Palindrome Partitioning DP Minimum Cuts in DSA Python - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of minimum cuts for palindrome partitioning

What is the output of the following code that calculates the minimum cuts needed to partition the string into palindromes?

DSA Python

def min_cut_palindrome(s):
    n = len(s)
    dp = [0] * n
    palindrome = [[False] * n for _ in range(n)]
    for i in range(n):
        min_cut = i
        for j in range(i + 1):
            if s[j] == s[i] and (i - j < 2 or palindrome[j + 1][i - 1]):
                palindrome[j][i] = True
                min_cut = 0 if j == 0 else min(min_cut, dp[j - 1] + 1)
        dp[i] = min_cut
    return dp[-1]

print(min_cut_palindrome("aab"))

A 1

B 3

C 2

D 0

💡 Hint

Think about how many cuts are needed to split "aab" into palindromes.

The string "aab" can be split into "aa" and "b" which are both palindromes. Only one cut is needed between the second and third character.

❓ Predict Output

intermediate

Minimum cuts for a palindrome string

What is the output of the code below for the input string "racecar"?

DSA Python

def min_cut_palindrome(s):
    n = len(s)
    dp = [0] * n
    palindrome = [[False] * n for _ in range(n)]
    for i in range(n):
        min_cut = i
        for j in range(i + 1):
            if s[j] == s[i] and (i - j < 2 or palindrome[j + 1][i - 1]):
                palindrome[j][i] = True
                min_cut = 0 if j == 0 else min(min_cut, dp[j - 1] + 1)
        dp[i] = min_cut
    return dp[-1]

print(min_cut_palindrome("racecar"))

A 1

B 0

C 3

D 6

💡 Hint

Is the whole string a palindrome?

The entire string "racecar" is a palindrome, so no cuts are needed.

🧠 Conceptual

advanced

Understanding the palindrome table in DP

In the palindrome partitioning DP solution, what does the 2D boolean table 'palindrome' represent?

A Whether the substring from index j to i is a palindrome

B The minimum cuts needed for substring from j to i

C The length of the longest palindrome substring ending at i

D The count of palindrome substrings between j and i

💡 Hint

Think about how the DP checks if a substring is palindrome.

The 'palindrome' table stores True if the substring s[j:i+1] is a palindrome, otherwise False.

❓ Predict Output

advanced

Minimum cuts for complex string

What is the output of the following code for the input string "abccbc"?

DSA Python

def min_cut_palindrome(s):
    n = len(s)
    dp = [0] * n
    palindrome = [[False] * n for _ in range(n)]
    for i in range(n):
        min_cut = i
        for j in range(i + 1):
            if s[j] == s[i] and (i - j < 2 or palindrome[j + 1][i - 1]):
                palindrome[j][i] = True
                min_cut = 0 if j == 0 else min(min_cut, dp[j - 1] + 1)
        dp[i] = min_cut
    return dp[-1]

print(min_cut_palindrome("abccbc"))

A 1

B 0

C 3

D 2

💡 Hint

Try to find palindrome partitions with minimum cuts.

One optimal partition is "a" | "bccb" | "c" with 2 cuts.

🧠 Conceptual

expert

Time complexity of palindrome partitioning DP

What is the time complexity of the dynamic programming solution for palindrome partitioning minimum cuts on a string of length n?

A O(n^3)

B O(n log n)

C O(n^2)

D O(n)

💡 Hint

Consider the nested loops and palindrome checks.

The DP uses two nested loops over n and constant time palindrome checks using precomputed table, resulting in O(n^2).

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