"itertools" --- 建立產生高效率迴圈之疊代器的函式
************************************************

======================================================================

這個模組實作了許多 *疊代器 (iterator)* 構建塊 (building block)，其靈感
來自 APL、Haskell 和 SML 的結構。每個構建塊都以適合 Python 的形式來重
新設計。

這個模組標準化了快速且高效率利用記憶體的核心工具集，這些工具本身或組合
使用都很有用。它們共同構成了一個「疊代器代數 (iterator algebra)」，使
得在純 Python 中簡潔且高效地建構專用工具成為可能。

例如，SML 提供了一個造表工具："tabulate(f)"，它產生一個序列 "f(0),
f(1), ..."。在 Python 中，可以透過結合 "map()" 和 "count()" 組成
"map(f, count())" 以達到同樣的效果。

這些工具及其內建的對等部分 (counterpart) 也可以很好地與 "operator" 模
組中的高速函式配合使用。例如，乘法運算子可以對映到兩個向量上以組成高效
率的內積："sum(starmap(operator.mul, zip(vec1, vec2, strict=True)))"。

**無限疊代器：**

+--------------------+-------------------+---------------------------------------------------+-------------------------------------------+
| 疊代器             | 引數              | 結果                                              | 範例                                      |
|====================|===================|===================================================|===========================================|
| "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，... 重複無限次或 n 次           | "repeat(10, 3) → 10 10 10"                |
+--------------------+-------------------+---------------------------------------------------+-------------------------------------------+

**在最短輸入序列 (shortest input sequence) 處終止的疊代器：**

+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| 疊代器                       | 引數                         | 結果                                              | 範例                                                          |
|==============================|==============================|===================================================|===============================================================|
| "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()"      | 可疊代物件                   | 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]，當 predicate 失敗時開始         | "dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8"               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "filterfalse()"              | predicate, seq               | 當 predicate(elem) 失敗時 seq 的元素              | "filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8"               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "groupby()"                  | iterable[, key]              | 根據 key(v) 的值分組的子疊代器                    |                                                               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "islice()"                   | seq, [start,] stop [, step]  | seq[start:stop:step] 的元素                       | "islice('ABCDEFG', 2, None) → C D E F G"                      |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "pairwise()"                 | 可疊代物件                   | (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]，直到 predicate 失敗               | "takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4"                 |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "tee()"                      | it, n                        | it1, it2, ... itn，將一個疊代器分成 n 個          |                                                               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "zip_longest()"              | p, q, ...                    | (p[0], q[0]), (p[1], q[1]), ...                   | "zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-"      |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+

**組合疊代器：**

+------------------------------------------------+----------------------+---------------------------------------------------------------+
| 疊代器                                         | 引數                 | 結果                                                          |
|================================================|======================|===============================================================|
| "product()"                                    | p, q, ... [repeat=1] | 笛卡爾乘積 (cartesian product)，相當於巢狀的 for 迴圈         |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "permutations()"                               | p[, r]               | 長度為 r 的元組，所有可能的定序，無重複元素                   |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "combinations()"                               | p, r                 | 長度為 r 的元組，按照排序過後的定序，無重複元素               |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "combinations_with_replacement()"              | p, r                 | 長度為 r 的元組，按照排序過後的定序，有重複元素               |
+------------------------------------------------+----------------------+---------------------------------------------------------------+

+------------------------------------------------+---------------------------------------------------------------+
| 範例                                           | 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 函式
=============

以下的模組函式都會建構並回傳疊代器。一些函式提供無限長度的串流
(stream)，因此應僅由截斷串流的函式或迴圈來存取它們。

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

   建立一個回傳累積和的疊代器，或其他二進位函式的累積結果。

   *function* 預設為加法。*function* 應接受兩個引數，即累積總和和來自
   *iterable* 的值。

   如果提供了 *initial* 值，則累積將從該值開始，並且輸出的元素數將比輸
   入的可疊代物件多一個。

   大致等價於：

      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

   *function* 引數可以被設定為 "min()" 以得到連續的最小值，設定為
   "max()" 以得到連續的最大值，或者設定為 "operator.mul()" 以得到連續
   的乘積。也可以透過累積利息和付款來建立攤銷表 (Amortization tables)
   ：

      >>> 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]

   可參見 "functools.reduce()"，其是個類似的函式，但僅回傳最終的累積值
   。

   在 3.2 版被加入.

   在 3.3 版的變更: 新增可選的 *function* 參數。

   在 3.8 版的變更: 新增可選的 *initial* 參數。

itertools.batched(iterable, n)

   將來自 *iterable* 的資料分批為長度為 *n* 的元組。最後一個批次可能比
   *n* 短。

   對輸入的可疊代物件進行迴圈，並將資料累積到大小為 *n* 的元組中。輸入
   是惰性地被消耗 (consumed lazily) 的，會剛好足夠填充一批的資料。一旦
   批次填滿或輸入的可疊代物件耗盡，就會 yield 出結果：

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

   大致等價於：

      def batched(iterable, n):
          # 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)):
              yield batch

   在 3.12 版被加入.

itertools.chain(*iterables)

   建立一個疊代器，從第一個可疊代物件回傳元素直到其耗盡，然後繼續處理
   下一個可疊代物件，直到所有可疊代物件都耗盡。用於將連續的序列做為單
   一序列處理。大致等價於：

      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()" 的另一個建構函式。從單個可疊代的引數中得到鏈接的輸入，該
   引數是惰性計算的。大致等價於：

      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)

   從輸入 *iterable* 中回傳長度為 *r* 的元素的子序列。

   輸出是 "product()" 的子序列，僅保留作為 *iterable* 子序列的條目。輸
   出的長度由 "math.comb()" 給定，當 "0 ≤ r ≤ n" 時，計算 "n! / r! /
   (n - r)!"，當 "r > n" 時為零。

   根據輸入值 *iterable* 的順序，組合的元組會按照字典順序輸出。如果輸
   入的 *iterable* 已經排序，則輸出的元組也將按排序的順序產生。

   元素是根據它們的位置（而非值）來決定其唯一性。如果輸入的元素都是獨
   特的，則每個組合內將不會有重複的值。

   大致等價於：

      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)

   回傳來自輸入 *iterable* 的長度為 *r* 的子序列，且允許個別元素重複多
   次。

   其輸出是一個 "product()" 的子序列，僅保留作為 *iterable* 子序列（可
   能有重複元素）的條目。當 "n > 0" 時，回傳的子序列數量為 "(n + r -
   1)! / r! / (n - 1)!"。

   根據輸入值 *iterable* 的順序，組合的元組會按照字典順序輸出。如果輸
   入的 *iterable* 已經排序，則輸出的元組也將按排序的順序產生。

   元素是根據它們的位置（而非值）來決定其唯一性。如果輸入的元素都是獨
   特的，生成的組合也將是獨特的。

   大致等價於：

      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)

   在 3.1 版被加入.

itertools.compress(data, selectors)

   建立一個疊代器，回傳 *data* 中對應 *selectors* 的元素為真的元素。當
   *data* 或 *selectors* 可疊代物件耗盡時停止。大致等價於：

      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)

   在 3.1 版被加入.

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

   建立一個疊代器，回傳從 *start* 開始的等差的值。可以與 "map()" 一起
   使用來產生連續的資料點，或與 "zip()" 一起使用來增加序列號。大致等價
   於：

      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())".

   在 3.1 版的變更: 新增 *step* 引數並允許非整數引數。

itertools.cycle(iterable)

   建立一個疊代器，回傳 *iterable* 中的元素並保存每個元素的副本。當可
   疊代物件耗盡時，從保存的副本中回傳元素。會無限次的重複。大致等價於
   ：

      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

   此 itertool 可能需要大量的輔助儲存空間（取決於可疊代物件的長度）。

itertools.dropwhile(predicate, iterable)

   建立一個疊代器，在 *predicate* 為真時丟棄 *iterable* 中的元素，之後
   回傳每個元素。大致等價於：

      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

   注意，在 predicate 首次變為 False 之前，這不會產生*任何*輸出，所以
   此 itertool 可能會有較長的啟動時間。

itertools.filterfalse(predicate, iterable)

   建立一個疊代器，過濾 *iterable* 中的元素，僅回傳 *predicate* 為
   False 值的元素。如果 *predicate* 是 "None"，則回傳為 False 的項目。
   大致等價於：

      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)

   建立一個疊代器，回傳 *iterable* 中連續的鍵和群組。*key* 是一個為每
   個元素計算鍵值的函式。如果其未指定或為 "None"，則 *key* 預設為一個
   識別性函式 (identity function)，並回傳未被更改的元素。一般來說，可
   疊代物件需要已經用相同的鍵函式進行排序。

   "groupby()" 的操作類似於 Unix 中的 "uniq" 過濾器。每當鍵函式的值發
   生變化時，它會產生一個 break 或新的群組（這就是為什麼通常需要使用相
   同的鍵函式對資料進行排序）。這種行為不同於 SQL 的 GROUP BY，其無論
   輸入順序如何都會聚合相同的元素。

   回傳的群組本身是一個與 "groupby()" 共享底層可疊代物件的疊代器。由於
   來源是共享的，當 "groupby()" 物件前進時，前一個群組將不再可見。因此
   ，如果之後需要該資料，應將其儲存為串列：

      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()" 大致等價於：

      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.

   大致等價於：

      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.

   大致等價於：

      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

   在 3.10 版被加入.

itertools.permutations(iterable, r=None)

   Return successive *r* length permutations of elements from the
   *iterable*.

   If *r* is not specified or is "None", then *r* defaults to the
   length of the *iterable* and all possible full-length permutations
   are generated.

   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.

   大致等價於：

      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)

   Cartesian product of input iterables.

   Roughly equivalent to nested for-loops in a generator expression.
   For example, "product(A, B)" returns the same as "((x,y) for x in A
   for y in B)".

   The nested loops cycle like an odometer with the rightmost element
   advancing on every iteration.  This pattern creates a lexicographic
   ordering so that if the input's iterables are sorted, the product
   tuples are emitted in sorted order.

   To compute the product of an iterable with itself, specify the
   number of repetitions with the optional *repeat* keyword argument.
   For example, "product(A, repeat=4)" means the same as "product(A,
   A, A, A)".

   This function is roughly equivalent to the following code, except
   that the actual implementation does not build up intermediate
   results in memory:

      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)

   Before "product()" runs, it completely consumes the input
   iterables, keeping pools of values in memory to generate the
   products.  Accordingly, it is only useful with finite inputs.

itertools.repeat(object[, times])

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

   大致等價於：

      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

   A common use for *repeat* is to supply a stream of constant values
   to *map* or *zip*:

      >>> 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)

   Return *n* independent iterators from a single iterable.

   大致等價於：

      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

   Once a "tee()" has been created, the original *iterable* should not
   be used anywhere else; otherwise, the *iterable* could get advanced
   without the tee objects being informed.

   "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.

   This itertool may require significant auxiliary storage (depending
   on how much temporary data needs to be stored). In general, if one
   iterator uses most or all of the data before another iterator
   starts, it is faster to use "list()" instead of "tee()".

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.

   大致等價於：

      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 Recipes
=================

This section shows recipes for creating an extended toolset using the
existing itertools as building blocks.

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)

   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
