This problem asks for the fewest cuts to split a string so that every piece is a palindrome. A direct brute-force solution fails due to overlapping subproblems, making dynamic programming the most efficient approach. GhostInterview guides you through precomputing palindromic substrings and applying state transitions to minimize cuts efficiently.
Problem Statement
Given a string s, partition s into substrings such that every substring is a palindrome. Your task is to compute the minimum number of cuts needed to achieve such a partitioning.
For example, given s = "aab", the optimal palindrome partitioning is ["aa","b"], which requires only one cut. Your solution must handle strings up to length 2000 consisting only of lowercase English letters.
Examples
Example 1
Input: s = "aab"
Output: 1
The palindrome partitioning ["aa","b"] could be produced using 1 cut.
Example 2
Input: s = "a"
Output: 0
Example details omitted.
Example 3
Input: s = "ab"
Output: 1
Example details omitted.
Constraints
- 1 <= s.length <= 2000
- s consists of lowercase English letters only.
Solution Approach
Precompute Palindrome Table
Create a 2D boolean table isPalindrome[i][j] where each entry indicates whether the substring s[i..j] is a palindrome. This avoids repeated palindrome checks during dynamic programming and ensures state transitions can reference results in constant time.
Dynamic Programming Array
Define dp[i] as the minimum number of cuts needed for the substring s[0..i]. Initialize dp[0] = 0 and iterate over all possible end indices j. For each palindrome s[k..j], update dp[j] = min(dp[j], dp[k-1]+1), capturing the state transition pattern efficiently.
Optimization and Early Termination
If the entire substring s[0..j] is a palindrome, set dp[j] = 0 immediately to avoid unnecessary calculations. Combine the precomputed palindrome table with dynamic programming to reduce overall complexity and handle worst-case input efficiently.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(n^2) due to filling the palindrome table and updating dp for each substring. Space complexity is O(n^2) for the table and O(n) for dp, optimizing lookups for state transitions in dynamic programming.
What Interviewers Usually Probe
- They may ask if you can precompute palindrome substrings to speed up checks.
- Expect questions about how to handle overlapping subproblems efficiently.
- They often probe understanding of the dp state definition and transitions.
Common Pitfalls or Variants
Common pitfalls
- Failing to precompute palindromic substrings leads to TLE for longer inputs.
- Incorrect dp initialization can cause off-by-one errors in cut counts.
- Assuming single-pass greedy partitioning works ignores overlapping subproblems.
Follow-up variants
- Return all possible palindrome partitions instead of the minimum cuts.
- Handle strings with uppercase letters or special characters in palindrome checks.
- Compute minimum cuts for palindromic substrings of a fixed maximum length.
How GhostInterview Helps
- Automatically tracks palindrome precomputation to save time in dynamic programming implementation.
- Guides through dp state definition and iterative updates to minimize cuts correctly.
- Identifies edge cases and ensures the solution handles maximum-length inputs efficiently.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the main dynamic programming pattern in Palindrome Partitioning II?
The pattern is state transition dp where dp[i] represents minimum cuts for s[0..i] and updates depend on precomputed palindrome substrings.
Can I optimize space usage for this problem?
Yes, you can reduce the 2D palindrome table to a 1D array for checking palindromes on the fly, though it may complicate updates.
What is the time complexity for large strings?
Filling the palindrome table is O(n^2), and computing dp is also O(n^2), so overall complexity remains quadratic for worst-case inputs.
Why can't a greedy approach work here?
Greedy cuts fail because local palindrome decisions may prevent a globally minimal cut solution, requiring full dp state tracking.
Are single-character substrings always considered palindromes?
Yes, any single character is a palindrome, which simplifies base cases in the dp array and precomputed table.
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Find all possible palindrome partitioning of a string using backtracking and dynamic programming.
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