new_contents: """
  https://cp-algorithms.com/string/z-function.html
  
  Z-function or Z algorithm
  
  Efficient algorithm for pattern occurrence in a string
  
  Time Complexity: O(n) - where n is the length of the string
  
  """
  
  
  def z_function(input_str: str) -> list:
      """
      For the given string this function computes value for each index,
      which represents the maximal length substring starting from the index
      and is the same as the prefix of the same size
  
      e.x.  for string 'abab' for second index value would be 2
  
      For the value of the first element the algorithm always returns 0
  
      >>> z_function("abracadabra")
      [0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 1]
      >>> z_function("aaaa")
      [0, 3, 2, 1]
      >>> z_function("zxxzxxz")
      [0, 0, 0, 4, 0, 0, 1]
      """
      z_result = [0] * len(input_str)
  
      # initialize interval's left pointer and right pointer
      left_pointer, right_pointer = 0, 0
  
      for i in range(1, len(input_str)):
          # case when current index is inside the interval
          if i <= right_pointer:
              min_edge = min(right_pointer - i + 1, z_result[i - left_pointer])
              z_result[i] = min_edge
  
          while go_next(i, z_result, input_str):
              z_result[i] += 1
  
          # if new index's result gives us more right interval,
          # we've to update left_pointer and right_pointer
          if i + z_result[i] - 1 > right_pointer:
              left_pointer, right_pointer = i, i + z_result[i] - 1
  
      return z_result
  
  
  def go_next(i, z_result, s):
      """
      Check if we have to move forward to the next characters or not
      """
      return i + z_result[i] < len(s) and s[z_result[i]] == s[i + z_result[i]]
  
  
  def find_pattern(pattern: str, input_str: str) -> int:
      """
      Example of using z-function for pattern occurrence
      Given function returns the number of times 'pattern'
      appears in 'input_str' as a substring
  
      >>> find_pattern("abr", "abracadabra")
      2
      >>> find_pattern("a", "aaaa")
      4
      >>> find_pattern("xz", "zxxzxxz")
      2
      """
      answer = 0
      # concatenate 'pattern' and 'input_str' and call z_function
      # with concatenated string
      z_result = z_function(pattern + input_str)
  
      for val in z_result:
          # if value is greater then length of the pattern string
          # that means this index is starting position of substring
          # which is equal to pattern string
          if val >= len(pattern):
              answer += 1
  
      return answer
  
  
  if __name__ == "__main__":
      import doctest
      doctest.testmod()
  
subject: Add Z-function algorithm implementation (#2067)