Palindrome Partitioning DP Minimum Cuts in DSA Javascript - Execution Trace

Concept Flow - Palindrome Partitioning DP Minimum Cuts

Start with string s

↓

Initialize DP arrays: cuts[

↓

For each end index j in s

↓

For each start index i <= j

↓

Check if s[i..j

Yes No↓

Update palindrome[i

↓

Update cuts[j

↓

Repeat for all j

↓

Return cuts[n-1

↓

Done

We check all substrings to find palindromes and update minimum cuts needed dynamically.

Execution Sample

DSA Javascript

const minCut = s => {
  const n = s.length;
  const cuts = Array(n).fill(0);
  const palindrome = Array.from({length: n}, () => Array(n).fill(false));
  for (let j = 0; j < n; j++) {
    let minCuts = j;
    for (let i = 0; i <= j; i++) {
      if (s[i] === s[j] && (j - i < 2 || palindrome[i + 1][j - 1])) {
        palindrome[i][j] = true;
        minCuts = i === 0 ? 0 : Math.min(minCuts, cuts[i - 1] + 1);
      }
    }
    cuts[j] = minCuts;
  }
  return cuts[n - 1];
};

This code finds the minimum cuts needed to partition string s into palindromes using dynamic programming.

Execution Table

Step	Operation	Indices i,j	Palindrome[i][j]	Cuts Array	Visual State (Palindrome substrings)
1	Initialize cuts and palindrome arrays	-	-	[0,0,0,0,0]	No palindromes marked yet
2	j=0: Check substring s[0..0]	i=0,j=0	true	[0,0,0,0,0]	"a" is palindrome
3	Update cuts[0]	-	-	[0,0,0,0,0]	Cuts at index 0 is 0
4	j=1: Check substring s[0..1]	i=0,j=1	false	[0,1,0,0,0]	"ab" not palindrome
5	j=1: Check substring s[1..1]	i=1,j=1	true	[0,1,0,0,0]	"b" palindrome
6	Update cuts[1]	-	-	[0,1,0,0,0]	Cuts at index 1 is 1
7	j=2: Check substring s[0..2]	i=0,j=2	true	[0,1,0,0,0]	"aba" is palindrome
8	j=2: Check substring s[1..2]	i=1,j=2	false	[0,1,0,0,0]	"ba" not palindrome
9	j=2: Check substring s[2..2]	i=2,j=2	true	[0,1,0,0,0]	"a" palindrome
10	Update cuts[2]	-	-	[0,1,0,0,0]	Cuts at index 2 updated to 0 ("aba" palindrome)
11	j=3: Check substring s[0..3]	i=0,j=3	false	[0,1,0,3,0]	"abac" not palindrome
12	j=3: Check substring s[1..3]	i=1,j=3	false	[0,1,0,3,0]	"bac" not palindrome
13	j=3: Check substring s[2..3]	i=2,j=3	false	[0,1,0,3,0]	"ac" not palindrome
14	j=3: Check substring s[3..3]	i=3,j=3	true	[0,1,0,1,0]	"c" palindrome
15	Update cuts[3]	-	-	[0,1,0,1,0]	Cuts at index 3 is 1
16	j=4: Check substring s[0..4]	i=0,j=4	false	[0,1,0,1,4]	"abaca" not palindrome
17	j=4: Check substring s[2..4]	i=2,j=4	true	[0,1,0,1,2]	"aca" is palindrome
18	j=4: Check substring s[3..4]	i=3,j=4	false	[0,1,0,1,2]	"ca" not palindrome
19	j=4: Check substring s[4..4]	i=4,j=4	true	[0,1,0,1,2]	"a" palindrome
20	Update cuts[4]	-	-	[0,1,0,1,2]	Cuts at index 4 updated to 2
21	Final state before return	-	-	[0,1,0,1,2]	All checks complete
22	Return cuts[n-1]	-	-	[0,1,0,1,2]	Minimum cuts needed: 2

💡 All substrings checked, minimum cuts found at cuts[n-1]

Variable Tracker

Variable	Start	After Step 3	After Step 6	After Step 10	After Step 15	After Step 18	Final
cuts	[0,0,0,0,0]	[0,0,0,0,0]	[0,1,0,0,0]	[0,1,0,0,0]	[0,1,0,1,0]	[0,1,0,1,2]	[0,1,0,1,2]
palindrome[0][0]	false	true	true	true	true	true	true
palindrome[1][1]	false	false	true	true	true	true	true
palindrome[2][2]	false	false	false	true	true	true	true
palindrome[3][3]	false	false	false	false	true	true	true
palindrome[2][4]	false	false	false	false	false	true	true

Key Moments - 3 Insights

Why do we check (j - i < 2 || palindrome[i + 1][j - 1]) to decide if s[i..j] is palindrome?

Because substrings of length 1 or 2 are simple palindromes if characters match, otherwise we rely on the smaller substring palindrome[i+1][j-1]. See execution_table rows 2, 5, and 9.

Why do we update cuts[j] with cuts[i-1] + 1 instead of just 1?

cuts[i-1] represents minimum cuts needed before substring s[i..j]. Adding 1 means cutting before i to separate palindrome s[i..j]. See execution_table rows 14 and 17.

Why is cuts[0] initialized to 0 and not 1?

Because a single character substring is always palindrome and needs zero cuts. See execution_table row 3.

Visual Quiz - 3 Questions

Look at the execution_table at step 10, what is the value of cuts[2]?

A 1

B 0

C 2

D 3

💡 Hint

Check the Cuts Array column at step 10 in execution_table.

At which step does palindrome[2][4] become true?

A Step 17

B Step 9

C Step 22

D Step 5

💡 Hint

Look at the Palindrome[i][j] column and Visual State for substring s[2..4].

If the string was "aaaaa", how would cuts[4] change compared to the current example?

A cuts[4] would be 4

B cuts[4] would be 2

C cuts[4] would be 0

D cuts[4] would be 1

💡 Hint

For a string of all same characters, the whole string is palindrome, so no cuts needed.

Concept Snapshot

Palindrome Partitioning Minimum Cuts:
- Use DP arrays: cuts[] for min cuts, palindrome[][] for palindrome checks
- palindrome[i][j] = true if s[i..j] is palindrome
- cuts[j] = min cuts needed for s[0..j]
- Update cuts[j] by checking all i ≤ j where s[i..j] is palindrome
- Result is cuts[n-1]

Full Transcript

This visualization shows how to find the minimum cuts needed to split a string into palindrome substrings using dynamic programming. We use two arrays: palindrome to mark if substring s[i..j] is palindrome, and cuts to store minimum cuts needed up to index j. For each end index j, we check all start indices i ≤ j. If s[i..j] is palindrome, we update cuts[j] to the minimum of its current value and cuts[i-1] + 1 (or 0 if i=0). The final answer is cuts[n-1]. The execution table traces each step, showing palindrome checks and cuts updates. Key moments clarify why we check inner substrings and how cuts are updated. The quiz tests understanding of cuts array and palindrome marking. This method efficiently finds minimum cuts by building on smaller palindrome substrings.

Practice

1. What does the term minimum cuts mean in the context of palindrome partitioning?

easy

A. The number of characters to remove to make the string a palindrome

B. The longest palindrome substring in the string

C. The total number of palindromic substrings in the string

D. The smallest number of cuts needed to split the string into palindromic substrings

Solution

1. Step 1: Understand palindrome partitioning

   Palindrome partitioning splits a string into substrings where each substring is a palindrome.

2. Step 2: Define minimum cuts

   Minimum cuts means the least number of cuts needed to divide the string so that every part is a palindrome.

3. Final Answer:

   The smallest number of cuts needed to split the string into palindromic substrings -> Option D

4. Quick Check:

   Minimum cuts = smallest cuts for palindrome parts [OK]

Hint: Minimum cuts split string into palindrome parts with fewest cuts [OK]

Common Mistakes:

• Confusing minimum cuts with longest palindrome substring
• Thinking minimum cuts count characters removed
• Mixing total palindrome substrings with minimum cuts

2. Which of the following is the correct way to initialize a 2D boolean array isPalindrome for a string s of length n in JavaScript?

easy

A. const isPalindrome = Array.from({length: n}, () => Array(n).fill(false));

B. const isPalindrome = Array(n).fill(Array(n).fill(false));

C. const isPalindrome = new Array(n).fill(new Array(n).fill(false));

D. const isPalindrome = Array(n).map(Array(n).fill(false));

Solution

1. Step 1: Understand 2D array initialization in JavaScript

   Using Array.from with a mapping function creates independent inner arrays.

2. Step 2: Compare options

   const isPalindrome = Array.from({length: n}, () => Array(n).fill(false)); correctly creates separate inner arrays filled with false. Options A and C create references to the same inner array, causing bugs. const isPalindrome = Array(n).map(() => Array(n).fill(false)); fails because map on uninitialized array does not work as expected.

3. Final Answer:

   const isPalindrome = Array.from({length: n}, () => Array(n).fill(false)); -> Option A

4. Quick Check:

   Use Array.from with function for independent inner arrays [OK]

Hint: Use Array.from with function to avoid shared references [OK]

Common Mistakes:

• Using fill with array directly causing shared references
• Using map on uninitialized arrays
• Not initializing inner arrays separately

3. What is the output of the following code snippet for s = "aab"?

function minCut(s) {
  const n = s.length;
  const isPalindrome = Array.from({length: n}, () => Array(n).fill(false));
  for (let i = 0; i < n; i++) {
    isPalindrome[i][i] = true;
  }
  for (let length = 2; length <= n; length++) {
    for (let start = 0; start <= n - length; start++) {
      let 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];
        }
      }
    }
  }
  const cuts = Array(n).fill(0);
  for (let i = 0; i < n; i++) {
    let minCut = i;
    for (let j = 0; j <= i; j++) {
      if (isPalindrome[j][i]) {
        minCut = j === 0 ? 0 : Math.min(minCut, cuts[j - 1] + 1);
      }
    }
    cuts[i] = minCut;
  }
  return cuts[n - 1];
}
console.log(minCut(s));

medium

A. 0

B. 1

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

   Minimum cuts needed: split "aab" as "aa" | "b" with 1 cut.

3. Final Answer:

   1 -> Option B

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindrome substrings, then find fewest cuts [OK]

Common Mistakes:

• Assuming no cuts needed for non-palindrome string
• Counting total palindromic substrings instead of cuts
• Misindexing palindrome checks

4. The following code attempts to compute minimum cuts for palindrome partitioning but has a bug. What is the error?

function minCut(s) {
  const n = s.length;
  const isPalindrome = Array(n).fill(Array(n).fill(false));
  for (let i = 0; i < n; i++) {
    isPalindrome[i][i] = true;
  }
  // ... rest of palindrome logic ...
  return 0;
}

medium

A. The function does not return the minimum cuts value

B. Loop for setting isPalindrome[i][i] should start from 1, not 0

C. isPalindrome initialization shares inner arrays causing incorrect updates

D. The string length n is not calculated correctly

Solution

1. Step 1: Analyze isPalindrome array initialization

   Using fill with an array creates references to the same inner array, so updating one row updates all rows.

2. Step 2: Understand impact on palindrome checks

   This causes incorrect palindrome marking because all rows share the same inner array, corrupting data.

3. Final Answer:

   isPalindrome initialization shares inner arrays causing incorrect updates -> Option C

4. Quick Check:

   Shared inner arrays cause bugs in 2D boolean array [OK]

Hint: Avoid fill with arrays; use Array.from with function [OK]

Common Mistakes:

• Ignoring shared reference problem in 2D arrays
• Thinking loop start index causes bug
• Overlooking missing return value as main error

5. Given the string s = "ababbbabbababa", what is the minimum number of cuts needed to partition it into palindromic substrings using dynamic programming?

hard

A. 3

B. 5

C. 4

D. 6

Solution

1. Step 1: Understand palindrome partitions for "ababbbabbababa"

   This string can be partitioned into palindromes with minimum cuts by checking all substrings and using DP to find minimal cuts.

2. Step 2: Use known result or DP approach

   The minimum cuts for this string is 3, as it can be split into palindromic parts like "a|babbbab|b|ababa" or similar partitions.

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Minimum cuts for complex palindrome string = 3 [OK]

Hint: Use DP palindrome checks and cuts array to find minimum cuts [OK]

Common Mistakes:

• Assuming minimum cuts equals number of palindromic substrings
• Miscounting cuts by off-by-one errors
• Not considering overlapping palindrome substrings