#### Min Distance up Bottom

p
```"""
Author  : Alexander Pantyukhin
Date    : October 14, 2022
This is an implementation of the up-bottom approach to find edit distance.
The implementation was tested on Leetcode: https://leetcode.com/problems/edit-distance/

Levinstein distance
Dynamic Programming: up -> down.
"""

import functools

def min_distance_up_bottom(word1: str, word2: str) -> int:
"""
>>> min_distance_up_bottom("intention", "execution")
5
>>> min_distance_up_bottom("intention", "")
9
>>> min_distance_up_bottom("", "")
0
>>> min_distance_up_bottom("zooicoarchaeologist", "zoologist")
10
"""
len_word1 = len(word1)
len_word2 = len(word2)

@functools.cache
def min_distance(index1: int, index2: int) -> int:
# if first word index overflows - delete all from the second word
if index1 >= len_word1:
return len_word2 - index2
# if second word index overflows - delete all from the first word
if index2 >= len_word2:
return len_word1 - index1
diff = int(word1[index1] != word2[index2])  # current letters not identical
return min(
1 + min_distance(index1 + 1, index2),
1 + min_distance(index1, index2 + 1),
diff + min_distance(index1 + 1, index2 + 1),
)

return min_distance(0, 0)

if __name__ == "__main__":
import doctest

doctest.testmod()
```  