add Levinstein distance with Dynamic Programming: up -> down approach (TheAlgorithms#7171)

* add Levinstein distance with Dynamic Programming: up -> down approach

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* add type hint

* fix flake8

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* Update dynamic_programming/min_distance_up_bottom.py

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* Update min_distance_up_bottom.py

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <[email protected]>

Author: 3 people

dynamic_programming/min_distance_up_bottom.py

@@ -0,0 +1,55 @@
+"""
+Author  : Alexander Pantyukhin
+Date    : October 14, 2022
+This is implementation Dynamic Programming up bottom approach
+to find edit distance.
+The aim is to demonstate up bottom approach for solving the task.
+The implementation was tested on the
+leetcode: https://leetcode.com/problems/edit-distance/
+"""
+
+"""
+Levinstein distance
+Dynamic Programming: up -> down.
+"""
+
+
+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
+    """
+
+    from functools import lru_cache
+
+    len_word1 = len(word1)
+    len_word2 = len(word2)
+
+    @lru_cache(maxsize=None)
+    def min_distance(index1: int, index2: int) -> int:
+        # if first word index is overflow - delete all from the second word
+        if index1 >= len_word1:
+            return len_word2 - index2
+        # if second word index is overflow - 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()