Palindrome Partitioning DP Minimum Cuts in DSA C# - Execution Trace

Concept Flow - Palindrome Partitioning DP Minimum Cuts

Start with full string

↓

Check all substrings palindrome?

↓

If substring is palindrome

↓

Update min cuts for substring

↓

Move to next substring

↓

Repeat until full string processed

↓

Return min cuts for full string

We check all substrings for palindrome, update minimum cuts needed, and repeat until the entire string is processed.

Execution Sample

DSA C#

string s = "aab";
int n = s.Length;
int[] cuts = new int[n];
bool[,] pal = new bool[n,n];
// Fill pal and cuts arrays
// Return cuts[n-1]

This code finds the minimum cuts needed to partition the string into palindromes.

Execution Table

Step	Operation	Substring Checked	Is Palindrome?	Cuts Array State	Visual State
1	Initialize cuts and pal arrays	-	-	[0,0,0]	s: a a b cuts: 0 0 0 pal: all false
2	Check substring s[0..0]	"a"	Yes	[0,0,0]	pal[0,0] = true
3	Update cuts for s[0..0]	a	Palindrome	[0,0,0]	cuts[0] = 0
4	Check substring s[1..1]	"a"	Yes	[0,0,0]	pal[1,1] = true
5	Update cuts for s[1..1]	a	Palindrome	[0,0,0]	cuts[1] = 0
6	Check substring s[0..1]	"aa"	Yes	[0,0,0]	pal[0,1] = true
7	Update cuts for s[0..1]	aa	Palindrome	[0,0,0]	cuts[1] = 0 (no cut needed)
8	Check substring s[2..2]	"b"	Yes	[0,0,0]	pal[2,2] = true
9	Update cuts for s[2..2]	b	Palindrome	[0,0,0]	cuts[2] = 0
10	Check substring s[1..2]	ab	No	[0,0,0]	pal[1,2] = false
11	Update cuts for s[1..2]	ab	Not palindrome	[0,0,1]	cuts[2] = min(cuts[2], cuts[1] + 1) = 1
12	Check substring s[0..2]	aab	No	[0,0,1]	pal[0,2] = false
13	Update cuts for s[0..2]	aab	Not palindrome	[0,0,1]	cuts[2] = min(cuts[2], cuts[0] + 1) = 1
14	Return minimum cuts for full string	aab	-	[0,0,1]	Minimum cuts = 1

💡 All substrings checked and minimum cuts updated; final answer is cuts[n-1]

Variable Tracker

Variable	Start	After Step 3	After Step 7	After Step 11	Final
cuts	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,1]	[0,0,1]
pal	all false	pal[0,0]=true	pal[0,1]=true	pal[1,2]=false	pal[0,0]=true, pal[0,1]=true, pal[1,1]=true, pal[2,2]=true

Key Moments - 3 Insights

Why do we set cuts[i] = 0 when substring s[0..i] is palindrome?

Because if the substring from start to i is palindrome, no cuts are needed, so cuts[i] = 0 as shown in steps 3 and 7.

Why do we update cuts[i] with cuts[j] + 1 when substring s[j+1..i] is palindrome?

Because we can cut after position j to partition the string, so minimum cuts at i is minimum of current cuts[i] and cuts[j] + 1, as seen in step 11.

Why do we check all substrings for palindrome before updating cuts?

Because knowing which substrings are palindrome helps decide where cuts can be made, ensuring minimum cuts, as done in steps 2,4,6,8,10,12.

Visual Quiz - 3 Questions

Look at the execution table, what is the value of cuts after step 7?

A [1,0,0]

B [0,1,0]

C [0,0,0]

D [0,0,1]

💡 Hint

Refer to the Cuts Array State column at step 7 in the execution_table.

At which step does the cuts array first change to include a 1?

A Step 9

B Step 11

C Step 7

D Step 13

💡 Hint

Check the Cuts Array State column for the first appearance of 1.

If the substring s[1..2] was palindrome, how would cuts[2] change at step 11?

A cuts[2] would remain 0

B cuts[2] would become 2

C cuts[2] would become 1

D cuts[2] would become 3

💡 Hint

Look at how palindrome substrings affect cuts updates in steps 3 and 7.

Concept Snapshot

Palindrome Partitioning DP Minimum Cuts:
- Use a 2D bool array pal[n,n] to mark palindromes.
- Use cuts[i] to store min cuts for s[0..i].
- If s[0..i] palindrome, cuts[i] = 0.
- Else cuts[i] = min(cuts[j] + 1) for j < i where s[j+1..i] palindrome.
- Result is cuts[n-1].

Full Transcript

This concept finds the minimum number of cuts needed to split a string into palindrome substrings using dynamic programming. We create a 2D array to mark which substrings are palindromes. We also maintain an array cuts where cuts[i] represents the minimum cuts needed for the substring from start to index i. If the substring from start to i is palindrome, no cuts are needed, so cuts[i] is zero. Otherwise, we check all possible partitions and update cuts[i] with the minimum cuts found. The process continues until the entire string is processed, and the final answer is cuts[n-1].

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