Palindrome Partitioning DP Minimum Cuts in DSA Typescript

Mental Model

We want to split a string into the fewest pieces so that each piece reads the same forwards and backwards.

Analogy: Imagine cutting a ribbon into the smallest number of palindromic segments, like cutting a patterned ribbon where each piece must have a symmetrical design.

s = "a b a c"
Indices: 0 1 2 3
We want cuts like: a | b | a | c or a b a | c
Each segment is a palindrome.

Dry Run Walkthrough

Input: string: "aab"

Goal: Find the minimum cuts needed to split "aab" into palindromic substrings

Step 1: Check all substrings for palindrome property

Palindrome table:
[true, true, false]
[ , true, false]
[ ,  , true]

Why: Knowing which substrings are palindromes helps decide where to cut

Step 2: Initialize cuts array with max cuts (worst case: cut before every char)

cuts = [0, 1, 2]

Why: Start with maximum cuts, then try to reduce

Step 3: At index 1 (substring "aa"), since it's palindrome, update cuts[1] to 0

cuts = [0, 0, 2]

Why: No cut needed between 0 and 1 because "aa" is palindrome

Step 4: At index 2 (substring "b"), check partitions: - "aab" not palindrome - cut after index 1: cuts[1] + 1 = 1 - cut after index 2: cuts[2] stays 2

cuts = [0, 0, 1]

Why: Minimum cuts for full string is 1

Result:

cuts array: [0, 0, 1]
Minimum cuts needed: 1

Annotated Code

DSA Typescript

class PalindromePartitioning {
  static minCut(s: string): number {
    const n = s.length;
    const palindrome: boolean[][] = Array.from({ length: n }, () => Array(n).fill(false));
    const cuts: number[] = Array(n).fill(0);

    // Step 1: Fill palindrome table
    for (let end = 0; end < n; end++) {
      let minCut = end; // max cuts
      for (let start = 0; start <= end; start++) {
        if (s[start] === s[end] && (end - start < 2 || palindrome[start + 1][end - 1])) {
          palindrome[start][end] = true;
          minCut = start === 0 ? 0 : Math.min(minCut, cuts[start - 1] + 1);
        }
      }
      cuts[end] = minCut;
    }
    return cuts[n - 1];
  }
}

// Driver code
const input = "aab";
const result = PalindromePartitioning.minCut(input);
console.log(result);

for (let end = 0; end < n; end++) {

iterate over substring end indices

let minCut = end;

initialize max cuts for current end index

for (let start = 0; start <= end; start++) {

check all possible start indices for palindrome

if (s[start] === s[end] && (end - start < 2 || palindrome[start + 1][end - 1])) {

check palindrome condition for substring s[start..end]

palindrome[start][end] = true;

mark substring as palindrome

minCut = start === 0 ? 0 : Math.min(minCut, cuts[start - 1] + 1);

update minimum cuts needed considering this palindrome

cuts[end] = minCut;

store minimum cuts for substring s[0..end]

Output

1

Complexity Analysis

Time: O(n^2) because we check all substrings and update cuts for each

Space: O(n^2) for palindrome table plus O(n) for cuts array

vs Alternative: Naive recursive approach is exponential; DP reduces it to polynomial time

Edge Cases

empty string

returns 0 cuts since no partition needed

DSA Typescript

const n = s.length;

string with all identical characters, e.g. "aaaa"

returns 0 cuts because whole string is palindrome

DSA Typescript

minCut = start === 0 ? 0 : Math.min(minCut, cuts[start - 1] + 1);

string with no palindromic substrings longer than 1, e.g. "abc"

returns length-1 cuts, splitting every character

DSA Typescript

let minCut = end;

When to Use This Pattern

When asked to partition a string into palindromic substrings with minimum cuts, use DP with palindrome substring precomputation to optimize.

Common Mistakes

Mistake: Not checking the palindrome condition correctly for substrings longer than 2 characters
Fix: Include the check palindrome[start + 1][end - 1] to confirm inner substring is palindrome

Mistake: Initializing cuts array incorrectly or not updating it properly
Fix: Initialize cuts[end] with max cuts and update it inside the inner loop considering palindrome partitions

Summary

Finds the minimum number of cuts needed to split a string into palindromic substrings.

Use when you need to partition strings into palindromes with minimal splits.

The key insight is to precompute palindrome substrings and use DP to track minimum cuts.