Palindrome Partitioning DP Minimum Cuts in DSA C - 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.

intermediate

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

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

intermediate

How do you check if a substring is a palindrome efficiently in DP?

Use a 2D boolean table where table[i][j] is true if substring from i to j is palindrome, built using smaller substrings results.

intermediate

Why do we add 1 to the minimum cuts when a palindrome is found in the substring?

Because cutting before the palindrome substring creates one more partition, so we add 1 to the cuts needed before that substring.

beginner

What is the base case for the minimum cuts DP array in Palindrome Partitioning?

If the substring from start to current index is a palindrome, minimum cuts is 0 because no cut is needed.

What does a minimum cut represent in Palindrome Partitioning?

A Number of cuts to split string into palindromes

B Length of the longest palindrome

C Number of palindromes in the string

D Total characters in the string

Minimum cuts are the fewest cuts needed to split the string into palindrome parts.

Which data structure helps to check palindrome substrings efficiently?

A Queue

B 2D boolean table

C Stack

D Linked list

A 2D boolean table stores palindrome status for all substrings.

If substring s[0..j] is palindrome, what is the minimum cut count at j?

A j

B 1

C 0

D j-1

No cuts needed if the whole substring is palindrome.

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

A O(log n)

B O(n)

C O(n^3)

D O(n^2)

Checking palindrome substrings and minimum cuts both take O(n^2) time.

When updating minimum cuts at index j, what do you add to the cuts at i-1?

A 1

B 0

C j

D i

Add 1 cut before the palindrome substring starting at i.

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

Think about breaking the problem into smaller parts and reusing results.

Describe the steps to check if a substring is palindrome using DP.

Start from smallest substrings and build up.

Practice

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

easy

A. The total number of substrings in the string

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

C. The length of the longest palindromic substring

D. The maximum number of palindromic 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 to divide the string so that every piece is a palindrome.

3. Final Answer:

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

4. Quick Check:

   Minimum cuts = smallest cuts for palindrome parts [OK]

Hint: Minimum cuts split string into 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 is the correct way to declare a 2D boolean array isPalindrome of size n x n in C for palindrome checks?

easy

A. bool isPalindrome[n][n];

B. bool isPalindrome = new bool[n][n];

C. bool* isPalindrome = malloc(n * n * sizeof(bool));

D. bool isPalindrome[n];

Solution

1. Step 1: Understand 2D array declaration in C

   In C, a 2D array of booleans with size n x n is declared as bool isPalindrome[n][n];

2. Step 2: Check other options

   bool isPalindrome = new bool[n][n]; uses Java-like syntax, bool* isPalindrome = malloc(n * n * sizeof(bool)); is a pointer but not a 2D array, bool isPalindrome[n]; is 1D array.

3. Final Answer:

   bool isPalindrome[n][n]; -> Option A

4. Quick Check:

   2D array in C = bool isPalindrome[n][n]; [OK]

Hint: Use bool isPalindrome[n][n]; for 2D boolean arrays in C [OK]

Common Mistakes:

• Using Java or C++ syntax in C
• Declaring 1D array instead of 2D
• Using malloc without proper pointer handling

3. Consider the following C code snippet for minimum cuts palindrome partitioning. What is the output for input string "aab"?

int minCut(char* s) {
    int n = strlen(s);
    int cuts[n+1];
    bool isPal[n][n];
    for (int i = 0; i < n; i++) {
        cuts[i] = i - 1;
        for (int j = 0; j <= i; j++) {
            if (s[j] == s[i] && (i - j < 2 || isPal[j+1][i-1])) {
                isPal[j][i] = true;
                cuts[i] = j == 0 ? 0 : fmin(cuts[i], cuts[j-1] + 1);
            } else {
                isPal[j][i] = false;
            }
        }
    }
    return cuts[n-1];
}
printf("%d", minCut("aab"));

medium

A. 2

B. 0

C. 1

D. 3

Solution

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

   For i=0 ('a'): cuts[0]= -1 initially, palindrome is 'a', cuts[0]=0.
   For i=1 ('a'): substring 'aa' is palindrome, cuts[1]=0.
   For i=2 ('b'): 'b' is palindrome, cuts[2] = min(cuts[2], cuts[1]+1) = 1.

2. Step 2: Final minimum cuts calculation

   Minimum cuts needed to partition "aab" into palindromes is 1 (split as "aa" | "b").

3. Final Answer:

   1 -> Option C

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Trace cuts array updates for each palindrome end [OK]

Common Mistakes:

• Assuming no cuts needed for non-palindromic parts
• Misunderstanding substring palindrome checks
• Incorrectly initializing cuts array

4. The following code snippet for palindrome partitioning minimum cuts has a bug. What is the error?

int minCut(char* s) {
    int n = strlen(s);
    int cuts[n];
    bool isPal[n][n];
    for (int i = 0; i < n; i++) {
        cuts[i] = i;
        for (int j = 0; j <= i; j++) {
            if (s[j] == s[i] && (i - j < 2 || isPal[j+1][i-1])) {
                isPal[j][i] = true;
                cuts[i] = j == 0 ? 0 : fmin(cuts[i], cuts[j-1] + 1);
            } else {
                isPal[j][i] = false;
            }
        }
    }
    return cuts[n-1];
}

medium

A. The condition (i - j < 2) is incorrect

B. isPal array is not initialized before use

C. The function returns cuts[0] instead of cuts[n-1]

D. cuts array should be of size n+1 and initialized with i-1

Solution

1. Step 1: Check cuts array size and initialization

   cuts array should be size n+1 to handle base case and initialized with i-1 to represent max cuts.

2. Step 2: Identify impact of incorrect cuts initialization

   Initializing cuts[i] = i causes wrong minimum cuts calculation because it doesn't consider zero-based indexing properly.

3. Final Answer:

   cuts array should be of size n+1 and initialized with i-1 -> Option D

4. Quick Check:

   Correct cuts size and init = n+1 and i-1 [OK]

Hint: Initialize cuts with i-1 and size n+1 [OK]

Common Mistakes:

• Using cuts size n instead of n+1
• Not initializing cuts properly
• Ignoring base case for cuts

5. Given the string "abcbm", what is the minimum number of cuts needed to partition it into palindromic substrings using dynamic programming?

hard

A. 2

B. 1

C. 3

D. 0

Solution

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

   "a" is palindrome, "bcb" is palindrome, "m" is palindrome.

2. Step 2: Calculate minimum cuts

   Partition as "a" | "bcb" | "m" requires 2 cuts.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Minimum cuts for "abcbm" = 2 [OK]

Hint: Look for longest palindromes to reduce cuts [OK]

Common Mistakes:

• Counting cuts as number of palindromes
• Missing middle palindrome "bcb"
• Assuming zero cuts for non-palindromic string