Palindrome Partitioning DP Minimum Cuts in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the goal of the Palindrome Partitioning problem with minimum cuts?

To split a given string into the fewest number of parts so that each part is a palindrome.

intermediate

How does dynamic programming help solve the Palindrome Partitioning problem?

It stores results of smaller subproblems (like whether substrings are palindromes and minimum cuts needed) to avoid repeated work and build up the solution efficiently.

intermediate

In the DP approach, what does the array that tracks minimum cuts represent?

Each position in the array holds the minimum number of cuts needed to partition the substring from the start up to that position into palindromes.

intermediate

What is the time complexity of the Palindrome Partitioning DP Minimum Cuts solution?

O(n²), where n is the length of the string, because it checks all substrings and calculates minimum cuts using DP.

intermediate

Why do we precompute a palindrome table in the DP solution?

To quickly check if any substring is a palindrome in constant time, which speeds up the calculation of minimum cuts.

What does a 'cut' mean in the Palindrome Partitioning problem?

A Splitting the string into two parts

B Changing a character in the string

C Reversing a substring

D Removing a character

A cut means splitting the string into parts so each part can be checked separately.

Which data structure is commonly used to store if substrings are palindromes?

A Stack

B Queue

C 2D boolean array

D Linked list

A 2D boolean array stores palindrome status for all substring start and end indices.

If the entire string is already a palindrome, what is the minimum number of cuts needed?

A Length of string

B 0

C Length of string minus 1

D 1

No cuts are needed if the whole string is a palindrome.

What is the main reason to use dynamic programming in this problem?

A To avoid checking the same substring multiple times

B To sort the string

C To reverse the string

D To count characters

DP helps by storing results of palindrome checks and minimum cuts to avoid repeated work.

What does the DP array for minimum cuts typically start with?

A Random values

B All ones

C All zeros

D Maximum possible cuts (like index value)

It starts with the worst case where each character is a cut, so initial values are set to the index.

Explain how to use dynamic programming to find the minimum cuts needed to partition a string into palindromes.

Think about building from smaller substrings to the whole string.

Describe why precomputing palindrome substrings helps in solving the minimum cuts problem efficiently.

Consider how checking palindrome status repeatedly affects performance.

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