Explain the complexity for the new improvement with precomputing.

Author: Fithi-Teklom

Sprint-2/improve_with_precomputing/common_prefix/common_prefix.py

@@ -26,3 +26,7 @@ def find_common_prefix(left: str, right: str) -> str:
         if left[i] != right[i]:
             return left[:i]
     return left[:min_length]
+
+# The original code compares every pair of strings, giving it (O(m^2)) pair comparisons. 
+# The new version sorts the strings and compares only neighbouring pairs, reducing the number of comparisons.
+# Its approximate time complexity improves from (O(m^2L)) to (O(mL\log m)).

Sprint-2/improve_with_precomputing/count_letters/count_letters.py

@@ -13,3 +13,8 @@ def count_letters(s: str) -> int:
 
 def is_upper_case(letter: str) -> bool:
     return letter == letter.upper()
+
+# 
+# The original code searches the string repeatedly, giving it (O(n^2)) time complexity.
+# The new version stores lowercase letters in a set first, so it can check them faster.
+# This reduces the expected time complexity to (O(n)), but uses extra memory.