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You are given a string s consisting only of lowercase English letters.

A string is called a palindrome if it reads the same from left to right as it does from right to left.

You need to create the lexicographically smallest palindrome by choosing some characters of s and changing them to any lowercase English letter.

Return the lexicographically smallest palindrome you can create.

Clarifying Questions

1. Can the input string be empty?
   • While not typical in most cases, we should handle the edge case where the input string might be empty.
2. What is the maximum length of the input string?
   • This is useful to know for space and time complexity considerations. Assuming it to be reasonably large but within typical constraints.
3. Are there any restrictions on how many characters we can change?
   • No restrictions are mentioned, so we can change any or all characters to achieve the goal.
4. What does lexicographically smallest mean in this context?
   • It means that when comparing two strings, the string that would appear first in a dictionary is considered smaller. For example, “a” is smaller than “b”.

Strategy

The goal is to transform the given string s into a palindrome while also ensuring it is the lexicographically smallest palindrome possible.

1. Palindrome Formation:
   • A string is a palindrome if it reads the same forwards and backwards.
2. Lexicographically Smallest:
   • To achieve this, we should prioritize changing characters to ‘a’ wherever possible, as ‘a’ is the smallest letter lexicographically.

Steps:

1. For each character in the first half of the string, compare it with the corresponding character in the second half.
2. If they are not the same, replace the larger character with the smaller one in the pair.
3. If both characters are the same, no change is needed.
4. After traversing and comparing/setting changes appropriately, we get our desired result.

Code

Here’s the Python code to achieve this:

def make_smallest_palindrome(s: str) -> str:
    s = list(s)  # Convert the string to a list to allow modifications
    n = len(s)
    
    for i in range(n // 2):
        j = n - i - 1
        if s[i] != s[j]:
            # Replace the larger character with the smaller one to keep lexicographically smallest
            if s[i] < s[j]:
                s[j] = s[i]
            else:
                s[i] = s[j]

    return ''.join(s)

# Example usage:
s = "egcfe"
result = make_smallest_palindrome(s)
print(result)  # Output should be "efcfe"

Time Complexity

• Time Complexity: O(n/2) which is equivalent to O(n)
  • We traverse half the string, performing constant-time operations for each character pair.
• Space Complexity: O(n)
  • We use extra space to store the list conversion of the string. The input and output sizes are both O(n).

This solution ensures we efficiently create the lexicographically smallest palindrome.

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