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.

Practice

1. What does the "Palindrome Partitioning DP Minimum Cuts" problem aim to find for a given string s?

easy

A. The minimum number of cuts needed to partition s into palindromic substrings

B. The maximum length of a palindrome substring in s

C. The total number of palindrome substrings in s

D. The longest palindrome substring in s

Solution

1. Step 1: Understand the problem goal

   The problem asks for the fewest cuts needed to split the string into parts where each part is a palindrome.

2. Step 2: Clarify what minimum cuts mean

   Minimum cuts means the smallest number of splits to make all parts palindromes.

3. Final Answer:

   The minimum number of cuts needed to partition s into palindromic substrings -> Option A

4. Quick Check:

   Minimum cuts = minimum splits for palindrome parts [OK]

Hint: Focus on splitting string into palindrome parts with fewest cuts [OK]

Common Mistakes:

• Confusing minimum cuts with longest palindrome substring
• Counting total palindrome substrings instead of cuts
• Thinking about maximum palindrome length

2. Which of the following TypeScript function signatures correctly represents a function to compute minimum palindrome partition cuts for a string s?

easy

A. function minCut(s: string): string[]

B. function minCut(s: number): string

C. function minCut(s: string): number

D. function minCut(s: boolean): number

Solution

1. Step 1: Identify input and output types

   Input is a string s, output is a number representing minimum cuts.

2. Step 2: Match correct signature

   Only function minCut(s: string): number matches input string and numeric output.

3. Final Answer:

   function minCut(s: string): number -> Option C

4. Quick Check:

   Input string, output number for cuts [OK]

Hint: Input string, output number for minimum cuts [OK]

Common Mistakes:

• Using wrong input type like number or boolean
• Returning array or string instead of number
• Mixing input and output types

3. What is the output of the following TypeScript code for input "aab"?

function minCut(s: string): number {
  const n = s.length;
  const dp = new Array(n).fill(0);
  const palindrome = Array.from({ length: n }, () => new Array(n).fill(false));

  for (let i = 0; i < n; i++) {
    let minCuts = i;
    for (let j = 0; j <= i; j++) {
      if (s[j] === s[i] && (i - j < 2 || palindrome[j + 1][i - 1])) {
        palindrome[j][i] = true;
        minCuts = j === 0 ? 0 : Math.min(minCuts, dp[j - 1] + 1);
      }
    }
    dp[i] = minCuts;
  }
  return dp[n - 1];
}
console.log(minCut("aab"));

medium

A. 0

B. 1

C. 2

D. 3

Solution

1. Step 1: Trace palindrome checks for "aab"

   Substrings: "a" (palindrome), "aa" (palindrome), "b" (palindrome). Minimum cuts for "aab" is 1 (cut between "aa" and "b").

2. Step 2: Check dp array updates

   dp[0]=0 ("a"), dp[1]=0 ("aa"), dp[2]=1 (cut before "b").

3. Final Answer:

   1 -> Option B

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindrome substrings and update dp for minimum cuts [OK]

Common Mistakes:

• Assuming no cuts needed for "aab"
• Counting total palindromes instead of cuts
• Incorrect dp index updates

4. Identify the bug in this TypeScript code snippet for minimum palindrome partition cuts:

function minCut(s: string): number {
  const n = s.length;
  const dp = new Array(n).fill(0);
  const palindrome = Array.from({ length: n }, () => new Array(n).fill(false));

  for (let i = 0; i < n; i++) {
    dp[i] = i;
    for (let j = 0; j <= i; j++) {
      if (s[j] === s[i] && (i - j < 2 || palindrome[j + 1][i - 1])) {
        palindrome[j][i] = true;
        dp[i] = Math.min(dp[i], dp[j - 1] + 1);
      }
    }
  }
  return dp[n - 1];
}

medium

A. dp[i] should be initialized to 0 instead of i

B. Inner loop should run from j = 0 to j <= i

C. palindrome array initialization is incorrect

D. dp[j] should be dp[j - 1] when updating dp[i]

Solution

1. Step 1: Analyze dp update logic

   When substring s[j..i] is palindrome, dp[i] should consider dp[j-1] + 1 cuts, not dp[j] + 1.

2. Step 2: Identify correct index for dp

   dp[j-1] represents cuts before substring start; dp[j] is incorrect and causes off-by-one error.

3. Final Answer:

   dp[j] should be dp[j - 1] when updating dp[i] -> Option D

4. Quick Check:

   Use dp[j-1] for correct minimum cuts [OK]

Hint: Use dp[j-1] when substring starts at j to avoid off-by-one error [OK]

Common Mistakes:

• Using dp[j] instead of dp[j-1]
• Incorrect loop bounds causing missed palindromes
• Wrong initialization of dp array

5. Given the string "abccbc", what is the minimum number of cuts needed to partition it into palindromic substrings using the DP approach?

hard

A. 2

B. 1

C. 3

D. 0

Solution

1. Step 1: Identify palindromic substrings in "abccbc"

   Palindromes include "a", "b", "cc", "bcb", "c". Partitioning as "a" | "bccb" | "c" is possible with 2 cuts.

2. Step 2: Calculate minimum cuts

   Minimum cuts = 2 because splitting into "a" | "bccb" | "c" yields all palindromes with 2 cuts.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Minimum cuts for "abccbc" = 2 [OK]

Hint: Look for largest palindrome substrings to minimize cuts [OK]

Common Mistakes:

• Assuming 0 cuts because string is not fully palindrome
• Counting each character as separate palindrome causing more cuts
• Missing palindrome "bccb" in middle