Palindrome Partitioning DP Minimum Cuts in DSA C++ - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of minimum cuts for palindrome partitioning

What is the output of the following C++ code that calculates the minimum cuts needed to partition the string into palindromes?

DSA C++

int minCut(string s) {
    int n = s.size();
    vector<vector<bool>> dp(n, vector<bool>(n, false));
    vector<int> cuts(n, 0);
    for (int i = 0; i < n; i++) {
        int minCut = i;
        for (int j = 0; j <= i; j++) {
            if (s[j] == s[i] && (i - j < 2 || dp[j + 1][i - 1])) {
                dp[j][i] = true;
                minCut = j == 0 ? 0 : min(minCut, cuts[j - 1] + 1);
            }
        }
        cuts[i] = minCut;
    }
    return cuts[n - 1];
}

int main() {
    string s = "aab";
    cout << minCut(s) << endl;
    return 0;
}

A 1

B 0

C 2

D 3

💡 Hint

Think about how many cuts are needed to split "aab" into palindromes.

The string "aab" can be partitioned as "aa" | "b" with 1 cut. So the minimum cuts needed is 1.

🧠 Conceptual

intermediate

Understanding the DP table in palindrome partitioning

In the palindrome partitioning DP solution, what does the 2D boolean table dp[i][j] represent?

A The number of palindrome partitions between i and j

B The minimum cuts needed for substring from i to j

C The length of the longest palindrome substring between i and j

D Whether the substring from index i to j is a palindrome

💡 Hint

Think about what information helps decide if a substring is palindrome.

dp[i][j] is true if the substring s[i..j] is a palindrome, false otherwise. This helps in deciding cuts.

🔧 Debug

advanced

Identify the error in the DP palindrome partitioning code

What error will the following code produce when run?

DSA C++

int minCut(string s) {
    int n = s.size();
    vector<vector<bool>> dp(n, vector<bool>(n, false));
    vector<int> cuts(n, 0);
    for (int i = 0; i < n; i++) {
        int minCut = i;
        for (int j = 0; j <= i; j++) {
            if (s[j] == s[i] && (i - j < 2 || dp[j + 1][i - 1])) {
                dp[j][i] = true;
                minCut = j == 0 ? 0 : min(minCut, cuts[j]);
            }
        }
        cuts[i] = minCut;
    }
    return cuts[n - 1];
}

int main() {
    string s = "aab";
    cout << minCut(s) << endl;
    return 0;
}

A Out of range error accessing cuts[j]

B No error, outputs 1

C Runtime error due to invalid dp access

D Compilation error due to missing header

💡 Hint

Check the index used to access cuts vector inside the inner loop.

The code uses cuts[j] where j can be 0, but cuts[j-1] is needed to avoid out of range when j=0.

🚀 Application

advanced

Minimum cuts for palindrome partitioning of a complex string

What is the minimum number of cuts needed to partition the string "ababbbabbababa" into palindromes?

A 3

B 5

C 4

D 6

💡 Hint

Try to find palindrome substrings and count cuts between them.

The minimum cuts needed is 4 for this string based on palindrome partitions.

🧠 Conceptual

expert

Time complexity of palindrome partitioning minimum cuts DP

What is the time complexity of the dynamic programming solution for palindrome partitioning minimum cuts on a string of length n?

A O(n^2)

B O(n^3)

C O(n log n)

D O(n)

💡 Hint

Consider the nested loops and palindrome checks.

The DP solution uses nested loops and palindrome checks leading to O(n^3) time complexity.

Practice

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

easy

A. The longest palindrome that can be formed by rearranging s.

B. The maximum length of a palindrome substring in s.

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

D. The total number of palindrome substrings 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 between characters so that all resulting substrings are palindromes.

3. Final Answer:

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

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 maximum palindrome length
• Counting total palindrome substrings instead of cuts
• Thinking about rearranging characters

2. Which of the following C++ code snippets correctly initializes a 2D vector isPalindrome of size n x n with false values for palindrome checks?

easy

A. std::vector<bool> isPalindrome(n, std::vector<bool>(n, false));

B. std::vector isPalindrome(n)(n, false);

C. std::vector isPalindrome(n, n, false);

D. std::vector isPalindrome(n, std::vector<bool>(n, false));

Solution

1. Step 1: Recall correct 2D vector initialization syntax

   In C++, a 2D vector is initialized as vector of vectors: vector v(rows, vector<bool>(cols, value)).

2. Step 2: Check each option

   std::vector isPalindrome(n, std::vector<bool>(n, false)); matches correct syntax. Options A, C, D have syntax errors or wrong types.

3. Final Answer:

   std::vector isPalindrome(n, std::vector<bool>(n, false)); -> Option D

4. Quick Check:

   Correct 2D vector init = std::vector isPalindrome(n, std::vector<bool>(n, false)); [OK]

Hint: Use vector of vectors with inner vector size and default value [OK]

Common Mistakes:

• Using wrong vector types or missing inner vector
• Trying to initialize 2D vector with wrong constructor
• Confusing parentheses and commas in initialization

3. What is the output of the following C++ code snippet that calculates minimum cuts for s = "aab"?

std::string s = "aab";
int n = s.size();
std::vector isPalindrome(n, std::vector<bool>(n, false));
for (int i = 0; i < n; i++) {
  isPalindrome[i][i] = true;
}
for (int length = 2; length <= n; length++) {
  for (int start = 0; start <= n - length; start++) {
    int end = start + length - 1;
    if (s[start] == s[end]) {
      if (length == 2) isPalindrome[start][end] = true;
      else isPalindrome[start][end] = isPalindrome[start + 1][end - 1];
    }
  }
}
std::vector<int> cuts(n, 0);
for (int i = 0; i < n; i++) {
  int minCut = i;
  for (int j = 0; j <= i; j++) {
    if (isPalindrome[j][i]) {
      minCut = j == 0 ? 0 : std::min(minCut, cuts[j - 1] + 1);
    }
  }
  cuts[i] = minCut;
}
std::cout << cuts[n - 1] << std::endl;

medium

A. 1

B. 0

C. 2

D. 3

Solution

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

   "a", "a", "b" are palindromes; "aa" is palindrome; "aab" is not.

2. Step 2: Calculate minimum cuts using DP

   Minimum cuts: "aa" | "b" needs 1 cut. So cuts[n-1] = 1.

3. Final Answer:

   1 -> Option A

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindrome substrings and count cuts between them [OK]

Common Mistakes:

• Assuming no cuts needed for non-palindrome whole string
• Misindexing palindrome checks
• Forgetting to update minimum cuts properly

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

for (int i = 0; i < n; i++) {
  int minCut = i;
  for (int j = 0; j <= i; j++) {
    if (isPalindrome[j][i]) {
      minCut = std::min(minCut, cuts[j] + 1);
    }
  }
  cuts[i] = minCut;
}

medium

A. Using cuts[j] instead of cuts[j - 1] causes incorrect indexing.

B. The outer loop should start from 1, not 0.

C. The inner loop should run from i to n, not 0 to i.

D. The isPalindrome check should be reversed to isPalindrome[i][j].

Solution

1. Step 1: Analyze the indexing in cuts array

   When j == 0, cuts[j - 1] is invalid, so we must handle j == 0 separately.

2. Step 2: Identify the bug in minCut update

   Using cuts[j] + 1 is wrong; it should be cuts[j - 1] + 1 except when j == 0.

3. Final Answer:

   Using cuts[j] instead of cuts[j - 1] causes incorrect indexing. -> Option A

4. Quick Check:

   Indexing cuts[j - 1] correctly fixes bug [OK]

Hint: Remember cuts[j - 1] for previous partition, handle j=0 separately [OK]

Common Mistakes:

• Ignoring boundary condition j=0
• Mixing up indices in cuts array
• Incorrect loop ranges

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

hard

A. 5

B. 3

C. 4

D. 6

Solution

1. Step 1: Understand the string and palindrome partitions

   The string "ababbbabbababa" can be partitioned into palindromes with minimum cuts by checking all substrings using DP.

2. Step 2: Use DP to find minimum cuts

   Known solution for this string is 3 cuts, for example: "a" | "babbbab" | "b" | "ababa".

3. Final Answer:

   3 -> Option B

4. Quick Check:

   Minimum cuts for given string = 3 [OK]

Hint: Use DP palindrome checks and track minimum cuts carefully [OK]

Common Mistakes:

• Underestimating cuts by missing longer palindrome substrings
• Overcounting cuts by splitting too often
• Not using DP to optimize palindrome checks