Add a page detailing the time complexity of operations on built-in types (#154363)
StanFromIreland <[email protected]>
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https://github.com/python/cpython/commit/c3f7c33dd7c6496d474979d5bd7f5dc22103dd22 commit: c3f7c33dd7c6496d474979d5bd7f5dc22103dd22 branch: main author: Stan Ulbrych <[email protected]> committer: StanFromIreland <[email protected]> date: 2026-08-24T19:57:26+01:00 summary: Add a page detailing the time complexity of operations on built-in types (#154363) Co-authored-by: Bénédikt Tran <[email protected]> Co-authored-by: Pieter Eendebak <[email protected]> Co-authored-by: Ned Batchelder <[email protected]> Co-authored-by: dgpb <[email protected]> files: A Doc/library/time-complexity.rst M Doc/faq/design.rst M Doc/faq/programming.rst M Doc/glossary.rst M Doc/library/index.rst M Doc/library/stdtypes.rst M Doc/tutorial/datastructures.rst diff --git a/Doc/faq/design.rst b/Doc/faq/design.rst index c914089e9806ec6..02a3a3a22070dae 100644 --- a/Doc/faq/design.rst +++ b/Doc/faq/design.rst @@ -430,6 +430,8 @@ tuples, but not lists, can be used as keys. Note, however, that a tuple is only hashable if all of its elements are hashable. +.. _how-are-lists-implemented: + How are lists implemented in CPython? ------------------------------------- @@ -445,6 +447,10 @@ cleverness is applied to improve the performance of appending items repeatedly; when the array must be grown, some extra space is allocated so the next few times don't require an actual resize. +See :ref:`time-complexity` for the costs of the various list operations. + + +.. _how-are-dictionaries-implemented: How are dictionaries implemented in CPython? -------------------------------------------- @@ -462,6 +468,8 @@ internal array where the value will be stored. Assuming that you're storing keys that all have different hash values, this means that dictionaries take constant time -- *O*\ (1), in Big-O notation -- to retrieve a key. +See :ref:`time-complexity` for the costs of the various dictionary operations. + Why must dictionary keys be immutable? -------------------------------------- diff --git a/Doc/faq/programming.rst b/Doc/faq/programming.rst index c2f8f72ee1f2c4b..4e1157e6ebe7296 100644 --- a/Doc/faq/programming.rst +++ b/Doc/faq/programming.rst @@ -1136,7 +1136,7 @@ What is the most efficient way to concatenate many strings together? :class:`str` and :class:`bytes` objects are immutable, therefore concatenating many strings together is inefficient as each concatenation creates a new object. In the general case, the total runtime cost is quadratic in the -total string length. +total string length. See :ref:`time-complexity` for more information. To accumulate many :class:`str` objects, the recommended idiom is to place them into a list and call :meth:`str.join` at the end:: diff --git a/Doc/glossary.rst b/Doc/glossary.rst index 4017f1e37f748e4..cd9d38b2fe4af29 100644 --- a/Doc/glossary.rst +++ b/Doc/glossary.rst @@ -942,7 +942,7 @@ Glossary list A built-in Python :term:`sequence`. Despite its name it is more akin to an array in other languages than to a linked list since access to - elements is *O*\ (1). + elements is *O*\ (1). See :ref:`time-complexity`. list comprehension A compact way to process all or part of the elements in a sequence and diff --git a/Doc/library/index.rst b/Doc/library/index.rst index 8fc77be520d4268..f28c03e2fae092f 100644 --- a/Doc/library/index.rst +++ b/Doc/library/index.rst @@ -44,6 +44,7 @@ the `Python Package Index <https://pypi.org>`_. stdtypes.rst exceptions.rst threadsafety.rst + time-complexity.rst text.rst binary.rst diff --git a/Doc/library/stdtypes.rst b/Doc/library/stdtypes.rst index 02b47ed5ca1dde2..31140555532f8f2 100644 --- a/Doc/library/stdtypes.rst +++ b/Doc/library/stdtypes.rst @@ -999,6 +999,9 @@ The ``in`` and ``not in`` operations have the same priorities as the comparison operations. The ``+`` (concatenation) and ``*`` (repetition) operations have the same priority as the corresponding numeric operations. [3]_ +See :ref:`time-complexity` for the costs of the various sequence +operations. + .. index:: triple: operations on; sequence; types pair: built-in function; len @@ -1121,6 +1124,8 @@ Notes: "end" values (which end depends on the sign of *k*). Note, *k* cannot be zero. If *k* is ``None``, it is treated like ``1``. +.. _typesseq-repeated-concatenation: + (6) Concatenating immutable sequences always results in a new object. This means that building up a sequence by repeated concatenation will have a @@ -5146,6 +5151,7 @@ computing mathematical operations such as intersection, union, difference, and symmetric difference. (For other containers see the built-in :class:`dict`, :class:`list`, and :class:`tuple` classes, and the :mod:`collections` module.) +See :ref:`time-complexity` for the costs of the various set operations. Like other collections, sets support ``x in set``, ``len(set)``, and ``for x in set``. Being an unordered collection, sets do not record element position or @@ -5370,6 +5376,8 @@ There are currently two standard mapping types, the :dfn:`dictionary` and (For other containers see the built-in :class:`list`, :class:`set`, and :class:`tuple` classes, and the :mod:`collections` module.) +See :ref:`time-complexity` for the costs of the various dictionary +operations. A dictionary's keys are *almost* arbitrary values. Values that are not :term:`hashable`, that is, values containing lists, dictionaries or other diff --git a/Doc/library/time-complexity.rst b/Doc/library/time-complexity.rst new file mode 100644 index 000000000000000..5ce02ac2761b621 --- /dev/null +++ b/Doc/library/time-complexity.rst @@ -0,0 +1,337 @@ +.. _time-complexity: + +=============================================== +Time complexity of operations on built-in types +=============================================== + +This page documents the time complexity of various operations on built-in types +in CPython. Other Python implementations may have different performance +characteristics. Additionally, the listed costs assume exact built-in types, as +instances of subclasses may have different costs. + +We use |big O notation|_ to describe how the running time of an operation grows +with the size of its inputs. Unless stated otherwise, *n* denotes the number of +elements currently in the container, and *k* is the value of a numeric +parameter, such as an index or a repeat count. + +.. |big O notation| replace:: Big *O* notation +.. _big O notation: https://en.wikipedia.org/wiki/Big_O_notation + + +:class:`!list` +============== + +Lists are mutable sequences; for more detail on the implementation see +:ref:`how-are-lists-implemented`. The largest costs come from growing beyond the +current allocation size (because everything must move), or from inserting or +deleting somewhere near the beginning (because everything after that must move). +If you need to add or remove at both ends, consider using a +:class:`collections.deque` instead. + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - Copy (``l.copy()``) + - *O*\ (*n*) + * - Append (``l.append(x)``) [1]_ + - *O*\ (1) + * - Pop (``l.pop(k)``) [1]_ [2]_ + - *O*\ (*n* - *k*) + * - Insert (``l.insert(k, x)``) [1]_ [2]_ + - *O*\ (*n* - *k*) + * - Get item (``l[k]``) + - *O*\ (1) + * - Set item (``l[k] = x``) + - *O*\ (1) + * - Delete item (``del l[k]``) [2]_ + - *O*\ (*n* - *k*) + * - Iteration + - *O*\ (*n*) + * - Get slice (``l[i:j]``) + - *O*\ (*j* - *i*) + * - Set slice (``l[i:j] = t``) [1]_ + - *O*\ (*j* - *i*) if len(*t*) == *j* - *i*, + otherwise *O*\ (*n* - *i* + len(*t*)) + * - Delete slice (``del l[i:j]``) + - *O*\ (*n* - *i*) + * - Extend (``l.extend(t)``) [1]_ [3]_ + - *O*\ (len(*t*)) + * - Sort (``l.sort()``) [4]_ + - *O*\ (*n* log *n*) + * - Concatenate (``l1 + l2``) + - *O*\ (len(*l1*) + len(*l2*)) + * - Multiply (``l * k``) + - *O*\ (*nk*) + * - ``x in l`` + - *O*\ (*n*) + * - ``min(l)``, ``max(l)`` + - *O*\ (*n*) + * - Get length (``len(l)``) [5]_ + - *O*\ (1) + + +:class:`!tuple` +=============== + +A :class:`tuple` is an :term:`immutable` sequence. Because a tuple can never +change, there are no insertion or deletion costs, and making a copy simply +returns the same object, so is constant time (*O*\ (1)). + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - Copy (``tuple(t)``) + - *O*\ (1) + * - Get item (``t[k]``) + - *O*\ (1) + * - Get slice (``t[i:j]``) + - *O*\ (*j* - *i*) + * - Concatenate (``t1 + t2``) + - *O*\ (len(*t1*) + len(*t2*)) + * - Multiply (``t * k``) + - *O*\ (*nk*) + * - Iteration + - *O*\ (*n*) + * - ``x in t`` + - *O*\ (*n*) + * - ``min(t)``, ``max(t)`` + - *O*\ (*n*) + * - Get length (``len(t)``) [5]_ + - *O*\ (1) + + +:class:`!dict`, :class:`!frozendict` +==================================== + +The times listed for dict objects are average-case times, as they assume the +hash function for the objects is sufficiently robust to make collisions +uncommon. They also assume the keys are well-distributed among the set of +possible keys. In the worst case, when every key hashes to the same value, +each of the *O*\ (1) operations below instead takes *O*\ (*n*) time. They also +assume that hashing and comparing a key is *O*\ (1). For more detail on the +implementation, see :ref:`how-are-dictionaries-implemented`. + +A :class:`frozendict` is immutable, so it does not support setting, deleting, +or updating items. The other operations below apply to it at the same costs. + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - ``key in d`` + - *O*\ (1) + * - Copy (``d.copy()``) [6]_ [7]_ + - *O*\ (*n*) + * - Get item (``d[key]``, ``d.get(key)``) + - *O*\ (1) + * - Set item (``d[key] = value``) [1]_ + - *O*\ (1) + * - Delete item (``del d[key]``, ``d.pop(key)``) + - *O*\ (1) + * - Update (``d.update(t)``, ``d |= t``) [1]_ [3]_ [7]_ + - *O*\ (len(*t*)) + * - Iteration [7]_ + - *O*\ (*n*) + * - Get length (``len(d)``) [5]_ + - *O*\ (1) + + +:class:`!set`, :class:`!frozenset` +================================== + +See :class:`dict` as the :class:`set` and :class:`frozenset` implementations are +similar, and the same caveats apply. +In the worst case, *O*\ (1) operations instead take *O*\ (*n*) time, +and operations that look up every element degrade accordingly. + +A :class:`frozenset` is :term:`immutable`, so it does not support adding, +discarding, or the in-place update operations. The others below apply to it at +the same costs. + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - ``x in s`` + - *O*\ (1) + * - Copy (``s.copy()``) [6]_ [7]_ + - *O*\ (*n*) + * - Add (``s.add(x)``) [1]_ + - *O*\ (1) + * - Discard (``s.discard(x)``, ``s.remove(x)``) + - *O*\ (1) + * - Union (``s1 | s2``, ``s1.union(s2)``) [7]_ + - *O*\ (len(*s1*) + len(*s2*)) + * - Update (``s1 |= s2``, ``s1.update(s2)``) [1]_ [7]_ + - *O*\ (len(*s2*)) + * - Intersection (``s1 & s2``, ``s1.intersection(s2)``) [7]_ [8]_ + - *O*\ (min(len(*s1*), len(*s2*))) + * - Intersection update (``s1 &= s2``, ``s1.intersection_update(s2)``) [1]_ [7]_ [8]_ + - *O*\ (min(len(*s1*), len(*s2*))) + * - Difference (``s1 - s2``, ``s1.difference(s2)``) [7]_ [9]_ + - *O*\ (len(*s1*)) + * - Difference update (``s1 -= s2``, ``s1.difference_update(s2)``) [1]_ [7]_ [8]_ + - *O*\ (min(len(*s1*), len(*s2*))) + * - Symmetric difference (``s1 ^ s2``, ``s1.symmetric_difference(s2)``) [7]_ + - *O*\ (len(*s1*) + len(*s2*)) + * - Symmetric difference update (``s1 ^= s2``, ``s1.symmetric_difference_update(s2)``) [1]_ [7]_ + - *O*\ (len(*s2*)) + * - Get length (``len(s)``) [5]_ + - *O*\ (1) + + +:class:`!str`, :class:`!bytes`, :class:`!bytearray` +=================================================== + +:class:`str` and :class:`bytes` objects are immutable sequences of characters and +bytes, respectively. As with tuples, copying one returns the original object. +A :class:`bytearray` is mutable, and additionally supports the mutating operations +of :class:`list` (except :meth:`!sort`), at the same costs. However, deleting at +the front with ``del`` (``del b[0]``, ``del b[:k]``) only advances the start of +the buffer instead of moving the remaining bytes, and is amortized *O*\ (1). + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - Get item (``s[k]``) + - *O*\ (1) + * - Get slice (``s[i:j]``) + - *O*\ (*j* - *i*) + * - Concatenate (``s + t``) [10]_ + - *O*\ (len(*s*) + len(*t*)) + * - Multiply (``s * k``) + - *O*\ (*nk*) + * - Substring search (``x in s``, ``s.find(x)``, ``s.index(x)``) [11]_ + - *O*\ (*n*) + * - Reverse substring search (``s.rfind(x)``, ``s.rindex(x)``) [11]_ [12]_ + - *O*\ (*n* × len(*x*)) + * - Encode or decode [13]_ + - *O*\ (*n*) + * - Iteration + - *O*\ (*n*) + * - Get length (``len(s)``) [5]_ + - *O*\ (1) + + +:class:`!memoryview` +==================== + +:class:`memoryview` objects allow Python code to access the internal data +of an object that supports the :ref:`buffer protocol <bufferobjects>` without +copying. In particular, slicing a memory view returns a new view onto the same +buffer. + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - Create (``memoryview(obj)``) + - *O*\ (1) + * - Get item (``v[k]``) + - *O*\ (1) + * - Get slice (``v[i:j]``) + - *O*\ (1) + * - Index (``v.index(x)``) [11]_ [14]_ + - *O*\ (*n*) + * - Count (``v.count(x)``) [14]_ + - *O*\ (*n*) + * - Convert to bytes (``v.tobytes()``, ``bytes(v)``) + - *O*\ (*n*) + * - Get length (``len(v)``) [5]_ + - *O*\ (1) + + +:class:`!range` +=============== + +A :class:`range` object computes its items on demand from its *start*, *stop* and +*step* values, so most operations do not depend on the length of the range. + +.. list-table:: + :header-rows: 1 + + * - Operation + - Complexity + * - Get item (``r[k]``) + - *O*\ (1) + * - Get slice (``r[i:j]``) + - *O*\ (1) + * - ``x in r`` [15]_ + - *O*\ (1) + * - Index and count (``r.index(x)``, ``r.count(x)``) [15]_ + - *O*\ (1) + * - Iteration + - *O*\ (*n*) + * - ``min(r)``, ``max(r)`` + - *O*\ (*n*) + * - Get length (``len(r)``) [5]_ + - *O*\ (1) + + +Notes +===== + +.. [1] Amortized. An individual operation may occasionally be *O*\ (*n*) + when the underlying storage is resized, but this cost is spread over + many operations, depending on the history of the container. + +.. [2] Popping or deleting the element at index *k* of a list of size *n* + shifts all elements after *k* one slot to the left, moving *n* - *k* - 1 + elements; inserting at index *k* shifts the elements from *k* onwards one + slot to the right, moving *n* - *k* elements. The worst case is index 0, + where the whole rest of the list has to be moved; the average case, an + index in the middle of the list, takes *O*\ (*n*/2) = *O*\ (*n*) + operations; and operating at the end of the list moves nothing and is + *O*\ (1). + +.. [3] Plus the cost of iterating over *t*, which may be expensive for an + arbitrary iterable. + +.. [4] This is the worst case scenario. Sorting is adaptive and input that is + already sorted or reverse-sorted takes only *O*\ (*n*) comparisons. + See :source:`Objects/listsort.txt` for more information. + +.. [5] The number of elements is stored in the object, so ``len()`` does + not need to count them. + +.. [6] Copying a :class:`frozendict` or a :class:`frozenset` is *O*\ (1) as it + returns the original object. + +.. [7] These operations scan the container's internal hash table, which is + not shrunk when elements are removed. After removing most elements, they + still take time proportional to the container's former size, until a + later insertion triggers a resize. + +.. [8] *O*\ (len(*t*)) if *t* is not a set. + +.. [9] *O*\ (len(*s*) + len(*t*)) if *t* is not a set. + +.. [10] Each concatenation builds a new object, so building a string by + concatenating many pieces in a loop is quadratic in the total length. + See the :ref:`note on concatenating immutable sequences + <typesseq-repeated-concatenation>` for alternatives. + +.. [11] With *start* and *end* arguments, *n* is the length of the region + searched rather than of *s*, and unlike slicing nothing is copied. + +.. [12] This is the worst case. Reverse searches are *O*\ (*n*) on typical + input. Forward searches instead use a more elaborate algorithm with a + linear worst case, described in + :source:`Objects/stringlib/stringlib_find_two_way_notes.txt`. + +.. [13] This assumes a codec that does a constant amount of work per character. + +.. [14] These unpack and compare each element individually, so they are much + slower than the equivalent :class:`bytes` methods. + +.. [15] Assuming :class:`int` or :class:`bool` arguments. For other types, + the range is searched like any other sequence in *O*\ (*n*) time. diff --git a/Doc/tutorial/datastructures.rst b/Doc/tutorial/datastructures.rst index 276e31a3056f0ee..77dd8a6e66d5cc4 100644 --- a/Doc/tutorial/datastructures.rst +++ b/Doc/tutorial/datastructures.rst @@ -167,7 +167,8 @@ It is also possible to use a list as a queue, where the first element added is the first element retrieved ("first-in, first-out"); however, lists are not efficient for this purpose. While appends and pops from the end of list are fast, doing inserts or pops from the beginning of a list is slow (because all -of the other elements have to be shifted by one). +of the other elements have to be shifted by one). See +:ref:`time-complexity` for more information. To implement a queue, use :class:`collections.deque` which was designed to have fast appends and pops from both ends. 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