input: def longest_common_subsequence(string1, string2):
      """ Finds the longest common subsequence between two strings.
      >>> longest_common_subsequence("ABCDF", "ABECD")
      'ABCD'
      >>> longest_common_subsequence("ABC", "DEF")
      ''
      """
output:     m, n = len(string1), len(string2)
      dp = [[0] * (n + 1) for _ in range(m + 1)]
      
      for i in range(1, m + 1):
          for j in range(1, n + 1):
              if string1[i - 1] == string2[j - 1]:
                  dp[i][j] = dp[i - 1][j - 1] + 1
              else:
                  dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])
      
      lcs = []
      i, j = m, n
      while i > 0 and j > 0:
          if string1[i - 1] == string2[j - 1]:
              lcs.append(string1[i - 1])
              i -= 1
              j -= 1
          elif dp[i - 1][j] > dp[i][j - 1]:
              i -= 1
          else:
              j -= 1
      
      return ''.join(reversed(lcs))