StanFromIreland · Aug 24, 2026 · Jul 8, 2026 · Jul 8, 2026 · Jul 21, 2026 · Jul 21, 2026
diff --git 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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -5112,6 +5115,7 @@ 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
@@ -5336,6 +5340,8 @@ 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
@@ -0,0 +1,308 @@
+.. _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 input. Unless stated otherwise, *n* denotes the number of
+elements currently in the container, and *k* is either the value of a parameter
+or the number of elements in the parameter.
+
+For a pragmatic approach to assessing time complexity, see Ned Batchelder's
+`Big-O: How Code Slows as Data Grows <https://nedbatchelder.com/text/bigo>`__
+talk and blog post.
+
+.. |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 (``s.copy()``)
+     - *O*\ (*n*)
+   * - Append (``s.append(x)``) [1]_
+     - *O*\ (1)
+   * - Pop (``s.pop(k)``) [1]_ [2]_
+     - *O*\ (*n* - *k*)
+   * - Insert (``s.insert(k, x)``) [1]_ [2]_
+     - *O*\ (*n* - *k*)
+   * - Get item (``s[k]``)
+     - *O*\ (1)
+   * - Set item (``s[k] = x``)
+     - *O*\ (1)
+   * - Delete item (``del s[k]``) [2]_
+     - *O*\ (*n* - *k*)
+   * - Iteration
+     - *O*\ (*n*)
+   * - Get slice (``s[i:j]``)
+     - *O*\ (*k*)
+   * - Set slice (``s[i:j] = t``)
+     - *O*\ (*k* + *n*)
+   * - Delete slice (``del s[i:j]``)
+     - *O*\ (*n*)
+   * - Extend (``s.extend(t)``) [1]_
+     - *O*\ (*k*)
+   * - Sort (``s.sort()``) [3]_
+     - *O*\ (*n* log *n*)
+   * - Multiply (``s * k``)
+     - *O*\ (*nk*)
+   * - ``x in s``
+     - *O*\ (*n*)
+   * - ``min(s)``, ``max(s)``
+     - *O*\ (*n*)
+   * - Get length (``len(s)``) [4]_
+     - *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 is constant
+time as the same object is returned.
+
+.. list-table::
+   :header-rows: 1
+
+   * - Operation
+     - Complexity
+   * - Copy (``tuple(s)``)
+     - *O*\ (1)
+   * - Get item (``s[k]``)
+     - *O*\ (1)
+   * - Get slice (``s[i:j]``)
+     - *O*\ (*k*)
+   * - Concatenate (``s + t``)
+     - *O*\ (*n* + *k*)
+   * - Multiply (``s * k``)
+     - *O*\ (*nk*)
+   * - Iteration
+     - *O*\ (*n*)
+   * - ``x in s``
+     - *O*\ (*n*)
+   * - ``min(s)``, ``max(s)``
+     - *O*\ (*n*)
+   * - Get length (``len(s)``) [4]_
+     - *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 used in parameters are selected uniformly
+at random from the set of all 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. For more detail on the implementation, see
+:ref:`how-are-dictionaries-implemented`.
+
+A :class:`frozendict` is immutable, so it does not support setting or deleting
+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()``) [5]_ [6]_
+     - *O*\ (*n*)
+   * - Get item (``d[key]``)
+     - *O*\ (1)
+   * - Set item (``d[key] = value``) [1]_
+     - *O*\ (1)
+   * - Delete item (``del d[key]``)
+     - *O*\ (1)
+   * - Iteration [6]_
+     - *O*\ (*n*)
+   * - Get length (``len(d)``) [4]_
+     - *O*\ (1)
+
+
+:class:`!set`, :class:`!frozenset`
+==================================
+
+See :class:`dict` as the :class:`set` and :class:`frozenset` implementations are
+intentionally very similar, and the same hash collision caveat applies.
+In the worst case, *O*\ (1) operations instead take *O*\ (*n*) time,
+and operations that look up every element of an operand 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)
+   * - Add (``s.add(x)``) [1]_
+     - *O*\ (1)
+   * - Discard (``s.discard(x)``)
+     - *O*\ (1)
+   * - Union (``s | t``) [6]_
+     - *O*\ (len(*s*) + len(*t*))
+   * - Intersection (``s & t``, ``s.intersection(t)``) [6]_ [7]_
+     - *O*\ (min(len(*s*), len(*t*)))
+   * - Difference (``s - t``, ``s.difference(t)``) [6]_ [8]_
+     - *O*\ (len(*s*))
+   * - Difference update (``s.difference_update(t)``) [1]_ [6]_ [7]_
+     - *O*\ (min(len(*s*), len(*t*)))
+   * - Symmetric difference (``s ^ t``) [6]_
+     - *O*\ (len(*s*) + len(*t*))
+   * - Symmetric difference update (``s.symmetric_difference_update(t)``) [1]_ [6]_
+     - *O*\ (len(*t*))
+   * - Get length (``len(s)``) [4]_
+     - *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 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*\ (*k*)
+   * - Concatenate (``s + t``)
+     - *O*\ (*n* + *k*)
+   * - Multiply (``s * k``)
+     - *O*\ (*nk*)
+   * - Substring search (``x in s``, ``s.find(x)``, ``s.index(x)``) [9]_
+     - *O*\ (*n*)
+   * - Encode or decode
+     - *O*\ (*n*)
+   * - Iteration
+     - *O*\ (*n*)
+   * - Get length (``len(s)``) [4]_
+     - *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)
+   * - Convert to bytes (``v.tobytes()``, ``bytes(v)``)
+     - *O*\ (*n*)
+   * - Get length (``len(v)``) [4]_
+     - *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 (``s[k]``)
+     - *O*\ (1)
+   * - Get slice (``s[i:j]``)
+     - *O*\ (1)
+   * - ``x in s`` [10]_
+     - *O*\ (1)
+   * - Index and count (``s.index(x)``, ``s.count(x)``) [10]_
+     - *O*\ (1)
+   * - Iteration
+     - *O*\ (*n*)
+   * - ``min(s)``, ``max(s)``
+     - *O*\ (*n*)
+   * - Get length (``len(s)``) [4]_
+     - *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] 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.
+
+.. [4] The number of elements is stored in the object, so ``len()`` does
+   not need to count them.
+
+.. [5] Copying a :class:`frozendict` is *O*\ (1) as it returns the original object.
+
+.. [6] 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.
+
+.. [7] *O*\ (len(*t*)) if *t* is not a set.
+
+.. [8] *O*\ (len(*s*) + len(*t*)) if *t* is not a set.
+
+.. [9] A naive substring search would need *O*\ (*nk*) comparisons in the
+   worst case, where *k* is the length of the substring searched for, but
+   CPython uses search algorithms with a linear worst case for forward
+   searches. See :source:`Objects/stringlib/stringlib_find_two_way_notes.txt`
+   for details.
+
+.. [10] Assuming :class:`int` or :class:`bool` arguments. For other types,
+   the range is searched like any other sequence in *O*\ (*n*) time.