,  2, 3])``

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: b45325e283b16ec8869aaea407de8256fc234f33
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-4-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: en_GB.UTF-8
LOCALE: en_GB.UTF-8

pandas: 0.22.0.dev0+201.gb45325e28
pytest: 3.0.6
pip: 9.0.1
setuptools: 33.1.1
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.18.1
pyarrow: None
xarray: None
IPython: 5.2.2
sphinx: None
patsy: 0.4.1+dev
dateutil: 2.6.0
pytz: 2016.10
blosc: None
bottleneck: 1.2.0
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.2.0
xlsxwriter: None
lxml: 3.7.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.8
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: None


</details>

No implicit casting of int -> float on full indexing of multiindex
#### Code Sample, a copy-pastable example if possible

```python

In [2]: mi = pd.MultiIndex.from_product([[0., 1.], ['begin', 'end']])

In [3]: mi.get_loc(1)
Out[3]: slice(2, 4, None)

In [4]: mi.get_indexer_for([(1, 'end')])
Out[4]: array([3])

In [5]: mi.get_loc((1, 'end'))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-5-eb387fc9b362> in <module>()
----> 1 mi.get_loc((1, 'end'))

/home/pietro/nobackup/repo/pandas/pandas/indexes/multi.py in get_loc(self, key, method)
   1962             key = _values_from_object(key)
   1963             key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
-> 1964             return self._engine.get_loc(key)
   1965 
   1966         # -- partial selection or non-unique index

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexEngine.get_loc (pandas/_libs/index.c:13171)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.MultiIndexEngine.get_loc (pandas/_libs/index.c:13018)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:23625)()

/home/pietro/nobackup/repo/pandas/pandas/_libs/hashtable_class_helper.pxi in pandas._libs.hashtable.MultiIndexHashTable.get_item (pandas/_libs/hashtable.c:23578)()

KeyError: (1, 'end')
```

#### Problem description

We implicitly cast if indexing only one level, and if asking a list of (complete) keys: we should do the same for a single complete key.

#### Expected Output

``` python
In [6]: mi.get_loc((1., 'end'))
Out[6]: 3
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: None
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.7.0-1-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.utf8
LOCALE: it_IT.UTF-8

pandas: 0.19.0+783.gcd35d22a0.dirty
pytest: 3.0.6
pip: 9.0.1
setuptools: 33.1.1
Cython: 0.25.2
numpy: 1.12.0
scipy: 0.18.1
xarray: 0.9.1
IPython: 5.1.0.dev
sphinx: 1.4.9
patsy: 0.3.0-dev
dateutil: 2.5.3
pytz: 2016.7
blosc: None
bottleneck: 1.2.0
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.1.2
xlsxwriter: 0.9.6
lxml: 3.7.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.8
s3fs: None
pandas_gbq: None
pandas_datareader: 0.2.1

</details>

MultiIndex (and not flat Index) casts bool to int
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex.from_product([['a', 'b'], [0, 1]])

In [3]: mi.get_loc(('a', True))
Out[3]: 1

In [4]: mi.get_level_values(1).get_loc(True)
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
~/nobackup/repo/pandas/pandas/core/indexes/base.py in get_loc(self, key, method, tolerance)
   2513             try:
-> 2514                 return self._engine.get_loc(key)
   2515             except KeyError:

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5834)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5617)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine._get_loc_duplicates (pandas/_libs/index.c:6064)()

~/nobackup/repo/pandas/pandas/_libs/index_class_helper.pxi in pandas._libs.index.Int64Engine._maybe_get_bool_indexer (pandas/_libs/index.c:19249)()

KeyError: True

During handling of the above exception, another exception occurred:

KeyError                                  Traceback (most recent call last)
<ipython-input-4-0c9f78315be2> in <module>()
----> 1 mi.get_level_values(1).get_loc(True)

~/nobackup/repo/pandas/pandas/core/indexes/base.py in get_loc(self, key, method, tolerance)
   2514                 return self._engine.get_loc(key)
   2515             except KeyError:
-> 2516                 return self._engine.get_loc(self._maybe_cast_indexer(key))
   2517 
   2518         indexer = self.get_indexer([key], method=method, tolerance=tolerance)

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5834)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc (pandas/_libs/index.c:5617)()

~/nobackup/repo/pandas/pandas/_libs/index.pyx in pandas._libs.index.IndexEngine._get_loc_duplicates (pandas/_libs/index.c:6064)()

~/nobackup/repo/pandas/pandas/_libs/index_class_helper.pxi in pandas._libs.index.Int64Engine._maybe_get_bool_indexer (pandas/_libs/index.c:19249)()

KeyError: True

In [5]: mi.get_indexer([('a', True), ('b', False)])
Out[5]: array([1, 2])

In [6]: mi.get_level_values(1).get_indexer([True, False])
Out[6]: array([-1, -1])

```

#### Problem description

From https://github.com/pandas-dev/pandas/issues/18519#issue-277042995 , just to keep track of it (and add a test in #19074 , which fixes it).

#### Expected Output

``KeyError: True`` in ``Out[3]: 1`` too, ``Out[5]:`` like ``Out[6]:``.

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: None
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-4-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.23.0.dev0+42.g93033151a
pytest: 3.2.3
pip: 9.0.1
setuptools: 36.7.0
Cython: 0.25.2
numpy: 1.12.1
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.4.1
dateutil: 2.6.1
pytz: 2017.2
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.3.0
xlsxwriter: 0.9.6
lxml: 4.1.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1

</details>



Functions to edit:
pandas/core/indexes/multi.py:MultiIndex._engine
pandas/core/indexes/multi.py:MultiIndex.get_indexer
pandas/core/indexes/multi.py:MultiIndex.get_loc
pandas/core/indexes/multi.py:MultiIndex.get_loc_level
Functions added: pandas/core/indexes/multi.py:MultiIndexUIntEngine
pandas/core/indexes/multi.py:MultiIndexUIntEngine._codes_to_ints
pandas/core/indexes/multi.py:MultiIndexPyIntEngine
pandas/core/indexes/multi.py:MultiIndexPyIntEngine._codes_to_ints

Patch:
diff --git a/doc/source/whatsnew/v0.23.0.txt b/doc/source/whatsnew/v0.23.0.txt
index 246eab386b2ab..efb4707649f08 100644
--- a/doc/source/whatsnew/v0.23.0.txt
+++ b/doc/source/whatsnew/v0.23.0.txt
@@ -380,6 +380,7 @@ Performance Improvements
 - Improved performance of ``DatetimeIndex`` and ``Series`` arithmetic operations with Business-Month and Business-Quarter frequencies (:issue:`18489`)
 - :func:`Series` / :func:`DataFrame` tab completion limits to 100 values, for better performance. (:issue:`18587`)
 - Improved performance of :func:`DataFrame.median` with ``axis=1`` when bottleneck is not installed (:issue:`16468`)
+- Improved performance of :func:`MultiIndex.get_loc` for large indexes, at the cost of a reduction in performance for small ones (:issue:`18519`)
 
 
 .. _whatsnew_0230.docs:
@@ -476,7 +477,11 @@ MultiIndex
 - Bug in :func:`MultiIndex.get_level_values` which would return an invalid index on level of ints with missing values (:issue:`17924`)
 - Bug in :func:`MultiIndex.remove_unused_levels` which would fill nan values (:issue:`18417`)
 - Bug in :func:`MultiIndex.from_tuples`` which would fail to take zipped tuples in python3 (:issue:`18434`)
--
+- Bug in :func:`MultiIndex.get_loc`` which would fail to automatically cast values between float and int (:issue:`18818`, :issue:`15994`)
+- Bug in :func:`MultiIndex.get_loc`` which would cast boolean to integer labels (:issue:`19086`)
+- Bug in :func:`MultiIndex.get_loc`` which would fail to locate keys containing ``NaN`` (:issue:`18485`)
+- Bug in :func:`MultiIndex.get_loc`` in large :class:`MultiIndex`, would fail when levels had different dtypes (:issue:`18520`)
+
 
 I/O
 ^^^
diff --git a/pandas/_libs/hashtable.pxd b/pandas/_libs/hashtable.pxd
index 014da22df3382..d735b3c0673b2 100644
--- a/pandas/_libs/hashtable.pxd
+++ b/pandas/_libs/hashtable.pxd
@@ -31,15 +31,6 @@ cdef class PyObjectHashTable(HashTable):
     cpdef get_item(self, object val)
     cpdef set_item(self, object key, Py_ssize_t val)
 
-cdef class MultiIndexHashTable(HashTable):
-    cdef:
-        kh_uint64_t *table
-        object mi
-
-    cpdef get_item(self, object val)
-    cpdef set_item(self, object key, Py_ssize_t val)
-    cdef inline void _check_for_collision(self, Py_ssize_t loc, object label)
-
 
 cdef class StringHashTable(HashTable):
     cdef kh_str_t *table
diff --git a/pandas/_libs/hashtable_class_helper.pxi.in b/pandas/_libs/hashtable_class_helper.pxi.in
index bd9dd1f9bae37..bca4e388f3279 100644
--- a/pandas/_libs/hashtable_class_helper.pxi.in
+++ b/pandas/_libs/hashtable_class_helper.pxi.in
@@ -899,139 +899,3 @@ cdef class PyObjectHashTable(HashTable):
                 count += 1
 
         return np.asarray(labels)
-
-
-cdef class MultiIndexHashTable(HashTable):
-
-    def __init__(self, size_hint=1):
-        self.table = kh_init_uint64()
-        self.mi = None
-        kh_resize_uint64(self.table, size_hint)
-
-    def __dealloc__(self):
-        if self.table is not NULL:
-            kh_destroy_uint64(self.table)
-            self.table = NULL
-
-    def __len__(self):
-        return self.table.size
-
-    def sizeof(self, deep=False):
-        """ return the size of my table in bytes """
-        return self.table.n_buckets * (sizeof(uint64_t) + # keys
-                                       sizeof(size_t) + # vals
-                                       sizeof(uint32_t)) # flags
-
-    def _check_for_collisions(self, int64_t[:] locs, object mi):
-        # validate that the locs map to the actual values
-        # provided in the mi
-        # we can only check if we *don't* have any missing values
-        # :<
-        cdef:
-            ndarray[int64_t] alocs
-
-        alocs = np.asarray(locs)
-        if (alocs != -1).all():
-
-            result = self.mi.take(locs)
-            if isinstance(mi, tuple):
-                from pandas import Index
-                mi = Index([mi])
-            if not result.equals(mi):
-                raise AssertionError(
-                    "hash collision\nlocs:\n{}\n"
-                    "result:\n{}\nmi:\n{}".format(alocs, result, mi))
-
-    cdef inline void _check_for_collision(self, Py_ssize_t loc, object label):
-        # validate that the loc maps to the actual value
-        # version of _check_for_collisions above for single label (tuple)
-
-        result = self.mi[loc]
-
-        if not all(l == r or (is_null_datetimelike(l)
-                              and is_null_datetimelike(r))
-                   for l, r in zip(result, label)):
-            raise AssertionError(
-                "hash collision\nloc:\n{}\n"
-                "result:\n{}\nmi:\n{}".format(loc, result, label))
-
-    def __contains__(self, object key):
-        try:
-            self.get_item(key)
-            return True
-        except (KeyError, ValueError, TypeError):
-            return False
-
-    cpdef get_item(self, object key):
-        cdef:
-            khiter_t k
-            uint64_t value
-            int64_t[:] locs
-            Py_ssize_t loc
-
-        value = self.mi._hashed_indexing_key(key)
-        k = kh_get_uint64(self.table, value)
-        if k != self.table.n_buckets:
-            loc = self.table.vals[k]
-            self._check_for_collision(loc, key)
-            return loc
-        else:
-            raise KeyError(key)
-
-    cpdef set_item(self, object key, Py_ssize_t val):
-        raise NotImplementedError
-
-    @cython.boundscheck(False)
-    def map_locations(self, object mi):
-        cdef:
-            Py_ssize_t i, n
-            ndarray[uint64_t] values
-            uint64_t val
-            int ret = 0
-            khiter_t k
-
-        self.mi = mi
-        n = len(mi)
-        values = mi._hashed_values
-
-        with nogil:
-            for i in range(n):
-                val = values[i]
-                k = kh_put_uint64(self.table, val, &ret)
-                self.table.vals[k] = i
-
-    @cython.boundscheck(False)
-    def lookup(self, object mi):
-        # look up with a target mi
-        cdef:
-            Py_ssize_t i, n
-            ndarray[uint64_t] values
-            int ret = 0
-            uint64_t val
-            khiter_t k
-            int64_t[:] locs
-
-        n = len(mi)
-        values = mi._hashed_values
-
-        locs = np.empty(n, dtype=np.int64)
-
-        with nogil:
-            for i in range(n):
-                val = values[i]
-                k = kh_get_uint64(self.table, val)
-                if k != self.table.n_buckets:
-                    locs[i] = self.table.vals[k]
-                else:
-                    locs[i] = -1
-
-        self._check_for_collisions(locs, mi)
-        return np.asarray(locs)
-
-    def unique(self, object mi):
-        raise NotImplementedError
-
-    def get_labels(self, object mi, ObjectVector uniques,
-                   Py_ssize_t count_prior, int64_t na_sentinel,
-                   bint check_null=True):
-        raise NotImplementedError
diff --git a/pandas/_libs/index.pyx b/pandas/_libs/index.pyx
index bfea4ff9915ac..6b23e487aad3a 100644
--- a/pandas/_libs/index.pyx
+++ b/pandas/_libs/index.pyx
@@ -26,11 +26,12 @@ from hashtable cimport HashTable
 from pandas._libs import algos, hashtable as _hash
 from pandas._libs.tslibs import period as periodlib
 from pandas._libs.tslib import Timestamp, Timedelta
+from pandas._libs.missing import checknull
 
 cdef int64_t iNaT = util.get_nat()
 
 
-cdef inline is_definitely_invalid_key(object val):
+cdef inline bint is_definitely_invalid_key(object val):
     if PyTuple_Check(val):
         try:
             hash(val)
@@ -585,70 +586,137 @@ cpdef convert_scalar(ndarray arr, object value):
     return value
 
 
-cdef class MultiIndexObjectEngine(ObjectEngine):
+cdef class BaseMultiIndexCodesEngine:
     """
-    provide the same interface as the MultiIndexEngine
-    but use the IndexEngine for computation
-
-    This provides good performance with samller MI's
+    Base class for MultiIndexUIntEngine and MultiIndexPyIntEngine, which
+    represent each label in a MultiIndex as an integer, by juxtaposing the bits
+    encoding each level, with appropriate offsets.
+
+    For instance: if 3 levels have respectively 3, 6 and 1 possible values,
+    then their labels can be represented using respectively 2, 3 and 1 bits,
+    as follows:
+     _ _ _ _____ _ __ __ __
+    |0|0|0| ... |0| 0|a1|a0| -> offset 0 (first level)
+     — — — ————— — —— —— ——
+    |0|0|0| ... |0|b2|b1|b0| -> offset 2 (bits required for first level)
+     — — — ————— — —— —— ——
+    |0|0|0| ... |0| 0| 0|c0| -> offset 5 (bits required for first two levels)
+     ‾ ‾ ‾ ‾‾‾‾‾ ‾ ‾‾ ‾‾ ‾‾
+    and the resulting unsigned integer representation will be:
+     _ _ _ _____ _ __ __ __ __ __ __
+    |0|0|0| ... |0|c0|b2|b1|b0|a1|a0|
+     ‾ ‾ ‾ ‾‾‾‾‾ ‾ ‾‾ ‾‾ ‾‾ ‾‾ ‾‾ ‾‾
+
+    Offsets are calculated at initialization, labels are transformed by method
+    _codes_to_ints.
+
+    Keys are located by first locating each component against the respective
+    level, then locating (the integer representation of) codes.
     """
-    def get_indexer(self, values):
-        # convert a MI to an ndarray
-        if hasattr(values, 'values'):
-            values = values.values
-        return super(MultiIndexObjectEngine, self).get_indexer(values)
+    def __init__(self, object levels, object labels,
+                 ndarray[uint64_t, ndim=1] offsets):
+        """
+        Parameters
+        ----------
+        levels : list-like of numpy arrays
+            Levels of the MultiIndex
+        labels : list-like of numpy arrays of integer dtype
+            Labels of the MultiIndex
+        offsets : numpy array of uint64 dtype
+            Pre-calculated offsets, one for each level of the index
+        """
 
-    cpdef get_loc(self, object val):
+        self.levels = levels
+        self.offsets = offsets
 
-        # convert a MI to an ndarray
-        if hasattr(val, 'values'):
-            val = val.values
-        return super(MultiIndexObjectEngine, self).get_loc(val)
+        # Transform labels in a single array, and add 1 so that we are working
+        # with positive integers (-1 for NaN becomes 0):
+        codes = (np.array(labels, dtype='int64').T + 1).astype('uint64',
+                                                               copy=False)
 
+        # Map each codes combination in the index to an integer unambiguously
+        # (no collisions possible), based on the "offsets", which describe the
+        # number of bits to switch labels for each level:
+        lab_ints = self._codes_to_ints(codes)
 
-cdef class MultiIndexHashEngine(ObjectEngine):
-    """
-    Use a hashing based MultiIndex impl
-    but use the IndexEngine for computation
+        # Initialize underlying index (e.g. libindex.UInt64Engine) with
+        # integers representing labels: we will use its get_loc and get_indexer
+        self._base.__init__(self, lambda: lab_ints, len(lab_ints))
 
-    This provides good performance with larger MI's
-    """
+    def _extract_level_codes(self, object target, object method=None):
+        """
+        Map the requested list of (tuple) keys to their integer representations
+        for searching in the underlying integer index.
+
+        Parameters
+        ----------
+        target : list-like of keys
+            Each key is a tuple, with a label for each level of the index.
+
+        Returns
+        ------
+        int_keys : 1-dimensional array of dtype uint64 or object
+            Integers representing one combination each
+        """
 
-    def _call_monotonic(self, object mi):
-        # defer these back to the mi iteself
-        return (mi.is_monotonic_increasing,
-                mi.is_monotonic_decreasing,
-                mi.is_unique)
+        level_codes = [lev.get_indexer(codes) + 1 for lev, codes
+                       in zip(self.levels, zip(*target))]
+        return self._codes_to_ints(np.array(level_codes, dtype='uint64').T)
+
+    def get_indexer(self, object target, object method=None,
+                    object limit=None):
+        lab_ints = self._extract_level_codes(target)
+
+        # All methods (exact, backfill, pad) directly map to the respective
+        # methods of the underlying (integers) index...
+        if method is not None:
+            # but underlying backfill and pad methods require index and keys
+            # to be sorted. The index already is (checked in
+            # Index._get_fill_indexer), sort (integer representations of) keys:
+            order = np.argsort(lab_ints)
+            lab_ints = lab_ints[order]
+            indexer = (getattr(self._base, 'get_{}_indexer'.format(method))
+                       (self, lab_ints, limit=limit))
+            indexer = indexer[order]
+        else:
+            indexer = self._base.get_indexer(self, lab_ints)
 
-    def get_backfill_indexer(self, other, limit=None):
-        # we coerce to ndarray-of-tuples
-        values = np.array(self._get_index_values())
-        return algos.backfill_object(values, other, limit=limit)
+        return indexer
 
-    def get_pad_indexer(self, other, limit=None):
-        # we coerce to ndarray-of-tuples
-        values = np.array(self._get_index_values())
-        return algos.pad_object(values, other, limit=limit)
+    def get_loc(self, object key):
+        if is_definitely_invalid_key(key):
+            raise TypeError("'{key}' is an invalid key".format(key=key))
+        if not PyTuple_Check(key):
+            raise KeyError(key)
+        try:
+            indices = [0 if checknull(v) else lev.get_loc(v) + 1
+                       for lev, v in zip(self.levels, key)]
+        except KeyError:
+            raise KeyError(key)
 
-    cpdef get_loc(self, object val):
-        if is_definitely_invalid_key(val):
-            raise TypeError("'{val}' is an invalid key".format(val=val))
+        # Transform indices into single integer:
+        lab_int = self._codes_to_ints(np.array(indices, dtype='uint64'))
 
-        self._ensure_mapping_populated()
-        if not self.unique:
-            return self._get_loc_duplicates(val)
+        return self._base.get_loc(self, lab_int)
 
-        try:
-            return self.mapping.get_item(val)
-        except TypeError:
-            raise KeyError(val)
+    def get_indexer_non_unique(self, object target):
+        # This needs to be overridden just because the default one works on
+        # target._values, and target can be itself a MultiIndex.
 
-    def get_indexer(self, values):
-        self._ensure_mapping_populated()
-        return self.mapping.lookup(values)
+        lab_ints = self._extract_level_codes(target)
+        indexer = self._base.get_indexer_non_unique(self, lab_ints)
+
+        return indexer
+
+    def __contains__(self, object val):
+        # Default __contains__ looks in the underlying mapping, which in this
+        # case only contains integer representations.
+        try:
+            self.get_loc(val)
+            return True
+        except (KeyError, TypeError, ValueError):
+            return False
 
-    cdef _make_hash_table(self, n):
-        return _hash.MultiIndexHashTable(n)
 
 # Generated from template.
 include "index_class_helper.pxi"
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 797774832aaa5..510f7245cebd8 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -45,6 +45,87 @@
          target_klass='MultiIndex or list of tuples'))
 
 
+class MultiIndexUIntEngine(libindex.BaseMultiIndexCodesEngine,
+                           libindex.UInt64Engine):
+    """
+    This class manages a MultiIndex by mapping label combinations to positive
+    integers.
+    """
+    _base = libindex.UInt64Engine
+
+    def _codes_to_ints(self, codes):
+        """
+        Transform combination(s) of uint64 in one uint64 (each), in a strictly
+        monotonic way (i.e. respecting the lexicographic order of integer
+        combinations): see BaseMultiIndexCodesEngine documentation.
+
+        Parameters
+        ----------
+        codes : 1- or 2-dimensional array of dtype uint64
+            Combinations of integers (one per row)
+
+        Returns
+        ------
+        int_keys : scalar or 1-dimensional array, of dtype uint64
+            Integer(s) representing one combination (each)
+        """
+        # Shift the representation of each level by the pre-calculated number
+        # of bits:
+        codes <<= self.offsets
+
+        # Now sum and OR are in fact interchangeable. This is a simple
+        # composition of the (disjunct) significant bits of each level (i.e.
+        # each column in "codes") in a single positive integer:
+        if codes.ndim == 1:
+            # Single key
+            return np.bitwise_or.reduce(codes)
+
+        # Multiple keys
+        return np.bitwise_or.reduce(codes, axis=1)
+
+
+class MultiIndexPyIntEngine(libindex.BaseMultiIndexCodesEngine,
+                            libindex.ObjectEngine):
+    """
+    This class manages those (extreme) cases in which the number of possible
+    label combinations overflows the 64 bits integers, and uses an ObjectEngine
+    containing Python integers.
+    """
+    _base = libindex.ObjectEngine
+
+    def _codes_to_ints(self, codes):
+        """
+        Transform combination(s) of uint64 in one Python integer (each), in a
+        strictly monotonic way (i.e. respecting the lexicographic order of
+        integer combinations): see BaseMultiIndexCodesEngine documentation.
+
+        Parameters
+        ----------
+        codes : 1- or 2-dimensional array of dtype uint64
+            Combinations of integers (one per row)
+
+        Returns
+        ------
+        int_keys : int, or 1-dimensional array of dtype object
+            Integer(s) representing one combination (each)
+        """
+
+        # Shift the representation of each level by the pre-calculated number
+        # of bits. Since this can overflow uint64, first make sure we are
+        # working with Python integers:
+        codes = codes.astype('object') << self.offsets
+
+        # Now sum and OR are in fact interchangeable. This is a simple
+        # composition of the (disjunct) significant bits of each level (i.e.
+        # each column in "codes") in a single positive integer (per row):
+        if codes.ndim == 1:
+            # Single key
+            return np.bitwise_or.reduce(codes)
+
+        # Multiple keys
+        return np.bitwise_or.reduce(codes, axis=1)
+
+
 class MultiIndex(Index):
     """
     A multi-level, or hierarchical, index object for pandas objects
@@ -687,16 +768,25 @@ def _get_level_number(self, level):
 
     @cache_readonly
     def _engine(self):
-
-        # choose our engine based on our size
-        # the hashing based MultiIndex for larger
-        # sizes, and the MultiIndexOjbect for smaller
-        # xref: https://github.com/pandas-dev/pandas/pull/16324
-        l = len(self)
-        if l > 10000:
-            return libindex.MultiIndexHashEngine(lambda: self, l)
-
-        return libindex.MultiIndexObjectEngine(lambda: self.values, l)
+        # Calculate the number of bits needed to represent labels in each
+        # level, as log2 of their sizes (including -1 for NaN):
+        sizes = np.ceil(np.log2([len(l) + 1 for l in self.levels]))
+
+        # Sum bit counts, starting from the _right_....
+        lev_bits = np.cumsum(sizes[::-1])[::-1]
+
+        # ... in order to obtain offsets such that sorting the combination of
+        # shifted codes (one for each level, resulting in a unique integer) is
+        # equivalent to sorting lexicographically the codes themselves. Notice
+        # that each level needs to be shifted by the number of bits needed to
+        # represent the _previous_ ones:
+        offsets = np.concatenate([lev_bits[1:], [0]]).astype('uint64')
+
+        # Check the total number of bits needed for our representation:
+        if lev_bits[0] > 64:
+            # The levels would overflow a 64 bit uint - use Python integers:
+            return MultiIndexPyIntEngine(self.levels, self.labels, offsets)
+        return MultiIndexUIntEngine(self.levels, self.labels, offsets)
 
     @property
     def values(self):
@@ -1885,16 +1975,11 @@ def get_indexer(self, target, method=None, limit=None, tolerance=None):
             if tolerance is not None:
                 raise NotImplementedError("tolerance not implemented yet "
                                           'for MultiIndex')
-            indexer = self._get_fill_indexer(target, method, limit)
+            indexer = self._engine.get_indexer(target, method, limit)
         elif method == 'nearest':
             raise NotImplementedError("method='nearest' not implemented yet "
                                       'for MultiIndex; see GitHub issue 9365')
         else:
-            # we may not compare equally because of hashing if we
-            # don't have the same dtypes
-            if self._inferred_type_levels != target._inferred_type_levels:
-                return Index(self.values).get_indexer(target.values)
-
             indexer = self._engine.get_indexer(target)
 
         return _ensure_platform_int(indexer)
@@ -2131,17 +2216,6 @@ def _maybe_to_slice(loc):
                            ''.format(keylen, self.nlevels))
 
         if keylen == self.nlevels and self.is_unique:
-
-            def _maybe_str_to_time_stamp(key, lev):
-                if lev.is_all_dates and not isinstance(key, Timestamp):
-                    try:
-                        return Timestamp(key, tz=getattr(lev, 'tz', None))
-                    except Exception:
-                        pass
-                return key
-
-            key = com._values_from_object(key)
-            key = tuple(map(_maybe_str_to_time_stamp, key, self.levels))
             return self._engine.get_loc(key)
 
         # -- partial selection or non-unique index
@@ -2274,34 +2348,9 @@ def partial_selection(key, indexer=None):
                     return indexer, maybe_droplevels(indexer, ilevels,
                                                      drop_level)
 
-                if len(key) == self.nlevels:
-
-                    if self.is_unique:
-
-                        # here we have a completely specified key, but are
-                        # using some partial string matching here
-                        # GH4758
-                        all_dates = ((l.is_all_dates and
-                                      not isinstance(k, compat.string_types))
-                                     for k, l in zip(key, self.levels))
-                        can_index_exactly = any(all_dates)
-                        if (any(l.is_all_dates
-                                for k, l in zip(key, self.levels)) and
-                                not can_index_exactly):
-                            indexer = self.get_loc(key)
-
-                            # we have a multiple selection here
-                            if (not isinstance(indexer, slice) or
-                                    indexer.stop - indexer.start != 1):
-                                return partial_selection(key, indexer)
-
-                            key = tuple(self[indexer].tolist()[0])
-
-                        return (self._engine.get_loc(
-                            com._values_from_object(key)), None)
-
-                    else:
-                        return partial_selection(key)
+                if len(key) == self.nlevels and self.is_unique:
+                    # Complete key in unique index -> standard get_loc
+                    return (self._engine.get_loc(key), None)
                 else:
                     return partial_selection(key)
             else:

Test patch: diff --git a/pandas/tests/indexes/test_multi.py b/pandas/tests/indexes/test_multi.py
index 9664d73651185..aedc957ec67da 100644
--- a/pandas/tests/indexes/test_multi.py
+++ b/pandas/tests/indexes/test_multi.py
@@ -1258,6 +1258,17 @@ def test_get_loc_level(self):
         assert result == expected
         assert new_index.equals(index.droplevel(0))
 
+    @pytest.mark.parametrize('level', [0, 1])
+    @pytest.mark.parametrize('null_val', [np.nan, pd.NaT, None])
+    def test_get_loc_nan(self, level, null_val):
+        # GH 18485 : NaN in MultiIndex
+        levels = [['a', 'b'], ['c', 'd']]
+        key = ['b', 'd']
+        levels[level] = np.array([0, null_val], dtype=type(null_val))
+        key[level] = null_val
+        idx = MultiIndex.from_product(levels)
+        assert idx.get_loc(tuple(key)) == 3
+
     def test_get_loc_missing_nan(self):
         # GH 8569
         idx = MultiIndex.from_arrays([[1.0, 2.0], [3.0, 4.0]])
@@ -1266,6 +1277,38 @@ def test_get_loc_missing_nan(self):
         pytest.raises(KeyError, idx.get_loc, np.nan)
         pytest.raises(KeyError, idx.get_loc, [np.nan])
 
+    @pytest.mark.parametrize('dtype1', [int, float, bool, str])
+    @pytest.mark.parametrize('dtype2', [int, float, bool, str])
+    def test_get_loc_multiple_dtypes(self, dtype1, dtype2):
+        # GH 18520
+        levels = [np.array([0, 1]).astype(dtype1),
+                  np.array([0, 1]).astype(dtype2)]
+        idx = pd.MultiIndex.from_product(levels)
+        assert idx.get_loc(idx[2]) == 2
+
+    @pytest.mark.parametrize('level', [0, 1])
+    @pytest.mark.parametrize('dtypes', [[int, float], [float, int]])
+    def test_get_loc_implicit_cast(self, level, dtypes):
+        # GH 18818, GH 15994 : as flat index, cast int to float and vice-versa
+        levels = [['a', 'b'], ['c', 'd']]
+        key = ['b', 'd']
+        lev_dtype, key_dtype = dtypes
+        levels[level] = np.array([0, 1], dtype=lev_dtype)
+        key[level] = key_dtype(1)
+        idx = MultiIndex.from_product(levels)
+        assert idx.get_loc(tuple(key)) == 3
+
+    def test_get_loc_cast_bool(self):
+        # GH 19086 : int is casted to bool, but not vice-versa
+        levels = [[False, True], np.arange(2, dtype='int64')]
+        idx = MultiIndex.from_product(levels)
+
+        assert idx.get_loc((0, 1)) == 1
+        assert idx.get_loc((1, 0)) == 2
+
+        pytest.raises(KeyError, idx.get_loc, (False, True))
+        pytest.raises(KeyError, idx.get_loc, (True, False))
+
     def test_slice_locs(self):
         df = tm.makeTimeDataFrame()
         stacked = df.stack()
diff --git a/pandas/tests/test_multilevel.py b/pandas/tests/test_multilevel.py
index 424ba6aab9a56..9582264a8c716 100644
--- a/pandas/tests/test_multilevel.py
+++ b/pandas/tests/test_multilevel.py
@@ -1590,6 +1590,38 @@ def test_unstack_group_index_overflow(self):
         result = s.unstack(4)
         assert result.shape == (500, 2)
 
+    def test_pyint_engine(self):
+        # GH 18519 : when combinations of codes cannot be represented in 64
+        # bits, the index underlying the MultiIndex engine works with Python
+        # integers, rather than uint64.
+        N = 5
+        keys = [tuple(l) for l in [[0] * 10 * N,
+                                   [1] * 10 * N,
+                                   [2] * 10 * N,
+                                   [np.nan] * N + [2] * 9 * N,
+                                   [0] * N + [2] * 9 * N,
+                                   [np.nan] * N + [2] * 8 * N + [0] * N]]
+        # Each level contains 4 elements (including NaN), so it is represented
+        # in 2 bits, for a total of 2*N*10 = 100 > 64 bits. If we were using a
+        # 64 bit engine and truncating the first levels, the fourth and fifth
+        # keys would collide; if truncating the last levels, the fifth and
+        # sixth; if rotating bits rather than shifting, the third and fifth.
+
+        for idx in range(len(keys)):
+            index = MultiIndex.from_tuples(keys)
+            assert index.get_loc(keys[idx]) == idx
+
+            expected = np.arange(idx + 1, dtype='int64')
+            result = index.get_indexer([keys[i] for i in expected])
+            tm.assert_numpy_array_equal(result, expected)
+
+        # With missing key:
+        idces = range(len(keys))
+        expected = np.array([-1] + list(idces), dtype='int64')
+        missing = tuple([0, 1] * 5 * N)
+        result = index.get_indexer([missing] + [keys[i] for i in idces])
+        tm.assert_numpy_array_equal(result, expected)
+
     def test_getitem_lowerdim_corner(self):
         pytest.raises(KeyError, self.frame.loc.__getitem__,
                       (('bar', 'three'), 'B'))