# `itertools` --- 効率的なループ用のイテレータ生成関数群

このモジュールは イテレータ を構築する部品を実装しています。プログラム言語 APL, Haskell, SML からアイデアを得ていますが、 Python に適した形に修正されています。

このモジュールは、高速でメモリ効率に優れ、単独でも組合せても使用することのできるツールを標準化したものです。同時に、このツール群は "イテレータの代数" を構成していて、pure Python で簡潔かつ効率的なツールを作れるようにしています。

例えば、SML の作表ツール `tabulate(f)` は `f(0), f(1), ...` のシーケンスを作成します。同じことを Python では `map()` と `count()` を組合せて `map(f, count())` という形で実現できます。

これらのツールと組み込み関数は `operator` モジュール内の高速な関数とともに使うことで見事に動作します。例えば、乗算演算子を2つのベクトルにわたってマップすることで効率的な内積計算を実現できます: `sum(starmap(operator.mul, zip(vec1, vec2, strict=True)))`。

無限イテレータ:

イテレータ | 引数 | 結果 | 使用例
--- | --- | --- | ---
`count()` | [start[, step]] | start, start+step, start+2*step, ... | `count(10) → 10 11 12 13 14 ...`
`cycle()` | p | p0, p1, ... plast, p0, p1, ... | `cycle('ABCD') → A B C D A B C D ...`
`repeat()` | elem [,n] | elem, elem, elem, ... 無限もしくは n 回 | `repeat(10, 3) → 10 10 10`

一番短い入力シーケンスで止まるイテレータ:

イテレータ | 引数 | 結果 | 使用例
--- | --- | --- | ---
`accumulate()` | p [,func] | p0, p0+p1, p0+p1+p2, ... | `accumulate([1,2,3,4,5]) → 1 3 6 10 15`
`batched()` | p, n | (p0, p1, ..., p_n-1), ... | `batched('ABCDEFG', n=3) → ABC DEF G`
`chain()` | p, q, ... | p0, p1, ... plast, q0, q1, ... | `chain('ABC', 'DEF') → A B C D E F`
`chain.from_iterable()` | iterable | p0, p1, ... plast, q0, q1, ... | `chain.from_iterable(['ABC', 'DEF']) → A B C D E F`
`compress()` | data, selectors | (d[0] if s[0]), (d[1] if s[1]), ... | `compress('ABCDEF', [1,0,1,0,1,1]) → A C E F`
`dropwhile()` | predicate, seq | seq[n], seq[n+1], starting when predicate fails | `dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8`
`filterfalse()` | predicate, seq | elements of seq where predicate(elem) fails | `filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8`
`groupby()` | iterable[, key] | key(v) の値でグループ化したサブイテレータ |
`islice()` | seq, [start,] stop [, step] | seq[start:stop:step] | `islice('ABCDEFG', 2, None) → C D E F G`
`pairwise()` | iterable | (p[0], p[1]), (p[1], p[2]) | `pairwise('ABCDEFG') → AB BC CD DE EF FG`
`starmap()` | func, seq | func(*seq[0]), func(*seq[1]), ... | `starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000`
`takewhile()` | predicate, seq | seq[0], seq[1], until predicate fails | `takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4`
`tee()` | it, n | it1, it2 , ... itn 一つのイテレータを n 個に分ける |
`zip_longest()` | p, q, ... | (p[0], q[0]), (p[1], q[1]), ... | `zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-`

組合せイテレータ:

イテレータ | 引数 | 結果
--- | --- | ---
`product()` | p, q, ... [repeat=1] | デカルト積、ネストしたforループと等価
`permutations()` | p[, r] | 長さrのタプル列、重複なしのあらゆる並び
`combinations()` | p, r | 長さrのタプル列、ソートされた順で重複なし
`combinations_with_replacement()` | p, r | 長さrのタプル列、ソートされた順で重複あり

使用例 | 結果
--- | ---
`product('ABCD', repeat=2)` | `AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD`
`permutations('ABCD', 2)` | `AB AC AD BA BC BD CA CB CD DA DB DC`
`combinations('ABCD', 2)` | `AB AC AD BC BD CD`
`combinations_with_replacement('ABCD', 2)` | `AA AB AC AD BB BC BD CC CD DD`

## Itertool関数

以下の関数は全て、イテレータを作成して返します。無限長のストリームのイテレータを返す関数もあり、この場合にはストリームを中断するような関数かループ処理から使用しなければなりません。

itertools.accumulate(iterable[, function, *, initial=None])

  Make an iterator that returns accumulated sums or accumulated results from other binary functions.

  The function defaults to addition. The function should accept two arguments, an accumulated total and a value from the iterable.

  If an initial value is provided, the accumulation will start with that value and the output will have one more element than the input iterable.

  およそ次と等価です:

      def accumulate(iterable, function=operator.add, *, initial=None):
          'Return running totals'
          # accumulate([1,2,3,4,5]) → 1 3 6 10 15
          # accumulate([1,2,3,4,5], initial=100) → 100 101 103 106 110 115
          # accumulate([1,2,3,4,5], operator.mul) → 1 2 6 24 120

          iterator = iter(iterable)
          total = initial
          if initial is None:
              try:
                  total = next(iterator)
              except StopIteration:
                  return

          yield total
          for element in iterator:
              total = function(total, element)
              yield total

  The function argument can be set to `min()` for a running minimum, `max()` for a running maximum, or `operator.mul()` for a running product. Amortization tables can be built by accumulating interest and applying payments:

      >>> data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
      >>> list(accumulate(data, max))              # running maximum
      [3, 4, 6, 6, 6, 9, 9, 9, 9, 9]
      >>> list(accumulate(data, operator.mul))     # running product
      [3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]

      # Amortize a 5% loan of 1000 with 10 annual payments of 90
      >>> update = lambda balance, payment: round(balance * 1.05) - payment
      >>> list(accumulate(repeat(90, 10), update, initial=1_000))
      [1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

  最終的な累積値だけを返す類似の関数については `functools.reduce()` を見てください。

  Added in version 3.2.

  バージョン 3.3 で変更: Added the optional function parameter.

  バージョン 3.8 で変更: オプションの initial パラメータが追加されました。

itertools.batched(iterable, n, *, strict=False)

  iterable から得られるデータを n 個ごとに一つのタプルにまとめます。 一番最後のバッチは n 個より少なくなる可能性があります。

  If strict is true, will raise a `ValueError` if the final batch is shorter than n.

  入力の iterable から要素を一つづつ取り出し、サイズが n になるまでタプルに溜め込みます。 入力の iterable は遅延評価され、次のタプルを作るのに必要なだけ要素が取り出されます。 タプルは、個数が n に到達するか、入力の iterable が尽きるとすぐに出力されます:

      >>> flattened_data = ['roses', 'red', 'violets', 'blue', 'sugar', 'sweet']
      >>> unflattened = list(batched(flattened_data, 2))
      >>> unflattened
      [('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

  およそ次と等価です:

      def batched(iterable, n, *, strict=False):
          # batched('ABCDEFG', 3) → ABC DEF G
          if n < 1:
              raise ValueError('n must be at least one')
          iterator = iter(iterable)
          while batch := tuple(islice(iterator, n)):
              if strict and len(batch) != n:
                  raise ValueError('batched(): incomplete batch')
              yield batch

  Added in version 3.12.

  バージョン 3.13 で変更: Added the strict option.

itertools.chain(*iterables)

  先頭の iterable の全要素を返し、次に2番目の iterable の全要素を返し、と全 iterable の要素を返すイテレータを作成します。連続したシーケンスを一つのシーケンスとして扱う場合に使用します。およそ次と等価です:

      def chain(*iterables):
          # chain('ABC', 'DEF') → A B C D E F
          for iterable in iterables:
              yield from iterable

classmethod chain.from_iterable(iterable)

  `chain()` のためのもう一つのコンストラクタです。遅延評価される iterable 引数一つから連鎖した入力を受け取ります。この関数は、以下のコードとほぼ等価です:

      def from_iterable(iterables):
          # chain.from_iterable(['ABC', 'DEF']) → A B C D E F
          for iterable in iterables:
              yield from iterable

itertools.combinations(iterable, r)

  入力 iterable の要素からなる長さ r の部分列を返します。

  The output is a subsequence of `product()` keeping only entries that are subsequences of the iterable. The length of the output is given by `math.comb()` which computes `n! / r! / (n - r)!` when `0 ≤ r ≤ n` or zero when `r > n`.

  The combination tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

  Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within each combination.

  およそ次と等価です:

      def combinations(iterable, r):
          # combinations('ABCD', 2) → AB AC AD BC BD CD
          # combinations(range(4), 3) → 012 013 023 123

          pool = tuple(iterable)
          n = len(pool)
          if r > n:
              return
          indices = list(range(r))

          yield tuple(pool[i] for i in indices)
          while True:
              for i in reversed(range(r)):
                  if indices[i] != i + n - r:
                      break
              else:
                  return
              indices[i] += 1
              for j in range(i+1, r):
                  indices[j] = indices[j-1] + 1
              yield tuple(pool[i] for i in indices)

itertools.combinations_with_replacement(iterable, r)

  入力 iterable から、それぞれの要素が複数回現れることを許して、長さ r の要素の部分列を返します。

  The output is a subsequence of `product()` that keeps only entries that are subsequences (with possible repeated elements) of the iterable. The number of subsequence returned is `(n + r - 1)! / r! / (n - 1)!` when `n > 0`.

  The combination tuples are emitted in lexicographic order according to the order of the input iterable. if the input iterable is sorted, the output tuples will be produced in sorted order.

  Elements are treated as unique based on their position, not on their value. If the input elements are unique, the generated combinations will also be unique.

  およそ次と等価です:

      def combinations_with_replacement(iterable, r):
          # combinations_with_replacement('ABC', 2) → AA AB AC BB BC CC

          pool = tuple(iterable)
          n = len(pool)
          if not n and r:
              return
          indices = [0] * r

          yield tuple(pool[i] for i in indices)
          while True:
              for i in reversed(range(r)):
                  if indices[i] != n - 1:
                      break
              else:
                  return
              indices[i:] = [indices[i] + 1] * (r - i)
              yield tuple(pool[i] for i in indices)

  Added in version 3.1.

itertools.compress(data, selectors)

  Make an iterator that returns elements from data where the corresponding element in selectors is true. Stops when either the data or selectors iterables have been exhausted. Roughly equivalent to:

      def compress(data, selectors):
          # compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
          return (datum for datum, selector in zip(data, selectors) if selector)

  Added in version 3.1.

itertools.count(start=0, step=1)

  Make an iterator that returns evenly spaced values beginning with start. Can be used with `map()` to generate consecutive data points or with `zip()` to add sequence numbers. Roughly equivalent to:

      def count(start=0, step=1):
          # count(10) → 10 11 12 13 14 ...
          # count(2.5, 0.5) → 2.5 3.0 3.5 ...
          n = start
          while True:
              yield n
              n += step

  When counting with floating-point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as: `(start + step * i for i in count())`.

  バージョン 3.1 で変更: step 引数が追加され、非整数の引数が許されるようになりました。

itertools.cycle(iterable)

  Make an iterator returning elements from the iterable and saving a copy of each. When the iterable is exhausted, return elements from the saved copy. Repeats indefinitely. Roughly equivalent to:

      def cycle(iterable):
          # cycle('ABCD') → A B C D A B C D A B C D ...
          saved = []
          for element in iterable:
              yield element
              saved.append(element)
          while saved:
              for element in saved:
                  yield element

  This itertool may require significant auxiliary storage (depending on the length of the iterable).

itertools.dropwhile(predicate, iterable)

  Make an iterator that drops elements from the iterable while the predicate is true and afterwards returns every element. Roughly equivalent to:

      def dropwhile(predicate, iterable):
          # dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8

          iterator = iter(iterable)
          for x in iterator:
              if not predicate(x):
                  yield x
                  break

          for x in iterator:
              yield x

  Note this does not produce any output until the predicate first becomes false, so this itertool may have a lengthy start-up time.

itertools.filterfalse(predicate, iterable)

  Make an iterator that filters elements from the iterable returning only those for which the predicate returns a false value. If predicate is `None`, returns the items that are false. Roughly equivalent to:

      def filterfalse(predicate, iterable):
          # filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8
          if predicate is None:
              predicate = bool
          for x in iterable:
              if not predicate(x):
                  yield x

itertools.groupby(iterable, key=None)

  同じキーをもつような要素からなる iterable 中のグループに対して、キーとグループを返すようなイテレータを作成します。key は各要素に対するキー値を計算する関数です。キーを指定しない場合や `None` にした場合、key 関数のデフォルトは恒等関数になり要素をそのまま返します。通常、iterable は同じキー関数でソート済みである必要があります。

  `groupby()` の操作は Unix の `uniq` フィルターと似ています。 key 関数の値が変わるたびに休止または新しいグループを生成します (このために通常同じ key 関数でソートしておく必要があるのです)。この動作は SQL の入力順に関係なく共通の要素を集約する GROUP BY とは違います。

  返されるグループはそれ自体がイテレータで、 `groupby()` と iterable を共有しています。もととなる iterable を共有しているため、 `groupby()` オブジェクトの要素取り出しを先に進めると、それ以前の要素であるグループは見えなくなってしまいます。従って、データが後で必要な場合にはリストの形で保存しておく必要があります:

      groups = []
      uniquekeys = []
      data = sorted(data, key=keyfunc)
      for k, g in groupby(data, keyfunc):
          groups.append(list(g))      # Store group iterator as a list
          uniquekeys.append(k)

  `groupby()` はおよそ次と等価です:

      def groupby(iterable, key=None):
          # [k for k, g in groupby('AAAABBBCCDAABBB')] → A B C D A B
          # [list(g) for k, g in groupby('AAAABBBCCD')] → AAAA BBB CC D

          keyfunc = (lambda x: x) if key is None else key
          iterator = iter(iterable)
          exhausted = False

          def _grouper(target_key):
              nonlocal curr_value, curr_key, exhausted
              yield curr_value
              for curr_value in iterator:
                  curr_key = keyfunc(curr_value)
                  if curr_key != target_key:
                      return
                  yield curr_value
              exhausted = True

          try:
              curr_value = next(iterator)
          except StopIteration:
              return
          curr_key = keyfunc(curr_value)

          while not exhausted:
              target_key = curr_key
              curr_group = _grouper(target_key)
              yield curr_key, curr_group
              if curr_key == target_key:
                  for _ in curr_group:
                      pass

itertools.islice(iterable, stop)
itertools.islice(iterable, start, stop[, step])

  Make an iterator that returns selected elements from the iterable. Works like sequence slicing but does not support negative values for start, stop, or step.

  If start is zero or `None`, iteration starts at zero. Otherwise, elements from the iterable are skipped until start is reached.

  If stop is `None`, iteration continues until the iterator is exhausted, if at all. Otherwise, it stops at the specified position.

  If step is `None`, the step defaults to one. Elements are returned consecutively unless step is set higher than one which results in items being skipped.

  およそ次と等価です:

      def islice(iterable, *args):
          # islice('ABCDEFG', 2) → A B
          # islice('ABCDEFG', 2, 4) → C D
          # islice('ABCDEFG', 2, None) → C D E F G
          # islice('ABCDEFG', 0, None, 2) → A C E G

          s = slice(*args)
          start = 0 if s.start is None else s.start
          stop = s.stop
          step = 1 if s.step is None else s.step
          if start < 0 or (stop is not None and stop < 0) or step <= 0:
              raise ValueError

          indices = count() if stop is None else range(max(start, stop))
          next_i = start
          for i, element in zip(indices, iterable):
              if i == next_i:
                  yield element
                  next_i += step

itertools.pairwise(iterable)

  Return successive overlapping pairs taken from the input iterable.

  The number of 2-tuples in the output iterator will be one fewer than the number of inputs. It will be empty if the input iterable has fewer than two values.

  およそ次と等価です:

      def pairwise(iterable):
          # pairwise('ABCDEFG') → AB BC CD DE EF FG
          iterator = iter(iterable)
          a = next(iterator, None)
          for b in iterator:
              yield a, b
              a = b

  Added in version 3.10.

itertools.permutations(iterable, r=None)

  Return successive r length permutations of elements from the iterable.

  r が指定されない場合や `None` の場合、r はデフォルトで iterable の長さとなり、可能な最長の順列の全てが生成されます。

  The output is a subsequence of `product()` where entries with repeated elements have been filtered out. The length of the output is given by `math.perm()` which computes `n! / (n - r)!` when `0 ≤ r ≤ n` or zero when `r > n`.

  The permutation tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

  Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within a permutation.

  およそ次と等価です:

      def permutations(iterable, r=None):
          # permutations('ABCD', 2) → AB AC AD BA BC BD CA CB CD DA DB DC
          # permutations(range(3)) → 012 021 102 120 201 210

          pool = tuple(iterable)
          n = len(pool)
          r = n if r is None else r
          if r > n:
              return

          indices = list(range(n))
          cycles = list(range(n, n-r, -1))
          yield tuple(pool[i] for i in indices[:r])

          while n:
              for i in reversed(range(r)):
                  cycles[i] -= 1
                  if cycles[i] == 0:
                      indices[i:] = indices[i+1:] + indices[i:i+1]
                      cycles[i] = n - i
                  else:
                      j = cycles[i]
                      indices[i], indices[-j] = indices[-j], indices[i]
                      yield tuple(pool[i] for i in indices[:r])
                      break
              else:
                  return

itertools.product(*iterables, repeat=1)

  入力イテラブルのデカルト積です。

  ジェネレータ式の入れ子になった for ループとおよそ等価です。たとえば `product(A, B)` は `((x,y) for x in A for y in B)` と同じものを返します。

  入れ子ループは走行距離計と同じように右端の要素がイテレーションごとに更新されていきます。このパターンは辞書式順序を作り出し、入力のイテレート可能オブジェクトたちがソートされていれば、直積タプルもソートされた順に出てきます。

  イテラブル自身との直積を計算するためには、オプションの repeat キーワード引数に繰り返し回数を指定します。たとえば `product(A, repeat=4)` は `product(A, A, A, A)` と同じ意味です。

  この関数は以下のコードとおよそ等価ですが、実際の実装ではメモリ中に中間結果を作りません:

      def product(*iterables, repeat=1):
          # product('ABCD', 'xy') → Ax Ay Bx By Cx Cy Dx Dy
          # product(range(2), repeat=3) → 000 001 010 011 100 101 110 111

          pools = [tuple(pool) for pool in iterables] * repeat

          result = [[]]
          for pool in pools:
              result = [x+[y] for x in result for y in pool]

          for prod in result:
              yield tuple(prod)

  `product()` は動作する前に、入力のイテラブルを完全に読み取り、直積を生成するためにメモリ内に値を蓄えます。したがって、入力が有限の場合に限り有用です。

itertools.repeat(object[, times])

  Make an iterator that returns object over and over again. Runs indefinitely unless the times argument is specified.

  およそ次と等価です:

      def repeat(object, times=None):
          # repeat(10, 3) → 10 10 10
          if times is None:
              while True:
                  yield object
          else:
              for i in range(times):
                  yield object

  repeat は map や zip に定数のストリームを与えるためによく利用されます:

      >>> list(map(pow, range(10), repeat(2)))
      [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

itertools.starmap(function, iterable)

  Make an iterator that computes the function using arguments obtained from the iterable. Used instead of `map()` when argument parameters have already been "pre-zipped" into tuples.

  The difference between `map()` and `starmap()` parallels the distinction between `function(a,b)` and `function(*c)`. Roughly equivalent to:

      def starmap(function, iterable):
          # starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000
          for args in iterable:
              yield function(*args)

itertools.takewhile(predicate, iterable)

  Make an iterator that returns elements from the iterable as long as the predicate is true. Roughly equivalent to:

      def takewhile(predicate, iterable):
          # takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4
          for x in iterable:
              if not predicate(x):
                  break
              yield x

  Note, the element that first fails the predicate condition is consumed from the input iterator and there is no way to access it. This could be an issue if an application wants to further consume the input iterator after takewhile has been run to exhaustion. To work around this problem, consider using more-iterools before_and_after() instead.

itertools.tee(iterable, n=2)

  一つの iterable から n 個の独立したイテレータを返します。

  およそ次と等価です:

      def tee(iterable, n=2):
          iterator = iter(iterable)
          shared_link = [None, None]
          return tuple(_tee(iterator, shared_link) for _ in range(n))

      def _tee(iterator, link):
          try:
              while True:
                  if link[1] is None:
                      link[0] = next(iterator)
                      link[1] = [None, None]
                  value, link = link
                  yield value
          except StopIteration:
              return

  一度 `tee()` が生成されたら、もとの iterable を他で使ってはいけません。さもなければ、 `tee()` オブジェクトの知らない間に iterable が先の要素に進んでしまうことになります。

  `tee` iterators are not threadsafe. A `RuntimeError` may be raised when simultaneously using iterators returned by the same `tee()` call, even if the original iterable is threadsafe.

  `tee()` はかなり大きなメモリ領域を使用するかもしれません (使用するメモリ量はiterableの大きさに依存します)。一般には、一つのイテレータが他のイテレータよりも先にほとんどまたは全ての要素を消費するような場合には、 `tee()` よりも `list()` を使った方が高速です。

itertools.zip_longest(*iterables, fillvalue=None)

  Make an iterator that aggregates elements from each of the iterables.

  If the iterables are of uneven length, missing values are filled-in with fillvalue. If not specified, fillvalue defaults to `None`.

  Iteration continues until the longest iterable is exhausted.

  およそ次と等価です:

      def zip_longest(*iterables, fillvalue=None):
          # zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

          iterators = list(map(iter, iterables))
          num_active = len(iterators)
          if not num_active:
              return

          while True:
              values = []
              for i, iterator in enumerate(iterators):
                  try:
                      value = next(iterator)
                  except StopIteration:
                      num_active -= 1
                      if not num_active:
                          return
                      iterators[i] = repeat(fillvalue)
                      value = fillvalue
                  values.append(value)
              yield tuple(values)

  If one of the iterables is potentially infinite, then the `zip_longest()` function should be wrapped with something that limits the number of calls (for example `islice()` or `takewhile()`).

## Itertools レシピ

この節では、既存の itertools を素材としてツールセットを拡張するためのレシピを示します。

The primary purpose of the itertools recipes is educational. The recipes show various ways of thinking about individual tools — for example, that `chain.from_iterable` is related to the concept of flattening. The recipes also give ideas about ways that the tools can be combined — for example, how `starmap()` and `repeat()` can work together. The recipes also show patterns for using itertools with the `operator` and `collections` modules as well as with the built-in itertools such as `map()`, `filter()`, `reversed()`, and `enumerate()`.

A secondary purpose of the recipes is to serve as an incubator. The `accumulate()`, `compress()`, and `pairwise()` itertools started out as recipes. Currently, the `sliding_window()`, `iter_index()`, and `sieve()` recipes are being tested to see whether they prove their worth.

Substantially all of these recipes and many, many others can be installed from the more-itertools project found on the Python Package Index:

    python -m pip install more-itertools

Many of the recipes offer the same high performance as the underlying toolset. Superior memory performance is kept by processing elements one at a time rather than bringing the whole iterable into memory all at once. Code volume is kept small by linking the tools together in a functional style. High speed is retained by preferring "vectorized" building blocks over the use of for-loops and generators which incur interpreter overhead.

    import collections
    import contextlib
    import functools
    import math
    import operator
    import random

    def take(n, iterable):
        "Return first n items of the iterable as a list."
        return list(islice(iterable, n))

    def prepend(value, iterable):
        "Prepend a single value in front of an iterable."
        # prepend(1, [2, 3, 4]) → 1 2 3 4
        return chain([value], iterable)

    def tabulate(function, start=0):
        "Return function(0), function(1), ..."
        return map(function, count(start))

    def repeatfunc(func, times=None, *args):
        "Repeat calls to func with specified arguments."
        if times is None:
            return starmap(func, repeat(args))
        return starmap(func, repeat(args, times))

    def flatten(list_of_lists):
        "Flatten one level of nesting."
        return chain.from_iterable(list_of_lists)

    def ncycles(iterable, n):
        "Returns the sequence elements n times."
        return chain.from_iterable(repeat(tuple(iterable), n))

    def tail(n, iterable):
        "Return an iterator over the last n items."
        # tail(3, 'ABCDEFG') → E F G
        return iter(collections.deque(iterable, maxlen=n))

    def consume(iterator, n=None):
        "Advance the iterator n-steps ahead. If n is None, consume entirely."
        # Use functions that consume iterators at C speed.
        if n is None:
            collections.deque(iterator, maxlen=0)
        else:
            next(islice(iterator, n, n), None)

    def nth(iterable, n, default=None):
        "Returns the nth item or a default value."
        return next(islice(iterable, n, None), default)

    def quantify(iterable, predicate=bool):
        "Given a predicate that returns True or False, count the True results."
        return sum(map(predicate, iterable))

    def first_true(iterable, default=False, predicate=None):
        "Returns the first true value or the *default* if there is no true value."
        # first_true([a,b,c], x) → a or b or c or x
        # first_true([a,b], x, f) → a if f(a) else b if f(b) else x
        return next(filter(predicate, iterable), default)

    def all_equal(iterable, key=None):
        "Returns True if all the elements are equal to each other."
        # all_equal('4٤௪౪໔', key=int) → True
        return len(take(2, groupby(iterable, key))) <= 1

    def unique_justseen(iterable, key=None):
        "Yield unique elements, preserving order. Remember only the element just seen."
        # unique_justseen('AAAABBBCCDAABBB') → A B C D A B
        # unique_justseen('ABBcCAD', str.casefold) → A B c A D
        if key is None:
            return map(operator.itemgetter(0), groupby(iterable))
        return map(next, map(operator.itemgetter(1), groupby(iterable, key)))

    def unique_everseen(iterable, key=None):
        "Yield unique elements, preserving order. Remember all elements ever seen."
        # unique_everseen('AAAABBBCCDAABBB') → A B C D
        # unique_everseen('ABBcCAD', str.casefold) → A B c D
        seen = set()
        if key is None:
            for element in filterfalse(seen.__contains__, iterable):
                seen.add(element)
                yield element
        else:
            for element in iterable:
                k = key(element)
                if k not in seen:
                    seen.add(k)
                    yield element

    def unique(iterable, key=None, reverse=False):
       "Yield unique elements in sorted order. Supports unhashable inputs."
       # unique([[1, 2], [3, 4], [1, 2]]) → [1, 2] [3, 4]
       return unique_justseen(sorted(iterable, key=key, reverse=reverse), key=key)

    def sliding_window(iterable, n):
        "Collect data into overlapping fixed-length chunks or blocks."
        # sliding_window('ABCDEFG', 4) → ABCD BCDE CDEF DEFG
        iterator = iter(iterable)
        window = collections.deque(islice(iterator, n - 1), maxlen=n)
        for x in iterator:
            window.append(x)
            yield tuple(window)

    def grouper(iterable, n, *, incomplete='fill', fillvalue=None):
        "Collect data into non-overlapping fixed-length chunks or blocks."
        # grouper('ABCDEFG', 3, fillvalue='x') → ABC DEF Gxx
        # grouper('ABCDEFG', 3, incomplete='strict') → ABC DEF ValueError
        # grouper('ABCDEFG', 3, incomplete='ignore') → ABC DEF
        iterators = [iter(iterable)] * n
        match incomplete:
            case 'fill':
                return zip_longest(*iterators, fillvalue=fillvalue)
            case 'strict':
                return zip(*iterators, strict=True)
            case 'ignore':
                return zip(*iterators)
            case _:
                raise ValueError('Expected fill, strict, or ignore')

    def roundrobin(*iterables):
        "Visit input iterables in a cycle until each is exhausted."
        # roundrobin('ABC', 'D', 'EF') → A D E B F C
        # Algorithm credited to George Sakkis
        iterators = map(iter, iterables)
        for num_active in range(len(iterables), 0, -1):
            iterators = cycle(islice(iterators, num_active))
            yield from map(next, iterators)

    def partition(predicate, iterable):
        """Partition entries into false entries and true entries.

        If *predicate* is slow, consider wrapping it with functools.lru_cache().
        """
        # partition(is_odd, range(10)) → 0 2 4 6 8   and  1 3 5 7 9
        t1, t2 = tee(iterable)
        return filterfalse(predicate, t1), filter(predicate, t2)

    def subslices(seq):
        "Return all contiguous non-empty subslices of a sequence."
        # subslices('ABCD') → A AB ABC ABCD B BC BCD C CD D
        slices = starmap(slice, combinations(range(len(seq) + 1), 2))
        return map(operator.getitem, repeat(seq), slices)

    def iter_index(iterable, value, start=0, stop=None):
        "Return indices where a value occurs in a sequence or iterable."
        # iter_index('AABCADEAF', 'A') → 0 1 4 7
        seq_index = getattr(iterable, 'index', None)
        if seq_index is None:
            iterator = islice(iterable, start, stop)
            for i, element in enumerate(iterator, start):
                if element is value or element == value:
                    yield i
        else:
            stop = len(iterable) if stop is None else stop
            i = start
            with contextlib.suppress(ValueError):
                while True:
                    yield (i := seq_index(value, i, stop))
                    i += 1

    def iter_except(func, exception, first=None):
        "Convert a call-until-exception interface to an iterator interface."
        # iter_except(d.popitem, KeyError) → non-blocking dictionary iterator
        with contextlib.suppress(exception):
            if first is not None:
                yield first()
            while True:
                yield func()

The following recipes have a more mathematical flavor:

    def powerset(iterable):
        "powerset([1,2,3]) → () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
        s = list(iterable)
        return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

    def sum_of_squares(iterable):
        "Add up the squares of the input values."
        # sum_of_squares([10, 20, 30]) → 1400
        return math.sumprod(*tee(iterable))

    def reshape(matrix, cols):
        "Reshape a 2-D matrix to have a given number of columns."
        # reshape([(0, 1), (2, 3), (4, 5)], 3) →  (0, 1, 2), (3, 4, 5)
        return batched(chain.from_iterable(matrix), cols, strict=True)

    def transpose(matrix):
        "Swap the rows and columns of a 2-D matrix."
        # transpose([(1, 2, 3), (11, 22, 33)]) → (1, 11) (2, 22) (3, 33)
        return zip(*matrix, strict=True)

    def matmul(m1, m2):
        "Multiply two matrices."
        # matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]) → (49, 80), (41, 60)
        n = len(m2[0])
        return batched(starmap(math.sumprod, product(m1, transpose(m2))), n)

    def convolve(signal, kernel):
        """Discrete linear convolution of two iterables.
        Equivalent to polynomial multiplication.

        Convolutions are mathematically commutative; however, the inputs are
        evaluated differently.  The signal is consumed lazily and can be
        infinite. The kernel is fully consumed before the calculations begin.

        Article:  https://betterexplained.com/articles/intuitive-convolution/
        Video:    https://www.youtube.com/watch?v=KuXjwB4LzSA
        """
        # convolve([1, -1, -20], [1, -3]) → 1 -4 -17 60
        # convolve(data, [0.25, 0.25, 0.25, 0.25]) → Moving average (blur)
        # convolve(data, [1/2, 0, -1/2]) → 1st derivative estimate
        # convolve(data, [1, -2, 1]) → 2nd derivative estimate
        kernel = tuple(kernel)[::-1]
        n = len(kernel)
        padded_signal = chain(repeat(0, n-1), signal, repeat(0, n-1))
        windowed_signal = sliding_window(padded_signal, n)
        return map(math.sumprod, repeat(kernel), windowed_signal)

    def polynomial_from_roots(roots):
        """Compute a polynomial's coefficients from its roots.

           (x - 5) (x + 4) (x - 3)  expands to:   x³ -4x² -17x + 60
        """
        # polynomial_from_roots([5, -4, 3]) → [1, -4, -17, 60]
        factors = zip(repeat(1), map(operator.neg, roots))
        return list(functools.reduce(convolve, factors, [1]))

    def polynomial_eval(coefficients, x):
        """Evaluate a polynomial at a specific value.

        Computes with better numeric stability than Horner's method.
        """
        # Evaluate x³ -4x² -17x + 60 at x = 5
        # polynomial_eval([1, -4, -17, 60], x=5) → 0
        n = len(coefficients)
        if not n:
            return type(x)(0)
        powers = map(pow, repeat(x), reversed(range(n)))
        return math.sumprod(coefficients, powers)

    def polynomial_derivative(coefficients):
        """Compute the first derivative of a polynomial.

           f(x)  =  x³ -4x² -17x + 60
           f'(x) = 3x² -8x  -17
        """
        # polynomial_derivative([1, -4, -17, 60]) → [3, -8, -17]
        n = len(coefficients)
        powers = reversed(range(1, n))
        return list(map(operator.mul, coefficients, powers))

    def sieve(n):
        "Primes less than n."
        # sieve(30) → 2 3 5 7 11 13 17 19 23 29
        if n > 2:
            yield 2
        data = bytearray((0, 1)) * (n // 2)
        for p in iter_index(data, 1, start=3, stop=math.isqrt(n) + 1):
            data[p*p : n : p+p] = bytes(len(range(p*p, n, p+p)))
        yield from iter_index(data, 1, start=3)

    def factor(n):
        "Prime factors of n."
        # factor(99) → 3 3 11
        # factor(1_000_000_000_000_007) → 47 59 360620266859
        # factor(1_000_000_000_000_403) → 1000000000000403
        for prime in sieve(math.isqrt(n) + 1):
            while not n % prime:
                yield prime
                n //= prime
                if n == 1:
                    return
        if n > 1:
            yield n

    def totient(n):
        "Count of natural numbers up to n that are coprime to n."
        # https://mathworld.wolfram.com/TotientFunction.html
        # totient(12) → 4 because len([1, 5, 7, 11]) == 4
        for prime in set(factor(n)):
            n -= n // prime
        return n