BUG: MultiIndex.factorize raising with length-0 (#62281)

Author: jbrockmendel

doc/source/whatsnew/v3.0.0.rst

@@ -1005,6 +1005,7 @@ I/O
 - Bug in :meth:`DataFrame.to_string` that raised ``StopIteration`` with nested DataFrames. (:issue:`16098`)
 - Bug in :meth:`HDFStore.get` was failing to save data of dtype datetime64[s] correctly (:issue:`59004`)
 - Bug in :meth:`HDFStore.select` causing queries on categorical string columns to return unexpected results (:issue:`57608`)
+- Bug in :meth:`MultiIndex.factorize` incorrectly raising on length-0 indexes (:issue:`57517`)
 - Bug in :meth:`read_csv` causing segmentation fault when ``encoding_errors`` is not a string. (:issue:`59059`)
 - Bug in :meth:`read_csv` raising ``TypeError`` when ``index_col`` is specified and ``na_values`` is a dict containing the key ``None``. (:issue:`57547`)
 - Bug in :meth:`read_csv` raising ``TypeError`` when ``nrows`` and ``iterator`` are specified without specifying a ``chunksize``. (:issue:`59079`)

pandas/core/base.py

@@ -1298,7 +1298,11 @@ def factorize(
 
         if isinstance(self, ABCMultiIndex):
             # preserve MultiIndex
-            uniques = self._constructor(uniques)
+            if len(self) == 0:
+                # GH#57517
+                uniques = self[:0]
+            else:
+                uniques = self._constructor(uniques)
         else:
             from pandas import Index
 

pandas/tests/test_algos.py

@@ -582,6 +582,16 @@ def test_index_returned(self, index):
                 expected = expected.normalize()
         tm.assert_index_equal(result, expected, exact=True)
 
+    def test_factorize_multiindex_empty(self):
+        # GH#57517
+        mi = MultiIndex.from_product(
+            [Index([], name="a", dtype=object), Index([], name="i", dtype="f4")]
+        )
+        codes, uniques = mi.factorize()
+        exp_codes = np.array([], dtype=np.intp)
+        tm.assert_numpy_array_equal(codes, exp_codes)
+        tm.assert_index_equal(uniques, mi[:0])
+
     def test_dtype_preservation(self, any_numpy_dtype):
         # GH 15442
         if any_numpy_dtype in (tm.BYTES_DTYPES + tm.STRING_DTYPES):