Palindrome Partitioning DP Minimum Cuts in DSA Typescript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the goal of the Palindrome Partitioning problem?

To split a string into the fewest possible parts so that each part is a palindrome.

beginner

What does the 'minimum cuts' mean in Palindrome Partitioning?

It means the smallest number of cuts needed to divide the string into palindrome parts.

intermediate

How does dynamic programming help in Palindrome Partitioning?

It stores results of smaller parts (substrings) to avoid repeated work and find minimum cuts efficiently.

intermediate

What is the role of the 'isPalindrome' table in the DP solution?

It quickly tells if a substring is a palindrome, so we don't check repeatedly during partitioning.

intermediate

In the DP approach, what does the array 'cuts' represent?

cuts[i] stores the minimum cuts needed for the substring from start to index i.

What is the minimum cuts needed for a string that is already a palindrome?

A 0

B 1

C Length of string - 1

D Length of string

If the whole string is a palindrome, no cuts are needed.

Which data structure helps to avoid repeated palindrome checks in DP?

A Stack

B Queue

C 2D boolean table

D Linked list

A 2D boolean table stores palindrome status for substrings.

What does cuts[i] represent in the DP solution?

A Length of substring s[0..i]

B Minimum cuts for substring s[0..i]

C Number of palindromes in s[0..i]

D Maximum cuts for substring s[i..end]

cuts[i] stores minimum cuts needed for substring from start to i.

If substring s[j..i] is palindrome, how is cuts[i] updated?

A cuts[i] = cuts[j-1] + 1

B cuts[i] = cuts[j] + 1

C cuts[i] = cuts[i-1] + 1

D cuts[i] = cuts[j-1] - 1

We add one cut after position j-1 to separate palindrome s[j..i].

What is the time complexity of the DP solution for Palindrome Partitioning minimum cuts?

A O(n)

B O(n^3)

C O(n log n)

D O(n^2)

The DP solution runs in O(n^2) time using palindrome table and cuts array.

Explain how dynamic programming is used to find minimum cuts for palindrome partitioning.

Think about storing results for substrings and building up the answer.

Describe the role of the 'isPalindrome' 2D table and the 'cuts' array in the solution.

One helps check palindromes fast, the other tracks 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