Palindrome Partitioning II: Minimum Cuts for Palindromic Substrings

This challenge focuses on efficiently partitioning a given string into palindromic substrings. The goal is to find the minimum number of cuts required to achieve such a partitioning. This problem is a classic example of dynamic programming and has applications in string processing and data compression where finding optimal substructures is key.

Problem Description

Given a string s, you need to find the minimum number of cuts required to partition s such that every substring in the partition is a palindrome.

A palindrome is a string that reads the same forwards and backward.

Key Requirements:

• The entire string must be partitioned.
• Each substring in the partition must be a palindrome.
• The objective is to minimize the total number of cuts.

Expected Behavior:

The function should return an integer representing the minimum number of cuts. If the string itself is already a palindrome, the minimum cuts required is 0.

Edge Cases:

• An empty string: What should be the output for an empty input string?
• A string with a single character: This is always a palindrome.
• A string that is already a palindrome.

Examples

Example 1:

Input: s = "aab"
Output: 1
Explanation: The string can be partitioned as ["aa", "b"]. "aa" is a palindrome, and "b" is a palindrome. This requires one cut. Another partition is ["a", "a", "b"], which requires two cuts. The minimum is 1.

Example 2:

Input: s = "a"
Output: 0
Explanation: The string "a" is already a palindrome. No cuts are needed.

Example 3:

Input: s = "abacaba"
Output: 0
Explanation: The string "abacaba" is itself a palindrome. No cuts are needed.

Example 4:

Input: s = "forgeeksskeegfor"
Output: 2
Explanation: The string can be partitioned as ["for", "geeksskeeg", "for"]. "geeksskeeg" is a palindrome. This requires two cuts.

Constraints

• 1 <= s.length <= 2000
• s consists of lowercase English letters only.
• The solution should aim for a time complexity of O(n^2) where n is the length of the string.

Notes

• Consider how you can efficiently check if a substring is a palindrome. Pre-computation might be helpful.
• Think about how the minimum cuts for a prefix of the string relate to the minimum cuts for the entire string. This suggests a dynamic programming approach.
• The problem asks for the minimum number of cuts, not the number of partitions. If a string is partitioned into k substrings, it requires k-1 cuts.