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

Recall & Review

beginner

What is the goal of the Palindrome Partitioning problem?

To split a string into the fewest possible 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 previous results.

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' represent in palindrome partitioning?

A A division between two palindrome substrings

B A character removal

C A palindrome substring

D A character swap

A cut divides the string into parts, each part being a palindrome.

Which data structure helps store if substrings are palindromes in DP?

A 1D integer array

B 2D boolean array

C Stack

D Queue

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

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

A 0

B Length of string - 1

C 1

D Length of string

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

What is the initial value for minimum cuts at each position before DP updates?

A Position index

B Length of string

C Maximum cuts possible (position index)

D 0

Initialize with maximum cuts possible to find minimum by comparison.

Which approach is used to solve Palindrome Partitioning minimum cuts?

A Backtracking only

B Divide and Conquer

C Greedy

D Dynamic Programming

Dynamic Programming efficiently solves this by storing intermediate results.

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

Think about how to avoid repeated palindrome checks.

Describe the process to check if a substring is a palindrome using DP.

Start from smallest substrings and expand.

Practice

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

easy

A. The total number of substrings in the string

B. The maximum number of palindromes in the string

C. The length of the longest palindrome substring

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

Solution

1. Step 1: Understand palindrome partitioning

   Palindrome partitioning means splitting a string into parts where each part is a palindrome.

2. Step 2: Define minimum cuts

   Minimum cuts is the least number of cuts needed to divide the string so every piece 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 = least splits for palindrome parts [OK]

Common Mistakes:

• Confusing minimum cuts with longest palindrome length
• Thinking minimum cuts means maximum palindromes
• Counting total substrings instead of cuts

2. Which of the following Python code snippets correctly initializes a 2D list dp of size n x n with False values for palindrome checks?

easy

A. dp = [[False for _ in range(n)] for _ in range(n)]

B. dp = [[False]*n]*n

C. dp = [False]*n*n

D. dp = [False for _ in range(n)]

Solution

1. Step 1: Understand 2D list initialization

   To create a 2D list with independent inner lists, use nested list comprehensions.

2. Step 2: Analyze options

   dp = [[False for _ in range(n)] for _ in range(n)] uses nested list comprehension correctly; dp = [[False]*n]*n creates shallow copies causing shared references.

3. Final Answer:

   dp = [[False for _ in range(n)] for _ in range(n)] -> Option A

4. Quick Check:

   Nested list comprehension creates independent rows [OK]

Hint: Use nested list comprehension for 2D lists [OK]

Common Mistakes:

• Using [[False]*n]*n causes shared inner lists
• Using single list for 2D structure
• Confusing list multiplication with nested lists

3. What is the output of the following Python code snippet for minimum cuts in palindrome partitioning?

def minCut(s):
    n = len(s)
    dp = [0] * n
    for i in range(n):
        min_cut = i
        for j in range(i + 1):
            if s[j:i+1] == s[j:i+1][::-1]:
                min_cut = 0 if j == 0 else min(min_cut, dp[j-1] + 1)
        dp[i] = min_cut
    return dp[-1]

print(minCut('aab'))

medium

A. 0

B. 2

C. 1

D. 3

Solution

1. Step 1: Trace the dp array updates

   For 'aab': i=0: 'a' palindrome, dp[0]=0; i=1: 'aa' palindrome, dp[1]=0; i=2: 'aab' not palindrome, check substrings: 'b' palindrome, dp[2]=dp[1]+1=1.

2. Step 2: Final dp array and result

   dp = [0, 0, 1], so minimum cuts = 1.

3. Final Answer:

   1 -> Option C

4. Quick Check:

   Minimum cuts for 'aab' = 1 [OK]

Hint: Check palindrome substrings and update dp with min cuts [OK]

Common Mistakes:

• Not updating dp correctly for substrings
• Assuming entire string is palindrome without check
• Confusing indices in slicing

4. Identify the bug in the following code snippet for palindrome partitioning minimum cuts and select the correct fix.

def minCut(s):
    n = len(s)
    dp = [0]*n
    for i in range(n):
        min_cut = i
        for j in range(i):
            if s[j:i+1] == s[j:i+1][::-1]:
                min_cut = 0 if j == 0 else min(min_cut, dp[j-1] + 1)
        dp[i] = min_cut
    return dp[-1]

print(minCut('aab'))

medium

A. Change inner loop to for j in range(i+1)

B. Initialize dp with -1 instead of 0

C. Use dp[i] = min_cut + 1 instead of dp[i] = min_cut

D. Reverse the string before processing

Solution

1. Step 1: Analyze loop range error

   The inner loop runs from 0 to i-1, missing the case j = i, which is needed to check substring s[i:i+1].

2. Step 2: Correct loop range

   Changing inner loop to range(i+1) includes the substring ending at i, fixing the bug.

3. Final Answer:

   Change inner loop to for j in range(i+1) -> Option A

4. Quick Check:

   Include full substring checks with correct loop range [OK]

Hint: Inner loop must include i to check full substring [OK]

Common Mistakes:

• Using range(i) misses last substring
• Incorrect dp initialization
• Unnecessary string reversal

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. 5

B. 3

C. 4

D. 6

Solution

1. Step 1: Understand the problem and DP approach

   We want to split 'ababbbabbababa' into palindromic parts with minimum cuts. Using DP, we check all substrings and update minimum cuts.

2. Step 2: Known result from palindrome partitioning DP

   This classic example has minimum cuts = 3, partitioning as 'a|babbbab|b|ababa' or similar (4 parts, 3 cuts).

3. Final Answer:

   3 -> Option B

4. Quick Check:

   Minimum cuts for 'ababbbabbababa' = 3 [OK]

Hint: Use DP palindrome checks and track min cuts [OK]

Common Mistakes:

• Underestimating cuts by assuming longer palindromes
• Overcounting cuts by splitting too often
• Not using DP to optimize palindrome checks