Palindrome Partitioning DP Minimum Cuts in DSA C++ - Build from Scratch

Palindrome Partitioning DP Minimum Cuts

📖 Scenario: Imagine you have a word and you want to cut it into pieces so that every piece is a palindrome. You want to make as few cuts as possible.This is useful in text processing and understanding patterns in strings.

🎯 Goal: Build a program that finds the minimum number of cuts needed to split a given string into palindrome parts using dynamic programming.

📋 What You'll Learn

Create a string variable s with the exact value "aab"

Create a 2D boolean vector isPalindrome to store palindrome checks

Create a vector cuts to store minimum cuts needed for substrings

Use nested loops to fill isPalindrome for all substrings

Use a loop to calculate minimum cuts using isPalindrome

Print the minimum cuts needed for the whole string

💡 Why This Matters

🌍 Real World

Palindrome partitioning helps in text analysis, DNA sequence analysis, and data compression where palindromic patterns are important.

💼 Career

Understanding dynamic programming and string manipulation is essential for software engineering roles involving algorithms and problem solving.

1

Create the input string

Create a string variable called s and set it to "aab".

DSA C++

# Your code here

Hint

Use std::string to create the string variable s.

2

Setup palindrome check table and cuts vector

Create a 2D boolean vector called isPalindrome with size s.size() by s.size() initialized to false. Also create a vector of integers called cuts with size s.size() initialized to 0.

DSA C++

std::string s = "aab";
// Your code here

Hint

Use std::vector to create isPalindrome and cuts with the correct sizes.

3

Fill palindrome table and calculate minimum cuts

Use a nested for loop with variables end and start to fill isPalindrome for all substrings of s. Then use a for loop with variable i to calculate minimum cuts in cuts using isPalindrome. Use cuts[i] = 0 if the substring from 0 to i is a palindrome, otherwise find the minimum cuts by checking all possible partitions.

DSA C++

std::string s = "aab";
int n = s.size();
std::vector> isPalindrome(n, std::vector(n, false));
std::vector cuts(n, 0);
// Your code here

Hint

Use the palindrome property: a substring is palindrome if the first and last characters are equal and the inside substring is palindrome or length is less than 2.

Calculate minimum cuts by checking all partitions.

4

Print the minimum cuts needed

Print the minimum cuts needed for the whole string by printing cuts[s.size() - 1].

DSA C++

std::string s = "aab";
int n = s.size();
std::vector> isPalindrome(n, std::vector(n, false));
std::vector cuts(n, 0);

for (int end = 0; end < n; ++end) {
    for (int start = 0; start <= end; ++start) {
        if (s[start] == s[end] && (end - start < 2 || isPalindrome[start + 1][end - 1])) {
            isPalindrome[start][end] = true;
        }
    }
}

for (int i = 0; i < n; ++i) {
    if (isPalindrome[0][i]) {
        cuts[i] = 0;
    } else {
        int minCuts = i;
        for (int j = 1; j <= i; ++j) {
            if (isPalindrome[j][i] && cuts[j - 1] + 1 < minCuts) {
                minCuts = cuts[j - 1] + 1;
            }
        }
        cuts[i] = minCuts;
    }
}
// Your code here

Hint

Print the last element of cuts vector which holds the minimum cuts for the whole string.

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