131. Palindrome Partitioning - Detailed Explanation

Complexity Analysis

• Time Complexity:
  • Worst-Case: O(2^n * n), because in the worst-case scenario (e.g., when every substring is a palindrome, like in a string of identical characters), we explore nearly all possible partitionings.
• Space Complexity:
  • O(n) for the recursion stack, plus additional space for storing the current partition, and the final results, which in the worst-case may also be exponential in number.

Step-by-Step Walkthrough & Visual Example

Consider the example s = "aab":

1. Initial Call:
   Start with an empty partition and start = 0.

2. First Level (start = 0):

   • Check substring "a" (indices 0 to 0): It is a palindrome.
     • Add "a" to the current partition → current partition becomes ["a"].
     • Recurse with start = 1.
   • Also, check substring "aa" (indices 0 to 1): It is a palindrome.
     • Add "aa" to the current partition → current partition becomes ["aa"].
     • Recurse with start = 2.
   • Substring "aab" (indices 0 to 2) is not a palindrome.

3. Second Level (for partition ["a"], start = 1):

   • Check substring "a" (indices 1 to 1): It is a palindrome.
     • Add "a" to get ["a", "a"].
     • Recurse with start = 2.
   • Check substring "ab" (indices 1 to 2): Not a palindrome.

4. Third Level (for partition ["a", "a"], start = 2):

   • Check substring "b" (indices 2 to 2): It is a palindrome.
     • Add "b" to get ["a", "a", "b"].
     • Since start becomes 3 (end of string), add this partition to the result.

5. Backtracking:

   • Backtrack and try the partition starting with "aa" from the first level:
     • For partition ["aa"] (start = 2):
       • Check substring "b" (indices 2 to 2): It is a palindrome.
         • Add "b" → partition becomes ["aa", "b"].
         • Recurse with start = 3 and add this partition to the result.

Final Partitions:
The result will contain:

• [["a", "a", "b"], ["aa", "b"]]

A visual tree of recursion would look like:

          ""
         /   \
       "a"   "aa"
       /       \
    "a"         "b"
    /             \
  "b"             (end)
   (end)

Common Mistakes, Edge Cases & Alternative Variations

Common Mistakes:

• Incorrect Palindrome Check:
  Failing to correctly check if a substring is a palindrome (e.g., not handling even-length palindromes properly).
• Inefficient Backtracking:
  Not backtracking correctly by forgetting to remove the last element, which can cause incorrect partitions.
• Missing Base Case:
  Not handling the case when the entire string has been partitioned, leading to incomplete or incorrect results.

Edge Cases:

• Single Character String:
  The string "a" should return [["a"]].
• All Characters Same:
  For a string like "aaa", there are multiple valid partitions (e.g., ["a","a","a"], ["aa","a"], ["a","aa"], ["aaa"]).
• Empty String:
  Although the constraints specify at least one character, always consider what your function should do if the input is empty.

Alternative Variations:

• Minimum Palindrome Partitioning:
  Find the minimum number of cuts needed to partition the string such that every substring is a palindrome.

• Longest Palindromic Substring:
  Given a string, find the longest substring which is a palindrome.

• Palindrome Partitioning II:
  Variations where you may need to count the number of palindromic partitions or return the partition with specific constraints.

Related Problems for Further Practice

• Longest Palindromic Substring:
  Given a string, return the longest substring that is a palindrome.

• Minimum Cut for Palindrome Partitioning:
  Given a string, return the minimum number of cuts needed to partition it so that every substring is a palindrome.

• Palindrome Permutation:
  Check if any permutation of a given string is a palindrome.

• Subsets/Backtracking Problems:
  Practice other backtracking problems like generating all subsets or combinations.