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 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 DP?

By using a 2D boolean DP table where dp[i][j] is true if substring from i to j is palindrome, built using previously computed results.

intermediate

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

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

What does a minimum cut represent in palindrome partitioning?

A The number of palindromes in the string

B The longest palindrome substring

C The fewest splits to make all substrings palindromes

D The length of the shortest palindrome

Minimum cuts mean the fewest splits needed so that every piece is a palindrome.

Which data structure helps to quickly check if substrings are palindromes?

A Hash map

B 2D boolean DP table

C Queue

D Stack

A 2D boolean DP table stores palindrome status for substrings efficiently.

If the entire string is already a palindrome, what is the minimum cut count?

A Length of string - 1

B 1

C Length of string

D 0

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

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

A Index value (maximum cuts possible)

B Zero

C Negative one

D Length of string

We start with the maximum cuts possible, which is the index value.

What is the main reason to use dynamic programming in this problem?

A To avoid recomputing palindrome checks for substrings

B To sort the string

C To reverse the string

D To count characters

DP stores results of palindrome checks to avoid repeated work.

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

Think about how smaller problems combine to solve the bigger problem.

Describe the process to check if substrings are palindromes efficiently in this DP approach.

Consider how knowing smaller palindrome substrings helps check bigger ones.

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