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

↓

No cut needed

↓

If not palindrome

↓

Try all partitions

↓

Calculate min cuts using DP

↓

Return min cuts for full string

We check all substrings for palindrome, then use DP to find minimum cuts needed to partition the string into palindromes.

Execution Sample

DSA C++

string s = "aab";
int n = s.size();
vector<int> cuts(n, 0);
vector<vector<bool>> pal(n, vector<bool>(n, false));

Initialize string, cuts array, and palindrome table for DP.

Execution Table

Step	Operation	Substring Checked	Is Palindrome?	Cuts Array State	Visual State
1	Initialize cuts and palindrome table	-	-	[0,0,0]	pal and cuts empty, ready
2	Check substring s[0:0] = 'a'	a	Yes	[0,0,0]	pal[0][0] = true
3	Check substring s[1:1] = 'a'	a	Yes	[0,0,0]	pal[1][1] = true
4	Check substring s[2:2] = 'b'	b	Yes	[0,0,0]	pal[2][2] = true
5	Check substring s[0:1] = 'aa'	aa	Yes	[0,0,0]	pal[0][1] = true
6	Check substring s[1:2] = 'ab'	ab	No	[0,0,0]	pal[1][2] = false
7	Calculate cuts for i=0	-	-	[0,0,0]	cuts[0] = 0 (no cut)
8	Calculate cuts for i=1	-	-	[0,0,0]	cuts[1] = 0 (substring 'aa' palindrome)
9	Calculate cuts for i=2	-	-	[0,0,1]	cuts[2] = 1 (cut between 'aa' and 'b')
10	Return cuts[n-1]	-	-	[0,0,1]	Minimum cuts = 1

💡 All substrings checked and cuts calculated; minimum cuts for full string found.

Variable Tracker

Variable	Start	After Step 2	After Step 5	After Step 8	Final
cuts	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,1]
pal	all false	pal[0][0]=true	pal[0][1]=true	pal[0][1]=true	pal[0][0]=true, pal[0][1]=true, pal[1][1]=true, pal[2][2]=true

Key Moments - 3 Insights

Why do we mark single characters as palindrome first?

Because every single character is a palindrome by definition, marking them helps build larger palindrome substrings (see steps 2-4 in execution_table).

Why does cuts[1] become 0 when substring 'aa' is palindrome?

Because if the substring from start to i is palindrome, no cuts are needed (step 8). This means cuts[i] = 0.

Why is cuts[2] equal to 1 and not 0?

Because substring 'aab' is not palindrome, so we try partitions. The best cut is after 'aa' (step 9), so cuts[2] = cuts[1] + 1 = 1.

Visual Quiz - 3 Questions

Look at the execution_table at step 5, what is the palindrome status of substring 'aa'?

A No, it is not palindrome

B Yes, it is palindrome

C Not checked yet

D Depends on cuts array

💡 Hint

Check the 'Is Palindrome?' column at step 5 in execution_table.

At which step does cuts[2] get its final value?

A Step 7

B Step 8

C Step 9

D Step 10

💡 Hint

Look at the 'Cuts Array State' column in execution_table rows for step 9.

If substring 'ab' at step 6 was palindrome, how would cuts[2] change?

A cuts[2] would not change

B cuts[2] would be 1

C cuts[2] would be 0

D cuts[2] would be 2

💡 Hint

Making 'ab' palindrome allows partition 'a' | 'ab' (cuts[0] + 1 = 1), same minimum as before.

Concept Snapshot

Palindrome Partitioning Minimum Cuts:
- Use DP with a palindrome table pal[i][j]
- pal[i][j] = true if s[i..j] palindrome
- cuts[i] = 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: cuts[n-1]

Full Transcript

This visualization shows how to find the minimum cuts needed to partition a string into palindromes using dynamic programming. We start by marking all single characters as palindromes. Then we check longer substrings and mark them palindrome if they read the same forwards and backwards. Using this palindrome table, we calculate the minimum cuts needed for each prefix of the string. If the whole prefix is palindrome, no cuts are needed. Otherwise, we try all partitions and pick the minimum cuts. The final answer is the minimum cuts for the entire 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