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

Recall & Review

beginner

What is the goal of the Palindrome Partitioning problem with minimum cuts?

To split a given string into the fewest possible substrings such that each substring is a palindrome.

beginner

What does the DP array represent in the Palindrome Partitioning Minimum Cuts solution?

It stores the minimum number of cuts needed to partition the substring from the start up to each index into palindromes.

intermediate

How do we check if a substring is a palindrome efficiently in this DP approach?

We use a 2D boolean array to store palindrome status for substrings, so we don't recompute it repeatedly.

intermediate

Why do we initialize the minimum cuts array with the maximum possible cuts initially?

Because the worst case is cutting before every character, so initializing with max cuts helps us find the minimum by comparison.

intermediate

What is the time complexity of the Palindrome Partitioning Minimum Cuts DP solution?

O(n^2), where n is the length of the string, due to checking all substrings and updating the DP arrays.

What does a 'cut' mean in the Palindrome Partitioning problem?

A Splitting the string into two parts

B Removing a character from the string

C Reversing a substring

D Checking if a substring is palindrome

A cut means splitting the string into two parts to form substrings.

Which data structure helps avoid repeated palindrome checks in DP?

A 2D boolean array

B Queue

C Stack

D 1D integer array

A 2D boolean array stores palindrome status for substrings to avoid repeated checks.

If the entire string is already a palindrome, what is the minimum number of cuts?

A Length of string - 1

B 1

C 0

D Length of string

No cuts are needed if the whole string is a palindrome.

What is the initial value assigned to the minimum cuts array for each position?

A 0

B Maximum possible cuts (index value)

C Length of string

D Index value

We initialize with the maximum cuts possible (index value) to find the minimum later.

Which approach is used to solve the Palindrome Partitioning Minimum Cuts problem?

A Backtracking only

B Divide and Conquer

C Greedy

D Dynamic Programming

Dynamic Programming is used to store and reuse results for minimum cuts.

Explain how dynamic programming helps find the minimum cuts for palindrome partitioning.

Think about how you can build the solution step-by-step using smaller parts.

Describe the role of the palindrome boolean table in optimizing the solution.

Consider how checking palindromes repeatedly can slow down the solution.

Practice

1. What does the term minimum cuts mean in the context of palindrome partitioning of a string?

easy

A. The smallest number of cuts needed to split the string into palindromic substrings

B. The maximum number of palindromic substrings in the string

C. The length of the longest palindromic substring

D. The total number of substrings in the string

Solution

1. Step 1: Understand palindrome partitioning

   Palindrome partitioning means splitting a string into parts where each part is a palindrome.

2. Step 2: Define minimum cuts

   Minimum cuts is the least number of cuts needed so that every substring after cutting is a palindrome.

3. Final Answer:

   The smallest number of cuts needed to split the string into palindromic substrings -> Option A

4. Quick Check:

   Minimum cuts = smallest cuts to get palindromes [OK]

Hint: Minimum cuts = least splits for palindrome parts [OK]

Common Mistakes:

• Confusing minimum cuts with maximum palindromes
• Thinking minimum cuts means longest palindrome length
• Counting total substrings instead of cuts

2. Which of the following Java code snippets correctly initializes the DP array for minimum cuts in palindrome partitioning of a string s of length n?

easy

A. int[] cuts = new int[n]; Arrays.fill(cuts, -1);

B. int[] cuts = new int[n]; for (int i = 0; i <= n; i++) cuts[i] = 0;

C. int[] cuts = new int[n]; for (int i = 0; i < n; i++) cuts[i] = i;

D. int[] cuts = new int[n+1]; Arrays.fill(cuts, Integer.MAX_VALUE);

Solution

1. Step 1: Understand DP array initialization

   We initialize cuts array where cuts[i] = i means max cuts needed for substring s[0..i].

2. Step 2: Check code correctness

   int[] cuts = new int[n]; for (int i = 0; i < n; i++) cuts[i] = i; initializes cuts array of size n and sets cuts[i] = i correctly. Other options have errors like wrong size or wrong initialization.

3. Final Answer:

   int[] cuts = new int[n]; for (int i = 0; i < n; i++) cuts[i] = i; -> Option C

4. Quick Check:

   Initialize cuts with max cuts = index [OK]

Hint: Initialize cuts[i] = i for max cuts before optimization [OK]

Common Mistakes:

• Using wrong array size causing IndexOutOfBounds
• Filling cuts with zero instead of max cuts
• Using negative values which are invalid

3. What is the output of the following Java code snippet for string s = "aab"?

public class Solution {
  public static int minCut(String s) {
    int n = s.length();
    int[] cuts = new int[n];
    boolean[][] palindrome = new boolean[n][n];
    for (int i = 0; i < n; i++) cuts[i] = i;
    for (int end = 0; end < n; end++) {
      for (int start = 0; start <= end; start++) {
        if (s.charAt(start) == s.charAt(end) && (end - start < 2 || palindrome[start + 1][end - 1])) {
          palindrome[start][end] = true;
          cuts[end] = start == 0 ? 0 : Math.min(cuts[end], cuts[start - 1] + 1);
        }
      }
    }
    return cuts[n - 1];
  }
  public static void main(String[] args) {
    System.out.println(minCut("aab"));
  }
}

medium

A. 0

B. 1

C. 2

D. 3

Solution

1. Step 1: Analyze palindrome substrings in "aab"

   "a", "a", "b" are palindromes. "aa" is palindrome. Minimum cuts needed is 1 (cut between "aa" and "b").

2. Step 2: Trace DP updates

   cuts array updates to [0,0,1]. Final cuts[n-1] = 1.

3. Final Answer:

   1 -> Option B

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindromes and update cuts with min previous cuts +1 [OK]

Common Mistakes:

• Assuming no cuts needed for "aab"
• Counting cuts as number of palindromes
• Ignoring palindrome "aa" substring

4. Identify the bug in the following Java code snippet for palindrome partitioning minimum cuts:

public int minCut(String s) {
  int n = s.length();
  int[] cuts = new int[n];
  boolean[][] palindrome = new boolean[n][n];
  for (int i = 0; i <= n; i++) cuts[i] = i;
  for (int end = 0; end < n; end++) {
    for (int start = 0; start <= end; start++) {
      if (s.charAt(start) == s.charAt(end) && (end - start < 2 || palindrome[start + 1][end - 1])) {
        palindrome[start][end] = true;
        cuts[end] = start == 0 ? 0 : Math.min(cuts[end], cuts[start - 1] + 1);
      }
    }
  }
  return cuts[n - 1];
}

medium

A. The return statement should be 'cuts[n]' instead of 'cuts[n - 1]'

B. The palindrome check condition is incorrect

C. The cuts array is not initialized properly

D. The loop 'for (int i = 0; i <= n; i++)' causes IndexOutOfBoundsException

Solution

1. Step 1: Check array initialization loop

   cuts array size is n, but loop runs i from 0 to n inclusive, causing IndexOutOfBoundsException at i = n.

2. Step 2: Confirm other parts are correct

   Palindrome check and return statement are correct. Initialization loop must be 'i < n'.

3. Final Answer:

   The loop 'for (int i = 0; i <= n; i++)' causes IndexOutOfBoundsException -> Option D

4. Quick Check:

   Array size n but loop runs to n inclusive causes error [OK]

Hint: Loop to n-1 for array size n to avoid out-of-bounds [OK]

Common Mistakes:

• Using <= n instead of < n in loops
• Misunderstanding palindrome condition
• Returning wrong index from cuts array

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

hard

A. 4

B. 5

C. 3

D. 6

Solution

1. Step 1: Understand the problem and DP approach

   We find minimum cuts so that each substring is palindrome. The string "ababbbabbababa" has palindromic partitions with 4 cuts.

2. Step 2: Trace or recall known result

   Known minimum cuts for this string is 4, splitting into palindromes like "a|babbbab|b|ababa" or similar.

3. Final Answer:

   4 -> Option A

4. Quick Check:

   Minimum cuts for "ababbbabbababa" = 4 [OK]

Hint: Use DP to find min cuts; known result for this string is 4 [OK]

Common Mistakes:

• Guessing cuts without checking palindrome substrings
• Counting number of palindromes instead of cuts
• Confusing cuts with substring length