Palindrome Partitioning DP Minimum Cuts in DSA Python - Build from Scratch

Palindrome Partitioning DP Minimum Cuts

📖 Scenario: Imagine you have a word and want to split it into parts so that each part reads the same forwards and backwards. This is called palindrome partitioning. You want to find the smallest number of cuts needed to do this.

🎯 Goal: Build a program that finds the minimum number of cuts needed to split a given string into palindromic parts using dynamic programming.

📋 What You'll Learn

Create a string variable with the exact value 'noonabbad'

Create a 2D list called palindrome to store palindrome checks

Create a list called cuts to store minimum cuts for each position

Use nested loops to fill palindrome with True/False for substrings

Use a loop to calculate minimum cuts using the palindrome table

Print the minimum cuts needed for the whole string

💡 Why This Matters

🌍 Real World

Palindrome partitioning helps in text processing, DNA sequence analysis, and data compression where symmetrical patterns matter.

💼 Career

Understanding dynamic programming and string manipulation is important for software development, especially in algorithm design and coding interviews.

1

Create the input string

Create a string variable called s with the exact value 'noonabbad'.

DSA Python

# Create the string variable s with value 'noonabbad'
# Your code here

Hint

Use s = 'noonabbad' to create the string.

2

Setup palindrome table and cuts list

Create a 2D list called palindrome with size len(s) x len(s) filled with False. Also create a list called cuts of length len(s) filled with zeros.

DSA Python

s = 'noonabbad'
# Create palindrome 2D list and cuts list
# Your code here

Hint

Use list comprehension to create the 2D list and a list of zeros for cuts.

3

Fill palindrome table and calculate minimum cuts

Use nested loops with variables end from 0 to len(s)-1 and start from 0 to end. Inside, set palindrome[start][end] to True if s[start] == s[end] and either end - start <= 2 or palindrome[start+1][end-1] is True. Then, use a loop with variable i from 0 to len(s)-1 to calculate minimum cuts: if palindrome[0][i] is True, set cuts[i] to 0; else set cuts[i] to the minimum of cuts[j] + 1 for all j from 0 to i-1 where palindrome[j+1][i] is True.

DSA Python

s = 'noonabbad'
length = len(s)
palindrome = [[False] * length for _ in range(length)]
cuts = [0] * length
# Fill palindrome table and calculate cuts
# Your code here

Hint

Use nested loops to fill palindrome table. Then use a loop to find minimum cuts using the palindrome table.

4

Print the minimum cuts needed

Print the minimum cuts needed for the whole string by printing cuts[len(s) - 1].

DSA Python

s = 'noonabbad'
length = len(s)
palindrome = [[False] * length for _ in range(length)]
cuts = [0] * length

for end in range(length):
    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

for i in range(length):
    if palindrome[0][i]:
        cuts[i] = 0
    else:
        cuts[i] = min(cuts[j] + 1 for j in range(i) if palindrome[j + 1][i])

# Print the minimum cuts needed for the whole string
# Your code here

Hint

Print cuts[length - 1] to show the minimum cuts needed.

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