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#

using System;

class Program {
    static bool IsPalindrome(string s, int start, int end) {
        while (start < end) {
            if (s[start] != s[end]) return false;
            start++;
            end--;
        }
        return true;
    }

    static int MinCut(string s) {
        int n = s.Length;
        int[] dp = new int[n];
        for (int i = 0; i < n; i++) {
            dp[i] = i; // max cuts
            for (int j = 0; j <= i; j++) {
                if (IsPalindrome(s, j, i)) {
                    dp[i] = j == 0 ? 0 : Math.Min(dp[i], dp[j - 1] + 1);
                }
            }
        }
        return dp[n - 1];
    }

    static void Main() {
        string input = "aab";
        Console.WriteLine(MinCut(input));
    }
}

A 2

B 1

C 0

D 3

💡 Hint

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

The string "aab" can be split into "aa" and "b", both palindromes, with 1 cut.

❓ Predict Output

intermediate

Minimum cuts for palindrome partitioning of a longer string

What is the output of the following C# code for the input string "ababbbabbababa"?

DSA C#

using System;

class Program {
    static bool IsPalindrome(string s, int start, int end) {
        while (start < end) {
            if (s[start] != s[end]) return false;
            start++;
            end--;
        }
        return true;
    }

    static int MinCut(string s) {
        int n = s.Length;
        int[] dp = new int[n];
        for (int i = 0; i < n; i++) {
            dp[i] = i;
            for (int j = 0; j <= i; j++) {
                if (IsPalindrome(s, j, i)) {
                    dp[i] = j == 0 ? 0 : Math.Min(dp[i], dp[j - 1] + 1);
                }
            }
        }
        return dp[n - 1];
    }

    static void Main() {
        string input = "ababbbabbababa";
        Console.WriteLine(MinCut(input));
    }
}

A 4

B 5

C 3

D 2

💡 Hint

Try to find the minimum cuts by checking palindrome substrings carefully.

The minimum cuts for "ababbbabbababa" is 2, splitting into palindromes like "a", "babbbab", "bababa".

🧠 Conceptual

advanced

Understanding the DP array in palindrome partitioning

In the palindrome partitioning minimum cuts algorithm, what does the dp array represent at index i?

A The minimum number of cuts needed for the substring s[0..i] to be partitioned into palindromes

B The maximum number of palindrome substrings in s[0..i]

C The length of the longest palindrome substring ending at index i

D The total number of palindrome substrings in s[0..i]

💡 Hint

Think about what the algorithm tries to minimize at each position.

dp[i] stores the minimum cuts needed to partition the substring from start to i into palindromes.

❓ Predict Output

advanced

Output of minimum cuts for a palindrome string

What is the output of the following code for the input string "racecar"?

DSA C#

using System;

class Program {
    static bool IsPalindrome(string s, int start, int end) {
        while (start < end) {
            if (s[start] != s[end]) return false;
            start++;
            end--;
        }
        return true;
    }

    static int MinCut(string s) {
        int n = s.Length;
        int[] dp = new int[n];
        for (int i = 0; i < n; i++) {
            dp[i] = i;
            for (int j = 0; j <= i; j++) {
                if (IsPalindrome(s, j, i)) {
                    dp[i] = j == 0 ? 0 : Math.Min(dp[i], dp[j - 1] + 1);
                }
            }
        }
        return dp[n - 1];
    }

    static void Main() {
        string input = "racecar";
        Console.WriteLine(MinCut(input));
    }
}

A 6

B 1

C 0

D 3

💡 Hint

Is the entire string a palindrome?

Since "racecar" is a palindrome itself, no cuts are needed, so output is 0.

🧠 Conceptual

expert

Time complexity of palindrome partitioning minimum cuts algorithm

What is the time complexity of the standard dynamic programming solution for palindrome partitioning minimum cuts using the approach shown in the code?

A O(n^3)

B O(n^2)

C O(n)

D O(n^4)

💡 Hint

Consider the nested loops and palindrome checks.

The algorithm has two nested loops and a palindrome check inside, 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 minimum number of cuts needed to partition s into palindromic substrings

B. The longest palindromic substring in s

C. The total number of palindromic substrings in s

D. The maximum number of cuts possible 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 mean the smallest number of splits so that every substring after splitting is a palindrome.

3. Final Answer:

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

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 longest palindrome length
• Counting total palindromic substrings instead of cuts
• Thinking maximum cuts is the goal

2. Which of the following C# code snippets correctly initializes the DP array to store minimum cuts for a string s of length n?

easy

A. int[] cuts = new int[n + 1]; for (int i = 0; i < n; i++) cuts[i] = n;

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

C. int[] cuts = new int[n + 1]; for (int i = 0; i <= n; i++) cuts[i] = i - 1;

D. int[] cuts = new int[n + 1]; for (int i = 0; i <= n; i++) cuts[i] = 0;

Solution

1. Step 1: Understand DP array purpose

   The DP array stores minimum cuts needed for substring up to index i, so size is n+1.

2. Step 2: Initialize cuts array correctly

   Initialize cuts[i] = i - 1 means worst case: each character is a palindrome, so max cuts = i-1.

3. Final Answer:

   int[] cuts = new int[n + 1]; for (int i = 0; i <= n; i++) cuts[i] = i - 1; -> Option C

4. Quick Check:

   DP array size n+1, initialized to max cuts i-1 [OK]

Hint: Initialize cuts[i] = i - 1 for worst case cuts [OK]

Common Mistakes:

• Using array size n instead of n+1
• Initializing all cuts to zero
• Starting loop from 0 but using wrong index in initialization

3. What is the output of the following C# code snippet for input string "aab"?

string s = "aab";
int n = s.Length;
int[] cuts = new int[n + 1];
for (int i = 0; i <= n; i++) cuts[i] = i - 1;
for (int i = 1; i <= n; i++) {
    for (int j = 0; j < i; j++) {
        if (IsPalindrome(s, j, i - 1)) {
            cuts[i] = Math.Min(cuts[i], cuts[j] + 1);
        }
    }
}
Console.WriteLine(cuts[n]);

bool IsPalindrome(string str, int left, int right) {
    while (left < right) {
        if (str[left] != str[right]) return false;
        left++;
        right--;
    }
    return true;
}

medium

A. 0

B. 3

C. 2

D. 1

Solution

1. Step 1: Trace DP initialization and loops

   cuts array initialized as [-1,0,1,2] for indices 0 to 3. Then loops check palindromes and update cuts.

2. Step 2: Check palindrome partitions for "aab"

   Partitions: "aa" (palindrome) + "b" (palindrome). Minimum cuts = 1.

3. Final Answer:

   1 -> Option D

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindrome substrings and update cuts with minimum [OK]

Common Mistakes:

• Assuming no cuts needed for "aab"
• Counting total palindromic substrings instead of cuts
• Off-by-one errors in loops

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

string s = "abc";
int n = s.Length;
int[] cuts = new int[n + 1];
for (int i = 0; i < n; i++) cuts[i] = i - 1;
for (int i = 1; i <= n; i++) {
    for (int j = 0; j < i; j++) {
        if (IsPalindrome(s, j, i - 1)) {
            cuts[i] = Math.Min(cuts[i], cuts[j] + 1);
        }
    }
}
Console.WriteLine(cuts[n]);

bool IsPalindrome(string str, int left, int right) {
    while (left < right) {
        if (str[left] != str[right]) return false;
        left++;
        right--;
    }
    return true;
}

medium

A. The outer loop should start from 0 instead of 1

B. The cuts array is not initialized for index n, causing incorrect values

C. The IsPalindrome function has incorrect loop conditions

D. The inner loop should run until j <= i instead of j < i

Solution

1. Step 1: Check cuts array initialization

   The loop initializes cuts from i=0 to i<n, missing index n, so cuts[n] remains 0 default, incorrect.

2. Step 2: Understand impact on DP calculation

   Without initializing cuts[n], Math.Min compares with 0, causing wrong minimum cuts result.

3. Final Answer:

   The cuts array is not initialized for index n, causing incorrect values -> Option B

4. Quick Check:

   Initialize cuts array fully to avoid wrong DP values [OK]

Hint: Initialize cuts array fully from 0 to n inclusive [OK]

Common Mistakes:

• Not initializing cuts[n] index
• Incorrect loop ranges in palindrome check
• Misunderstanding loop start indices

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

hard

A. 3

B. 4

C. 5

D. 6

Solution

1. Step 1: Understand the DP approach for complex string

   Use DP to check all substrings and update minimum cuts for partitions where substrings are palindromes.

2. Step 2: Apply known result or simulate DP

   For "ababbbabbababa", minimum cuts found by DP is 3, splitting into palindromic parts like "a|babbbab|b|ababa".

3. Final Answer:

   3 -> Option A

4. Quick Check:

   DP minimum cuts for complex palindrome partition = 3 [OK]

Hint: Use DP to find minimum cuts by checking all palindrome substrings [OK]

Common Mistakes:

• Guessing cuts without DP verification
• Miscounting palindrome partitions
• Assuming cuts equal to half string length