Palindrome Partitioning DP Minimum Cuts in DSA C#

Mental Model

We want to split a string into parts where each part reads the same forwards and backwards, using the fewest splits possible.

Analogy: Imagine cutting a ribbon into pieces so that each piece is a perfect mirror image of itself, and you want to make as few cuts as possible.

s: a b a c d c b a
cuts: [0, ?, ?, ?, ?, ?, ?, ?]
palindrome check matrix: 2D grid showing which substrings are palindromes

Dry Run Walkthrough

Input: string: "abacdcba"

Goal: Find the minimum number of cuts needed to split the string so every piece is a palindrome

Step 1: Check all substrings of length 1 and mark them as palindrome

palindrome matrix marks single letters: a, b, a, c, d, c, b, a as true

Why: Every single letter is a palindrome by itself

Step 2: Check substrings of length 2 and mark palindromes

palindrome matrix marks "ba", "ac", "cd", "dc", "cb", "ba" as false; "aa" or "cc" if present would be true

Why: Two letters are palindrome only if they are the same

Step 3: Check substrings of length 3 and longer, update palindrome matrix

palindrome matrix marks "aba" and "cdc" as true, others false

Why: A substring is palindrome if ends match and inner substring is palindrome

Step 4: Initialize cuts array with max cuts (index)

cuts = [0,1,2,3,4,5,6,7]

Why: Worst case: cut before every character

Step 5: For each position, update cuts if substring from start is palindrome

cuts updated to fewer cuts where palindrome substrings found, e.g. cuts[2] = 0 because "aba" is palindrome

Why: No cut needed if substring from start is palindrome

Step 6: For each position, check all partitions and update cuts with minimum

cuts updated to minimum cuts needed, e.g. cuts[7] = 3 for "abacdcba"

Why: We find minimum cuts by checking all palindrome partitions

Result:

cuts array: [0,1,0,1,2,1,2,3]
Minimum cuts needed: 3

Annotated Code

DSA C#

using System;

class PalindromePartitioning {
    public static int MinCut(string s) {
        int n = s.Length;
        if (n == 0) return 0;
        bool[,] palindrome = new bool[n, n];
        int[] cuts = new int[n];

        // Mark all substrings of length 1 as palindrome
        for (int i = 0; i < n; i++) {
            palindrome[i, i] = true;
        }

        // Check substrings of length 2 and more
        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) {
                        palindrome[start, end] = true; // two equal chars
                    } else {
                        palindrome[start, end] = palindrome[start + 1, end - 1]; // inner substring palindrome
                    }
                } else {
                    palindrome[start, end] = false;
                }
            }
        }

        // Initialize cuts array with max cuts
        for (int i = 0; i < n; i++) {
            cuts[i] = i; // max cuts: cut before every char
        }

        // Calculate minimum cuts
        for (int i = 0; i < n; i++) {
            if (palindrome[0, i]) {
                cuts[i] = 0; // no cut needed if substring 0..i is palindrome
            } else {
                for (int j = 0; j < i; j++) {
                    if (palindrome[j + 1, i] && cuts[j] + 1 < cuts[i]) {
                        cuts[i] = cuts[j] + 1; // update min cuts
                    }
                }
            }
        }

        return cuts[n - 1];
    }

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

palindrome[i, i] = true;

Mark single characters as palindrome

if (s[start] == s[end]) { ... palindrome[start, end] = palindrome[start + 1, end - 1]; }

Mark substring palindrome if ends match and inner substring palindrome

cuts[i] = i;

Initialize cuts with max possible cuts

if (palindrome[0, i]) { cuts[i] = 0; }

No cut needed if substring from start is palindrome

if (palindrome[j + 1, i] && cuts[j] + 1 < cuts[i]) { cuts[i] = cuts[j] + 1; }

Update cuts with minimum by checking all palindrome partitions

Output

3

Complexity Analysis

Time: O(n^2) because we check all substrings and update cuts for each position

Space: O(n^2) for palindrome matrix and O(n) for cuts array

vs Alternative: Naive recursive approach is exponential; DP reduces it to polynomial time

Edge Cases

Empty string

Returns 0 cuts since no partitions needed

DSA C#

int n = s.Length; if n == 0 then cuts array is empty and function returns 0

String with all same characters, e.g. "aaaa"

Returns 0 cuts since whole string is palindrome

DSA C#

if (palindrome[0, i]) cuts[i] = 0;

String with no palindrome longer than 1, e.g. "abcd"

Returns n-1 cuts, cutting before every character

DSA C#

cuts[i] = i; initialization ensures max cuts

When to Use This Pattern

When asked to split a string into palindromic parts with minimum cuts, use DP with palindrome substring precomputation to optimize.

Common Mistakes

Mistake: Not precomputing palindrome substrings and checking palindrome on the fly causing repeated work
Fix: Use a DP table to store palindrome results for all substrings before calculating cuts

Mistake: Incorrectly initializing cuts array leading to wrong minimum cuts
Fix: Initialize cuts[i] = i to represent max cuts before optimization

Summary

Finds the minimum cuts needed to partition a string so every part is a palindrome.

Use when you need to split strings into palindromic substrings with fewest cuts.

Precompute palindrome substrings and use DP to find minimum cuts efficiently.

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