1. One-Liner
Palindrome partitioning minimizes cuts so every substring is a palindrome—dp[i] stores the best for prefix ending at i, using palindrome checks on intervals.
2. The Problem It Solves
Given s, split it into palindromic chunks with fewest separators (or enumerate all valid partitions in another variant). Used in text analytics and advanced string DP.
3. The Core Idea
Precompute is_pal[i][j] or expand while DP-ing; then dp[i]=min over j<i+1 pal[j..i] of dp[j-1]+1—try every last palindrome chunk ending at i.
4. How It Works (Step-by-Step)
| Step | What Happens |
|---|---|
| 1 | Palindrome table for all substrings O(n²) |
| 2 | dp[0]=0, dp[i] min cuts for s[0..i) |
| 3 | Transition over palindromic suffixes |
| 4 | Return dp[n]-1 cuts or dp[n] per indexing |
5. Dry Run Example
"aab" → partition "aa"|"b" → 1 cut (0-based definition dependent).
6. Key Properties
| Property | Value |
|---|---|
| Time | O(n²) or O(n³) naive |
| Related | Palindrome centers |
7. Where It Is Used
| Domain | Use |
|---|---|
| NLP | Tokenization heuristics |
| Interviews | Hard string DP |
8. Interview Tips
Also know generate all partitions via backtracking + palindrome memo.
9. Comparison with Other Algorithms
| Problem | Goal |
|---|---|
| Min cuts | Optimization |
| List all | Backtracking |
10. Complexity
| Metric | Value |
|---|---|
| Time | O(n²) with palindrome preprocessing |
| Space | O(n²) for table |
Implementation Example (PYTHON)
def min_palindrome_cuts(s: str) -> int:
n = len(s)
pal = [[False] * n for _ in range(n)]
for i in range(n - 1, -1, -1):
for j in range(i, n):
pal[i][j] = s[i] == s[j] and (j - i < 2 or pal[i + 1][j - 1])
dp = [0] * (n + 1)
for i in range(1, n + 1):
dp[i] = i
for j in range(i):
if pal[j][i - 1]:
dp[i] = min(dp[i], dp[j] + 1)
return dp[n] - 1