User Guide

Bitsets are immutable ordered sets drawn from a predefined finite sequence of hashable items.

They are implemented as pure Python integers representing the rank of the set in colexicographical order (a.k.a bit strings, powers of two, binary strings). Hence, they can be very space-efficient, e.g. if a large number of subsets from a collection needs to be present in memory. Furthermore, they can also be compared, intersected, etc. using normal bitwise operations of integers (&, |, ^, ~).


bitsets is a pure-python package that runs both under Python 2.7 and 3.4+. To install it with using pip, run the following command:

$ pip install bitsets

There are no other dependencies. Optionally, you can install Graphviz and this Python wrapper to be able to draw Hasse diagrams of all possible sets from a bitsets’ object pool (see Advanced Usage).


Use the bitset() function to create a class representing ordered subsets from a fixed set of items (the domain):

>>> from bitsets import bitset

>>> PYTHONS = ('Chapman', 'Cleese', 'Gilliam', 'Idle', 'Jones', 'Palin')

>>> Pythons = bitset('Pythons', PYTHONS)

The domain needs to be a hashable duplicate-free sequence of at least one item (usually, a tuple).

The resulting class is an integer (int or long depending on Python version) subclass, so its instances (being integers) are immutable and hashable and thus in many ways similar to Python’s built-in frozenset.

>>> from bitsets._compat import integer_types

>>> issubclass(Pythons, integer_types)

Each domain item is mapped to a power of two (their rank in colexicographical order):

>>> Pythons.fromint(0)

>>> [Pythons.fromint(1), Pythons.fromint(2), Pythons.fromint(4)]
[Pythons(['Chapman']), Pythons(['Cleese']), Pythons(['Gilliam'])]

>>> Pythons.fromint(3)
Pythons(['Chapman', 'Cleese'])

>>> Pythons.fromint(2 ** 6 - 1)
Pythons(['Chapman', 'Cleese', 'Gilliam', 'Idle', 'Jones', 'Palin'])

>>> Pythons.fromint((1 << 0) + (1 << 5))
Pythons(['Chapman', 'Palin'])

The class provides access to the minimal (infimum) and maximal (supremum) sets from its domain:

>>> Pythons.infimum

>>> Pythons.supremum
Pythons(['Chapman', 'Cleese', 'Gilliam', 'Idle', 'Jones', 'Palin'])

Basic usage

Bitsets can be created from members, bit strings, boolean sequences, and integers:

>>> Pythons(['Palin', 'Cleese'])
Pythons(['Cleese', 'Palin'])

>>> Pythons.frombits('101000')
Pythons(['Chapman', 'Gilliam'])

>>> Pythons.frombools([True, False, True, False, False, False])
Pythons(['Chapman', 'Gilliam'])

>>> Pythons.fromint(5)
Pythons(['Chapman', 'Gilliam'])

Members always occur in the definition order.

Bitsets cannot contain items other than those from their domain:

>>> Pythons(['Brian'])
Traceback (most recent call last):
KeyError: 'Brian'

>>> 'Spam' in Pythons(['Jones'])
Traceback (most recent call last):
KeyError: 'Spam'

Bitsets can be converted to members, bit strings, boolean sequences and integers:

>>> Pythons(['Chapman', 'Gilliam']).members()
('Chapman', 'Gilliam')

>>> Pythons(['Chapman', 'Gilliam']).bits()

>>> Pythons(['Chapman', 'Gilliam']).bools()
(True, False, True, False, False, False)

>>> int(Pythons(['Chapman', 'Gilliam']))


To facilitate sorting collections of bitsets, they have key methods for different sort orders (shortlex(), shortcolex(), longlex(), and longcolex()):

>>> Pythons(['Idle']).shortlex() < Pythons(['Palin']).shortlex()

These orderings are derived from the number of set members and the definition order of the items.

>>> Digits = bitset('Digits', '12345')
>>> onetwo = [d for d in Digits('12345').powerset() if d.count() in (1, 2)]

>>> shortlex = sorted(onetwo, key=lambda d: d.shortlex())
>>> [''.join(d) for d in shortlex]  
['1',  '2',  '3',  '4',  '5',
      '12', '13', '14', '15',
            '23', '24', '25',
                  '34', '35',

>>> shortcolex = sorted(onetwo, key=lambda d: d.shortcolex())
>>> [''.join(d) for d in shortcolex]  
['1',  '2',  '3',  '4',  '5',
 '13', '23',
 '14', '24', '34',
 '15', '25', '35', '45']

Sorting a collection of bitsets without the use of a key function will order them in colexicographical order.

>>> [''.join(d) for d in sorted(onetwo)]  
 '2', '12',
 '3', '13', '23',
 '4', '14', '24', '34',
 '5', '15', '25', '35', '45']


Iterate over a bitsets’ powerset() in short lexicographic order:

>>> for p in Pythons(['Palin', 'Idle']).powerset():
...     print(p.members())
('Idle', 'Palin')

This is the same order as generated by itertools recipes powerset(iterable).

frozenset compatibility

For convenience, bitsets provide the same methods as frozenset (i.e. issubset(), issuperset(), isdisjoint(), intersection(), union(), difference(), symmetric_difference(), __len__(), __iter__(), __bool__(), __contains__(), and as a non-op copy()).

>>> 'Cleese' in Pythons(['Idle'])

>>> 'Idle' in Pythons(['Idle'])

>>> Pythons(['Chapman', 'Idle']).intersection(Pythons(['Idle', 'Palin']))

Note, however that all the operators methods (+, -, &, | etc.) retain their integer semantics:

>>> print(Pythons(['Chapman', 'Idle']) - Pythons(['Idle']))

In tight loops it might be worth to use bitwise expressions (&, |, ^, ~) for set comparisons/operations instead of the frozenset-compatible methods:

>>> # is subset ?
>>> Pythons(['Idle']) & Pythons(['Chapman', 'Idle']) == Pythons(['Idle'])

Added functionality

Differing from frozenset, you can also retrieve the complement() set of a bitset:

>>> Pythons(['Idle']).complement()
Pythons(['Chapman', 'Cleese', 'Gilliam', 'Jones', 'Palin'])

>>> Pythons().complement().complement()

Test if a bitset is maximal (the supremum):

>>> Pythons(['Idle']).all()

>>> Pythons(['Chapman', 'Cleese', 'Gilliam', 'Idle', 'Jones', 'Palin']).all()

Test if a bitset is non-minimal (the infimum), same as bool(bitset):

>>> Pythons(['Idle']).any()

>>> Pythons().any()