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Solution
#!/bin/python3
import math
import os
// ... rest of solution available after purchaseExplanation
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To solve the problem of breaking a palindrome by changing exactly one character to create a string that is not a palindrome, is lexicographically smaller than the original, and is the smallest such string possible, you need to carefully analyze the properties of palindromes and the constraints of the problem.
Start by understanding what it means for a string to be a palindrome: it reads the same forwards and backwards. Your goal is to make a minimal change that breaks this symmetry and results in a strictly smaller string in alphabetical order.
Step 1: Consider the length and structure of the palindrome
If the palindrome length is 1, it’s impossible to break it into a non-palindrome by changing a single character because any single character is trivially a palindrome. Therefore, if the input string has length 1, return "IMPOSSIBLE" immediately.
Step 2: Focus on lexicographical order
Since the requirement is that the new string be alphabetically lower than the original, you should aim to decrease the earliest character possible to get the smallest lexicographical string.
Step 3: Iterating through the first half of the string
Because a palindrome is symmetric, changing a character in the first half affects the lexicographical order and is more impactful for getting a lexicographically smaller string. Iterate from the start up to the middle (not inclusive if odd length) and look for the first character that is not 'a'. Changing this character to 'a' will lower the string lex order and break the palindrome since the corresponding character on the other half remains unchanged.
Step 4: Handle all 'a' characters in the first half
If every character in the first half is 'a', then changing any of these to 'a' won't lower the string or break the palindrome. In this case, your only option is to change the last character in the string to 'b' to break the palindrome and maintain lexicographical order, since 'b' is the next smallest character after 'a'. This change breaks the palindrome because the last character is different from the first character, and it results in the lexicographically smallest possible string in this scenario.
Step 5: Ensuring only one character is changed
The problem states exactly one character must be changed. Be careful to only apply one change: either the earliest non-'a' character in the first half or the last character if all are 'a'.
Step 6: Verify the new string
After making the change, confirm that the new string is indeed not a palindrome and lexicographically smaller. The logic of picking the first non-'a' character or the last character ensures this, but double-checking the palindrome property helps avoid corner cases.
Step 7: Return the resulting string or "IMPOSSIBLE"
If the input string is length 1, or no change is possible to meet the criteria, return "IMPOSSIBLE". Otherwise, return the modified string.
Step 8: Edge cases to consider
- Single-character strings
- Strings composed entirely of 'a's
- Strings with odd lengths, where the middle character doesn’t affect palindrome breaking if changed
- Strings where the first non-'a' character appears late in the first half
Step 9: Efficiency and complexity
This solution requires only one pass through up to half the string, leading to O(n) time complexity where n is the string length, which is optimal for this problem.
Summary:
1. If string length is 1, return "IMPOSSIBLE".
2. Iterate over the first half of the string to find the first character not 'a'.
3. Change that character to 'a' and return the string immediately.
4. If all characters in the first half are 'a', change the last character to 'b'.
5. Return the resulting string or "IMPOSSIBLE" if no valid change can be made.
By following this stepwise logic, you can systematically break the palindrome into the lexicographically smallest possible non-palindromic string with exactly one character change.
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