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

Recall & Review

beginner

What is the goal of the Palindrome Partitioning problem?

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

intermediate

What does the DP array represent in the minimum cuts palindrome partitioning solution?

It stores the minimum number of cuts needed for the substring from the start up to each position.

intermediate

How do we check if a substring is a palindrome efficiently in DP?

By using a 2D boolean table where table[i][j] is true if substring from i to j is palindrome, built using smaller substrings.

beginner

Why do we initialize the minimum cuts array with the maximum cuts possible?

Because the worst case is cutting before every character, so initializing with max cuts helps find the minimum by comparison.

intermediate

What is the time complexity of the DP solution for palindrome partitioning minimum cuts?

O(n^2), where n is the length of the string, due to checking all substrings and updating cuts.

What does a 'cut' mean in palindrome partitioning?

A Reversing a substring

B Splitting the string into two parts

C Changing a character

D Checking if substring is palindrome

A cut splits the string into smaller parts to check if each part is a palindrome.

Which data structure helps store if substrings are palindromes?

A 2D boolean array

B Stack

C 1D integer array

D Queue

A 2D boolean array stores palindrome status for all substring ranges.

If the whole string is a palindrome, minimum cuts needed are:

A Length of string - 1

B Length of string

C 0

D 1

No cuts are needed if the entire string is already a palindrome.

What is the base case for minimum cuts DP array?

A 0 cuts for first character

B 0 cuts for empty string

C 1 cut for first character

D Length of string cuts

The first character alone needs 0 cuts since it's a palindrome.

Which approach is used to solve palindrome partitioning minimum cuts?

A Greedy

B Backtracking only

C Divide and Conquer

D Dynamic Programming

Dynamic Programming efficiently solves this by storing results of subproblems.

Explain how dynamic programming helps find the minimum cuts for palindrome partitioning.

Think about storing results to avoid repeated work.

Describe the process to check if substrings are palindromes efficiently in the DP solution.

Build from smaller to bigger substrings.

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 maximum length of a palindrome substring in s

C. The total number of palindrome substrings in s

D. The longest palindrome substring 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: Differentiate from other palindrome problems

   It is not about longest palindrome or counting palindromes, but minimizing cuts for palindrome partitions.

3. Final Answer:

   The minimum number of cuts needed to partition s into palindromic substrings -> Option A

4. Quick Check:

   Minimum cuts = minimum partitions [OK]

Hint: Focus on cuts that split string into palindrome parts [OK]

Common Mistakes:

• Confusing minimum cuts with longest palindrome length
• Thinking problem counts palindrome substrings
• Assuming problem finds longest palindrome substring

2. Which of the following Go code snippets correctly initializes a 2D boolean slice isPalindrome for a string s of length n?

easy

A. isPalindrome := [][]bool{{false}}
for i := 0; i < n; i++ {
isPalindrome[i][i] = true
}

B. var isPalindrome [][]bool
isPalindrome = make([]bool, n*n)

C. isPalindrome := make([][]bool, n)
for i := range isPalindrome {
isPalindrome[i] = make([]bool, n)
}

D. isPalindrome := make([]bool, n)
for i := 0; i < n; i++ {
isPalindrome[i] = true
}

Solution

1. Step 1: Understand 2D slice initialization in Go

   To create a 2D slice, first allocate the outer slice, then allocate each inner slice separately.

2. Step 2: Check each option for correct 2D boolean slice creation

   isPalindrome := make([][]bool, n) for i := range isPalindrome { isPalindrome[i] = make([]bool, n) } correctly creates a slice of slices with length n and initializes each inner slice with length n.

3. Final Answer:

   isPalindrome := make([][]bool, n) for i := range isPalindrome { isPalindrome[i] = make([]bool, n) -> Option C

4. Quick Check:

   2D slice needs nested make calls [OK]

Hint: Use nested make calls for 2D slices in Go [OK]

Common Mistakes:

• Trying to create 2D slice with single make call
• Confusing slice of bool with slice of slices
• Not initializing inner slices before use

3. What is the output of the following Go code snippet that calculates minimum cuts for palindrome partitioning of s = "aab"?

func minCut(s string) int {
    n := len(s)
    isPalindrome := make([][]bool, n)
    for i := range isPalindrome {
        isPalindrome[i] = make([]bool, n)
    }
    for i := n - 1; i >= 0; i-- {
        for j := i; j < n; j++ {
            if s[i] == s[j] && (j - i < 2 || isPalindrome[i+1][j-1]) {
                isPalindrome[i][j] = true
            }
        }
    }
    cuts := make([]int, n)
    for i := 0; i < n; i++ {
        minCut := i
        for j := 0; j <= i; j++ {
            if isPalindrome[j][i] {
                if j == 0 {
                    minCut = 0
                } else if cuts[j-1]+1 < minCut {
                    minCut = cuts[j-1] + 1
                }
            }
        }
        cuts[i] = minCut
    }
    return cuts[n-1]
}

fmt.Println(minCut("aab"))

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 using DP logic

   Minimum cuts needed: cut between "aa" and "b" -> 1 cut.

3. Final Answer:

   1 -> Option B

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindrome substrings, then count cuts between them [OK]

Common Mistakes:

• Assuming no cuts needed for "aab"
• Counting cuts as number of palindrome substrings
• Misunderstanding DP array meaning

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

func minCut(s string) int {
    n := len(s)
    isPalindrome := make([][]bool, n)
    for i := range isPalindrome {
        isPalindrome[i] = make([]bool, n)
    }
    for i := 0; i < n; i++ {
        for j := i; j < n; j++ {
            if s[i] == s[j] && (j - i < 2 || isPalindrome[i+1][j-1]) {
                isPalindrome[i][j] = true
            }
        }
    }
    cuts := make([]int, n)
    for i := 0; i < n; i++ {
        cuts[i] = i
        for j := 0; j <= i; j++ {
            if isPalindrome[j][i] {
                if j == 0 {
                    cuts[i] = 0
                } else if cuts[j-1]+1 < cuts[i] {
                    cuts[i] = cuts[j-1] + 1
                }
            }
        }
    }
    return cuts[n-1]
}

medium

A. The function returns cuts[0] instead of cuts[n-1]

B. The cuts array is not initialized with correct default values

C. The condition s[i] == s[j] is incorrect for palindrome check

D. The palindrome check loop should run from i = n-1 down to 0, not 0 to n-1

Solution

1. Step 1: Analyze palindrome DP filling order

   Palindrome DP requires filling from end to start (i from n-1 down to 0) to use isPalindrome[i+1][j-1] correctly.

2. Step 2: Identify impact of incorrect loop order

   Looping i from 0 to n-1 causes isPalindrome[i+1][j-1] to be uninitialized or incorrect, leading to wrong results.

3. Final Answer:

   The palindrome check loop should run from i = n-1 down to 0, not 0 to n-1 -> Option D

4. Quick Check:

   Fill palindrome DP bottom-up from right [OK]

Hint: Fill palindrome DP from right to left for correct subproblems [OK]

Common Mistakes:

• Filling palindrome DP from left to right
• Ignoring dependency on isPalindrome[i+1][j-1]
• Incorrect initialization of cuts array

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

hard

A. 4

B. 6

C. 5

D. 3

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: Apply DP minimum cuts algorithm

   Using the palindrome DP table and cuts array, the minimum cuts calculated for this string is 4.

3. Final Answer:

   4 -> Option A

4. Quick Check:

   Minimum cuts for "ababbbabbababa" = 4 [OK]

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

Common Mistakes:

• Underestimating cuts by ignoring some palindrome partitions
• Overcounting cuts by splitting unnecessarily
• Not using DP to optimize palindrome checks