Palindrome Partitioning DP Minimum Cuts in DSA Python

Mental Model

We want to split a string into the fewest pieces so that each piece reads the same forwards and backwards.

Analogy: Imagine cutting a ribbon into the least number of palindromic segments, like cutting a patterned ribbon where each piece must have a symmetrical design.

s = "a b a c"
Indices: 0 1 2 3
We want cuts like: a | b | a | c or a b a | c
Visual:
 a -> b -> a -> c -> null
Cuts separate palindrome parts.

Dry Run Walkthrough

Input: string: "aab"

Goal: Find the minimum cuts needed to split "aab" into palindromic substrings

Step 1: Check all substrings for palindrome property

Palindrome table:
[True, True, False]
[False, True, False]
[False, False, True]

Why: Knowing which substrings are palindromes helps decide where to cut

Step 2: Initialize cuts array with max cuts (worst case: cut before every char)

cuts = [0, 1, 2]

Why: Start with the assumption that each character is a separate palindrome

Step 3: For i=1 (substring 'aa'), update cuts since 'aa' is palindrome

cuts = [0, 0, 2]

Why: 'aa' is palindrome, so no cut needed between 0 and 1

Step 4: For i=2 (substring 'aab'), check palindromes ending at 2

cuts = [0, 0, 1]

Why: 'b' alone is palindrome, so cut after 'aa' gives 1 cut total

Result:

cuts array: [0, 0, 1]
Minimum cuts needed: 1

Annotated Code

DSA Python

class Solution:
    def minCut(self, s: str) -> int:
        n = len(s)
        # Create a 2D table to store palindrome info
        palindrome = [[False]*n for _ in range(n)]
        # cuts[i] = min cuts for substring s[:i+1]
        cuts = [0]*n

        for i in range(n):
            min_cut = i  # max cuts possible
            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
                    # If palindrome starts at 0, no cut needed
                    if j == 0:
                        min_cut = 0
                    else:
                        min_cut = min(min_cut, cuts[j-1] + 1)
            cuts[i] = min_cut
        return cuts[-1] if n else 0

# Driver code
sol = Solution()
input_str = "aab"
print(sol.minCut(input_str))

for i in range(n):

iterate over end index of substring

min_cut = i # max cuts possible

initialize max cuts for substring s[:i+1]

for j in range(i+1):

iterate over start index of substring

if s[j] == s[i] and (i - j < 2 or palindrome[j+1][i-1]):

check if substring s[j:i+1] is palindrome

palindrome[j][i] = True

mark substring as palindrome

if j == 0:

if palindrome starts at beginning, no cut needed

min_cut = 0

update min_cut to zero

else:

if palindrome starts after 0, add one cut before j

min_cut = min(min_cut, cuts[j-1] + 1)

choose minimum cuts between current and previous plus one

cuts[i] = min_cut

store minimum cuts for substring s[:i+1]

Output

1

Complexity Analysis

Time: O(n^2) because we check all substrings and update cuts accordingly

Space: O(n^2) for the palindrome table plus O(n) for cuts array

vs Alternative: Naive approach tries all partitions recursively with exponential time; DP reduces it to polynomial time

Edge Cases

empty string

returns 0 cuts since no partition needed

DSA Python

n = len(s)

string with all same characters like "aaa"

returns 0 cuts because whole string is palindrome

DSA Python

if j == 0: min_cut = 0

string with no palindromic substring longer than 1 like "abc"

returns length-1 cuts (2 for "abc")

DSA Python

min_cut = i

When to Use This Pattern

When asked to split a string into palindromic parts with minimum cuts, use DP with palindrome substring precomputation to optimize.

Common Mistakes

Mistake: Not checking the palindrome condition correctly for substrings longer than 2
Fix: Include the condition palindrome[j+1][i-1] to ensure inner substring is palindrome

Mistake: Initializing cuts array incorrectly leading to wrong minimum cuts
Fix: Initialize cuts[i] with max cuts possible (i) before minimizing

Summary

Finds the fewest cuts needed to split a string into palindromic substrings.

Use when you want to partition strings into palindromes with minimal splits.

Key insight: precompute palindrome substrings and use DP to track minimum cuts.

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