递归函数的记忆给出了错误的答案

Question

这是Edit distance问题的实施。

这是没有任何记忆的递归版本

 def edit_distance(str_a, str_b, len_a, len_b):
    # bot strs consumed
    if len_a == 0 and len_b == 0:
       return 0

    if len_a == 0:
       return len_b

    if len_b == 0:
       return len_a

    i_a = len_a-1
    i_b = len_b-1

    if str_a[i_a] == str_b[i_b]:
       return edit_distance(str_a, str_b, i_a, i_b)

    replace = edit_distance(str_a, str_b, i_a, i_b)
    delete = edit_distance(str_a, str_b, i_a, len_b)
    insert = edit_distance(str_a, str_b, len_a, i_b)

    return 1+min(replace, delete, insert)

现在这里是memoized版本，我在其中缓存调用结果。

def edit_distance_memo(str_a, str_b, len_a, len_b, cache):
  if cache[len_a][len_b] != -1:
    return cache[len_a][len_b]


  if len_a == 0 and len_b == 0:
    cache[len_a][len_b] = 0
    return 0

  if len_a == 0:
    cache[len_a][len_b] = len_b
    return len_b

  if len_b == 0:
    cache[len_a][len_b] = len_a
    return len_a

  if cache[len_a][len_b] != -1:
    return cache[len_a][len_b]

  i_a = len_a-1
  i_b = len_b-1

 if str_a[i_a] == str_b[i_b]:
   cache[len_a][len_b] =  edit_distance_memo(str_a, str_b, i_a, i_b, cache)
   return cache[len_a][len_b]

 replace = edit_distance_memo(str_a, str_b, i_a, i_b, cache)
 delete = edit_distance_memo(str_a, str_b, i_a, len_b, cache)
 insert = edit_distance_memo(str_a, str_b, len_a, i_b, cache)

 best_option = min(replace, delete, insert)

 cache[len_a][len_b] = 1 + best_option

 return cache[len_a][len_b]

这是调用代码：

 from time import time
 str1 = "Shakespeare"
 str2 = "shake spear"
 s1 =  time()
 print edit_distance(str1, str2, len(str1), len(str2)), "edit_distance"
 print "diff---", time()-s1

 rows = len("Shakespeare")+1
 columns = len("shake spear")+1

 cache = [[-1] * columns] * rows
 st  = time()
 print edit_distance_memo("Shakespeare", "shake spear",len("Shakespeare"), len("shake spear"), cache)

由于某种原因，记忆版本似乎给出了错误的答案（7而不是3）。这有什么不对？

提前致谢。