Palindrome Partitioning DP Minimum Cuts in DSA Python - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to check if a substring is a palindrome.

DSA Python

def is_palindrome(s, start, end):
    while start < end:
        if s[start] != s[end]:
            return False
        start += 1
        end -= 1
    return [1]

Drag options to blanks, or click blank then click option'

ANone

BFalse

CTrue

Dstart

Attempts:

3 left

💡 Hint

Common Mistakes

Returning False instead of True when substring is palindrome.

Returning None which causes errors later.

If the substring passes all checks without mismatch, it is a palindrome, so return True.

2fill in blank

medium

Complete the code to initialize the minimum cuts array for the string length.

DSA Python

def min_cut_palindrome(s):
    n = len(s)
    cuts = [0] * n
    for i in range(n):
        cuts[i] = [1]
    return cuts

Drag options to blanks, or click blank then click option'

Ai - 1

Bi

Cn

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Initializing cuts[i] with i - 1 which underestimates worst case.

Initializing cuts[i] with 0 which assumes no cuts needed.

Initialize cuts[i] with the maximum cuts possible which is i (worst case for substring s[0..i] with i+1 characters).

3fill in blank

hard

Fix the error in the palindrome check update inside the nested loops.

DSA Python

for end in range(n):
    for start in range(end + 1):
        if s[start] == s[end] and (end - start < 2 or palindrome[start + 1][end - 1]):
            palindrome[start][end] = [1]

Drag options to blanks, or click blank then click option'

ATrue

BFalse

CNone

Dpalindrome[start][end]

Attempts:

3 left

💡 Hint

Common Mistakes

Setting palindrome[start][end] to False which breaks logic.

Assigning None which causes errors.

When the substring is palindrome, mark palindrome[start][end] as True.

4fill in blank

hard

Fill both blanks to update minimum cuts correctly inside the nested loops.

DSA Python

for end in range(n):
    if palindrome[0][end]:
        cuts[end] = 0
    else:
        for start in range(1, end + 1):
            if palindrome[start][end]:
                cuts[end] = min(cuts[end], cuts[[1]] + [2])

Drag options to blanks, or click blank then click option'

Astart - 1

Bstart + 1

C1

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using start + 1 which goes beyond the current substring.

Adding 0 cuts instead of 1.

We cut before start, so use cuts[start - 1] plus 1 cut.

5fill in blank

hard

Fill all three blanks to complete the function that returns minimum cuts for palindrome partitioning.

DSA Python

def min_cut_palindrome(s):
    n = len(s)
    palindrome = [[False] * n for _ in range(n)]
    cuts = [0] * n
    for i in range(n):
        cuts[i] = i
    for end in range(n):
        for start in range(end + 1):
            if s[start] == s[end] and (end - start < 2 or palindrome[start + 1][end - 1]):
                palindrome[start][end] = True
        if palindrome[0][end]:
            cuts[end] = 0
        else:
            for start in range(1, end + 1):
                if palindrome[start][end]:
                    cuts[end] = min(cuts[end], cuts[[1]] + [2])
    return cuts[[3]]

Drag options to blanks, or click blank then click option'

Astart - 1

B1

Cn - 1

Dend

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong indices for cuts array.

Returning cuts[end] which is not accessible outside the loop.

cuts[end] is updated using cuts[start - 1] + 1, and final answer is cuts[n - 1].

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