# `itertools` --- Functions creating iterators for efficient looping

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`
`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()` | pred, seq | seq[n], seq[n+1], starting when pred fails | `dropwhile(lambda x: x<5, [1,4,6,4,1]) --> 6 4 1`
`filterfalse()` | pred, seq | elements of seq where pred(elem) is false | `filterfalse(lambda x: x%2, range(10)) --> 0 2 4 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()` | pred, seq | seq[0], seq[1], until pred fails | `takewhile(lambda x: x<5, [1,4,6,4,1]) --> 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 functions

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[, func, *, initial=None])

  Make an iterator that returns accumulated sums, or accumulated results of other binary functions (specified via the optional func argument).

  If func is supplied, it should be a function of two arguments. Elements of the input iterable may be any type that can be accepted as arguments to func. (For example, with the default operation of addition, elements may be any addable type including `Decimal` or `Fraction`.)

  Usually, the number of elements output matches the input iterable. However, if the keyword argument initial is provided, the accumulation leads off with the initial value so that the output has one more element than the input iterable.

  およそ次と等価です:

      def accumulate(iterable, func=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
          it = iter(iterable)
          total = initial
          if initial is None:
              try:
                  total = next(it)
              except StopIteration:
                  return
          yield total
          for element in it:
              total = func(total, element)
              yield total

  There are a number of uses for the func argument. It 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, operator.mul))     # running product
      [3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]
      >>> list(accumulate(data, max))              # running maximum
      [3, 4, 6, 6, 6, 9, 9, 9, 9, 9]

      # Amortize a 5% loan of 1000 with 4 annual payments of 90
      >>> cashflows = [1000, -90, -90, -90, -90]
      >>> list(accumulate(cashflows, lambda bal, pmt: bal*1.05 + pmt))
      [1000, 960.0, 918.0, 873.9000000000001, 827.5950000000001]

  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。

  バージョン 3.2 で追加.

  バージョン 3.3 で変更: Added the optional func 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.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 it in iterables:
              for element in it:
                  yield element

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 it in iterables:
              for element in it:
                  yield element

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 combination tuples are emitted in lexicographic ordering according to the order of the input iterable. So, 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. So if the input elements are unique, there will be no repeated values in 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)

  The code for `combinations()` can be also expressed as a subsequence of `permutations()` after filtering entries where the elements are not in sorted order (according to their position in the input pool):

      def combinations(iterable, r):
          pool = tuple(iterable)
          n = len(pool)
          for indices in permutations(range(n), r):
              if sorted(indices) == list(indices):
                  yield tuple(pool[i] for i in indices)

  The number of items returned is `n! / r! / (n-r)!` when `0 <= r <= n` or zero when `r > n`.

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 combination tuples are emitted in lexicographic ordering according to the order of the input iterable. So, 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. So 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)

  The code for `combinations_with_replacement()` can be also expressed as a subsequence of `product()` after filtering entries where the elements are not in sorted order (according to their position in the input pool):

      def combinations_with_replacement(iterable, r):
          pool = tuple(iterable)
          n = len(pool)
          for indices in product(range(n), repeat=r):
              if sorted(indices) == list(indices):
                  yield tuple(pool[i] for i in indices)

  The number of items returned is `(n+r-1)! / r! / (n-1)!` when `n > 0`.

  バージョン 3.1 で追加.

itertools.compress(data, selectors)

  Make an iterator that filters elements from data returning only those that have a corresponding element in selectors that evaluates to `True`. Stops when either the data or selectors iterables has been exhausted. Roughly equivalent to:

      def compress(data, selectors):
          # compress('ABCDEF', [1,0,1,0,1,1]) --> A C E F
          return (d for d, s in zip(data, selectors) if s)

  バージョン 3.1 で追加.

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

  Make an iterator that returns evenly spaced values starting with number start. Often used as an argument to `map()` to generate consecutive data points. Also, used 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

  Note, this member of the toolkit 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 as long as the predicate is true; afterwards, returns every element. Note, the iterator does not produce any output until the predicate first becomes false, so it may have a lengthy start-up time. Roughly equivalent to:

      def dropwhile(predicate, iterable):
          # dropwhile(lambda x: x<5, [1,4,6,4,1]) --> 6 4 1
          iterable = iter(iterable)
          for x in iterable:
              if not predicate(x):
                  yield x
                  break
          for x in iterable:
              yield x

itertools.filterfalse(predicate, iterable)

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

      def filterfalse(predicate, iterable):
          # filterfalse(lambda x: x%2, range(10)) --> 0 2 4 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:

      class groupby:
          # [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

          def __init__(self, iterable, key=None):
              if key is None:
                  key = lambda x: x
              self.keyfunc = key
              self.it = iter(iterable)
              self.tgtkey = self.currkey = self.currvalue = object()

          def __iter__(self):
              return self

          def __next__(self):
              self.id = object()
              while self.currkey == self.tgtkey:
                  self.currvalue = next(self.it)    # Exit on StopIteration
                  self.currkey = self.keyfunc(self.currvalue)
              self.tgtkey = self.currkey
              return (self.currkey, self._grouper(self.tgtkey, self.id))

          def _grouper(self, tgtkey, id):
              while self.id is id and self.currkey == tgtkey:
                  yield self.currvalue
                  try:
                      self.currvalue = next(self.it)
                  except StopIteration:
                      return
                  self.currkey = self.keyfunc(self.currvalue)

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

  Make an iterator that returns selected elements from the iterable. If start is non-zero, then elements from the iterable are skipped until start is reached. Afterward, elements are returned consecutively unless step is set higher than one which results in items being skipped. If stop is `None`, then iteration continues until the iterator is exhausted, if at all; otherwise, it stops at the specified position.

  If start is `None`, then iteration starts at zero. If step is `None`, then the step defaults to one.

  Unlike regular slicing, `islice()` does not support negative values for start, stop, or step. Can be used to extract related fields from data where the internal structure has been flattened (for example, a multi-line report may list a name field on every third line).

  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, stop, step = s.start or 0, s.stop or sys.maxsize, s.step or 1
          it = iter(range(start, stop, step))
          try:
              nexti = next(it)
          except StopIteration:
              # Consume *iterable* up to the *start* position.
              for i, element in zip(range(start), iterable):
                  pass
              return
          try:
              for i, element in enumerate(iterable):
                  if i == nexti:
                      yield element
                      nexti = next(it)
          except StopIteration:
              # Consume to *stop*.
              for i, element in zip(range(i + 1, stop), iterable):
                  pass

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
          a, b = tee(iterable)
          next(b, None)
          return zip(a, b)

  バージョン 3.10 で追加.

itertools.permutations(iterable, r=None)

  Return successive r length permutations of elements in 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 permutation tuples are emitted in lexicographic order according to the order of the input iterable. So, 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. So 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

  The code for `permutations()` can be also expressed as a subsequence of `product()`, filtered to exclude entries with repeated elements (those from the same position in the input pool):

      def permutations(iterable, r=None):
          pool = tuple(iterable)
          n = len(pool)
          r = n if r is None else r
          for indices in product(range(n), repeat=r):
              if len(set(indices)) == r:
                  yield tuple(pool[i] for i in indices)

  The number of items returned is `n! / (n-r)!` when `0 <= r <= n` or zero when `r > n`.

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(*args, 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 args] * 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 are already grouped in tuples from a single iterable (when the data has been "pre-zipped").

  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, [1,4,6,4,1]) --> 1 4
          for x in iterable:
              if predicate(x):
                  yield x
              else:
                  break

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。

  The following Python code helps explain what tee does (although the actual implementation is more complex and uses only a single underlying FIFO queue):

      def tee(iterable, n=2):
          it = iter(iterable)
          deques = [collections.deque() for i in range(n)]
          def gen(mydeque):
              while True:
                  if not mydeque:             # when the local deque is empty
                      try:
                          newval = next(it)   # fetch a new value and
                      except StopIteration:
                          return
                      for d in deques:        # load it to all the deques
                          d.append(newval)
                  yield mydeque.popleft()
          return tuple(gen(d) for d in deques)

  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 using simultaneously 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. Iteration continues until the longest iterable is exhausted. Roughly equivalent to:

      def zip_longest(*args, fillvalue=None):
          # zip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D-
          iterators = [iter(it) for it in args]
          num_active = len(iterators)
          if not num_active:
              return
          while True:
              values = []
              for i, it in enumerate(iterators):
                  try:
                      value = next(it)
                  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()`). If not specified, fillvalue defaults to `None`.

## 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 `compress()` and `range()` 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 `iter_index()` recipe is being tested to see whether it proves its 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 which helps eliminate temporary variables. High speed is retained by preferring "vectorized" building blocks over the use of for-loops and generators which incur interpreter overhead.

    import collections
    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 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:
            # feed the entire iterator into a zero-length deque
            collections.deque(iterator, maxlen=0)
        else:
            # advance to the empty slice starting at position n
            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 all_equal(iterable):
        "Returns True if all the elements are equal to each other"
        g = groupby(iterable)
        return next(g, True) and not next(g, False)

    def quantify(iterable, pred=bool):
        "Count how many times the predicate is True"
        return sum(map(pred, iterable))

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

    def batched(iterable, n):
        "Batch data into tuples of length n. The last batch may be shorter."
        # batched('ABCDEFG', 3) --> ABC DEF G
        if n < 1:
            raise ValueError('n must be at least one')
        it = iter(iterable)
        while batch := tuple(islice(it, n)):
            yield batch

    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
        args = [iter(iterable)] * n
        if incomplete == 'fill':
            return zip_longest(*args, fillvalue=fillvalue)
        if incomplete == 'strict':
            return zip(*args, strict=True)
        if incomplete == 'ignore':
            return zip(*args)
        else:
            raise ValueError('Expected fill, strict, or ignore')

    def sumprod(vec1, vec2):
        "Compute a sum of products."
        return sum(starmap(operator.mul, zip(vec1, vec2, strict=True)))

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

    def transpose(it):
        "Swap the rows and columns of the input."
        # transpose([(1, 2, 3), (11, 22, 33)]) --> (1, 11) (2, 22) (3, 33)
        return zip(*it, 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(sumprod, product(m1, transpose(m2))), n)

    def convolve(signal, kernel):
        # See:  https://betterexplained.com/articles/intuitive-convolution/
        # convolve(data, [0.25, 0.25, 0.25, 0.25]) --> Moving average (blur)
        # convolve(data, [1, -1]) --> 1st finite difference (1st derivative)
        # convolve(data, [1, -2, 1]) --> 2nd finite difference (2nd derivative)
        kernel = tuple(kernel)[::-1]
        n = len(kernel)
        window = collections.deque([0], maxlen=n) * n
        for x in chain(signal, repeat(0, n-1)):
            window.append(x)
            yield sumprod(kernel, window)

    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]
        expansion = [1]
        for r in roots:
            expansion = convolve(expansion, (1, -r))
        return list(expansion)

    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 = 2.5
        # polynomial_eval([1, -4, -17, 60], x=2.5) --> 8.125
        n = len(coefficients)
        if n == 0:
            return x * 0  # coerce zero to the type of x
        powers = map(pow, repeat(x), reversed(range(n)))
        return sumprod(coefficients, powers)

    def iter_index(iterable, value, start=0):
        "Return indices where a value occurs in a sequence or iterable."
        # iter_index('AABCADEAF', 'A') --> 0 1 4 7
        try:
            seq_index = iterable.index
        except AttributeError:
            # Slow path for general iterables
            it = islice(iterable, start, None)
            i = start - 1
            try:
                while True:
                    yield (i := i + operator.indexOf(it, value) + 1)
            except ValueError:
                pass
        else:
            # Fast path for sequences
            i = start - 1
            try:
                while True:
                    yield (i := seq_index(value, i+1))
            except ValueError:
                pass

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

    def factor(n):
        "Prime factors of n."
        # factor(99) --> 3 3 11
        for prime in sieve(math.isqrt(n) + 1):
            while True:
                quotient, remainder = divmod(n, prime)
                if remainder:
                    break
                yield prime
                n = quotient
                if n == 1:
                    return
        if n > 1:
            yield n

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

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

        Example:  repeatfunc(random.random)
        """
        if times is None:
            return starmap(func, repeat(args))
        return starmap(func, repeat(args, times))

    def triplewise(iterable):
        "Return overlapping triplets from an iterable"
        # triplewise('ABCDEFG') --> ABC BCD CDE DEF EFG
        for (a, _), (b, c) in pairwise(pairwise(iterable)):
            yield a, b, c

    def sliding_window(iterable, n):
        # sliding_window('ABCDEFG', 4) --> ABCD BCDE CDEF DEFG
        it = iter(iterable)
        window = collections.deque(islice(it, n), maxlen=n)
        if len(window) == n:
            yield tuple(window)
        for x in it:
            window.append(x)
            yield tuple(window)

    def roundrobin(*iterables):
        "roundrobin('ABC', 'D', 'EF') --> A D E B F C"
        # Recipe credited to George Sakkis
        num_active = len(iterables)
        nexts = cycle(iter(it).__next__ for it in iterables)
        while num_active:
            try:
                for next in nexts:
                    yield next()
            except StopIteration:
                # Remove the iterator we just exhausted from the cycle.
                num_active -= 1
                nexts = cycle(islice(nexts, num_active))

    def partition(pred, iterable):
        "Use a predicate to partition entries into false entries and true entries"
        # partition(is_odd, range(10)) --> 0 2 4 6 8   and  1 3 5 7 9
        t1, t2 = tee(iterable)
        return filterfalse(pred, t1), filter(pred, t2)

    def before_and_after(predicate, it):
        """ Variant of takewhile() that allows complete
            access to the remainder of the iterator.

            >>> it = iter('ABCdEfGhI')
            >>> all_upper, remainder = before_and_after(str.isupper, it)
            >>> ''.join(all_upper)
            'ABC'
            >>> ''.join(remainder)     # takewhile() would lose the 'd'
            'dEfGhI'

            Note that the first iterator must be fully
            consumed before the second iterator can
            generate valid results.
        """
        it = iter(it)
        transition = []
        def true_iterator():
            for elem in it:
                if predicate(elem):
                    yield elem
                else:
                    transition.append(elem)
                    return
        def remainder_iterator():
            yield from transition
            yield from it
        return true_iterator(), remainder_iterator()

    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 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 unique_everseen(iterable, key=None):
        "List unique elements, preserving order. Remember all elements ever seen."
        # unique_everseen('AAAABBBCCDAABBB') --> A B C D
        # unique_everseen('ABBcCAD', str.lower) --> A B c D
        seen = set()
        if key is None:
            for element in filterfalse(seen.__contains__, iterable):
                seen.add(element)
                yield element
            # For order preserving deduplication,
            # a faster but non-lazy solution is:
            #     yield from dict.fromkeys(iterable)
        else:
            for element in iterable:
                k = key(element)
                if k not in seen:
                    seen.add(k)
                    yield element
            # For use cases that allow the last matching element to be returned,
            # a faster but non-lazy solution is:
            #      t1, t2 = tee(iterable)
            #      yield from dict(zip(map(key, t1), t2)).values()

    def unique_justseen(iterable, key=None):
        "List unique elements, preserving order. Remember only the element just seen."
        # unique_justseen('AAAABBBCCDAABBB') --> A B C D A B
        # unique_justseen('ABBcCAD', str.lower) --> A B c A D
        return map(next, map(operator.itemgetter(1), groupby(iterable, key)))

    def iter_except(func, exception, first=None):
        """ Call a function repeatedly until an exception is raised.

        Converts a call-until-exception interface to an iterator interface.
        Like builtins.iter(func, sentinel) but uses an exception instead
        of a sentinel to end the loop.

        Examples:
            iter_except(functools.partial(heappop, h), IndexError)   # priority queue iterator
            iter_except(d.popitem, KeyError)                         # non-blocking dict iterator
            iter_except(d.popleft, IndexError)                       # non-blocking deque iterator
            iter_except(q.get_nowait, Queue.Empty)                   # loop over a producer Queue
            iter_except(s.pop, KeyError)                             # non-blocking set iterator

        """
        try:
            if first is not None:
                yield first()            # For database APIs needing an initial cast to db.first()
            while True:
                yield func()
        except exception:
            pass

    def first_true(iterable, default=False, pred=None):
        """Returns the first true value in the iterable.

        If no true value is found, returns *default*

        If *pred* is not None, returns the first item
        for which pred(item) is true.

        """
        # 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(pred, iterable), default)

    def nth_combination(iterable, r, index):
        "Equivalent to list(combinations(iterable, r))[index]"
        pool = tuple(iterable)
        n = len(pool)
        c = math.comb(n, r)
        if index < 0:
            index += c
        if index < 0 or index >= c:
            raise IndexError
        result = []
        while r:
            c, n, r = c*r//n, n-1, r-1
            while index >= c:
                index -= c
                c, n = c*(n-r)//n, n-1
            result.append(pool[-1-n])
        return tuple(result)