Palindrome Partitioning DP Minimum Cuts in DSA Python - Execution Trace

Concept Flow - Palindrome Partitioning DP Minimum Cuts

Start with full string

↓

Check all substrings palindrome?

↓

If palindrome: cuts=0

↓

Else: Try all partitions

↓

Calculate min cuts = min(left cuts + right cuts + 1)

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Store result in DP table

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Return min cuts for full string

We check all substrings to find palindromes, then use dynamic programming to find the minimum cuts needed to partition the string into palindromes.

Execution Sample

DSA Python

s = "aab"

# dp[i] = min cuts for s[:i]
dp = [0]*(len(s)+1)
for i in range(len(s)+1):
    dp[i] = i-1

for i in range(len(s)):
    for j in range(i, len(s)):
        if s[i:j+1] == s[i:j+1][::-1]:
            dp[j+1] = min(dp[j+1], dp[i]+1)

print(dp[len(s)])

This code calculates the minimum cuts needed to partition the string 'aab' into palindromes.

Execution Table

Step	Operation	Substring Checked	Is Palindrome?	DP Array State	Visual State
1	Initialize dp	-	-	[-1, 0, 1, 2]	dp = [-1,0,1,2] (index means substring length)
2	Check substring s[0:1] = 'a'	a	Yes	[-1, 0, 1, 2]	No change, dp[1] = min(dp[1], dp[0]+1) = min(0, -1+1)=0
3	Check substring s[0:2] = 'aa'	aa	Yes	[-1, 0, 0, 2]	dp[2] = min(dp[2], dp[0]+1) = min(1, -1+1)=0
4	Check substring s[0:3] = 'aab'	aab	No	[-1, 0, 0, 2]	No change
5	Check substring s[1:2] = 'a'	a	Yes	[-1, 0, 0, 2]	dp[2] = min(dp[2], dp[1]+1) = min(0, 0+1)=0 (no change)
6	Check substring s[1:3] = 'ab'	ab	No	[-1, 0, 0, 2]	No change
7	Check substring s[2:3] = 'b'	b	Yes	[-1, 0, 0, 1]	dp[3] = min(dp[3], dp[2]+1) = min(2, 0+1)=1
8	End	-	-	[-1, 0, 0, 1]	Minimum cuts for 'aab' is dp[3] = 1

💡 All substrings checked; dp array holds minimum cuts for each prefix; final answer dp[len(s)] = 1

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	Final
dp	[-1,0,1,2]	[-1,0,1,2]	[-1,0,0,2]	[-1,0,0,2]	[-1,0,0,2]	[-1,0,0,2]	[-1,0,0,1]	[-1,0,0,1]
substring	-	'a'	'aa'	'aab'	'a'	'ab'	'b'	-
is_palindrome	-	True	True	False	True	False	True	-

Key Moments - 3 Insights

Why do we initialize dp[i] = i-1 at the start?

Because in the worst case, each character is a palindrome itself, so the maximum cuts needed for substring length i is i-1. This is shown in Step 1 of the execution_table.

Why do we update dp[j+1] with dp[i] + 1 when substring s[i:j+1] is palindrome?

Because if s[i:j+1] is palindrome, we can cut before i, so total cuts are cuts needed for s[:i] plus one more cut. This is shown in Steps 3 and 7 where dp array updates.

Why do we check all substrings s[i:j+1] instead of just some?

Because any substring can be palindrome, and we need to find the minimum cuts considering all possible partitions. This exhaustive check is shown in the nested loops in the code and all substring checks in execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at Step 3, what is the dp array state?

A [-1, 0, 0, 2]

B [-1, 0, 1, 2]

C [-1, 1, 0, 2]

D [-1, 0, 0, 1]

💡 Hint

Check the 'DP Array State' column at Step 3 in execution_table.

At which step does the dp[3] value change from 2 to 1?

A Step 4

B Step 7

C Step 5

D Step 6

💡 Hint

Look at the 'DP Array State' column and 'Step' column in execution_table for dp[3] changes.

If the string was 'aaa' instead of 'aab', how would dp[3] change at the end?

A dp[3] would be 2

B dp[3] would be 1

C dp[3] would be 0

D dp[3] would be -1

💡 Hint

Consider that 'aaa' is a palindrome itself, so no cuts needed; check how dp updates for palindrome substrings.

Concept Snapshot

Palindrome Partitioning Minimum Cuts:
- Use dp array where dp[i] = min cuts for s[:i]
- Initialize dp[i] = i-1 (max cuts)
- For each substring s[i:j], if palindrome, dp[j+1] = min(dp[j+1], dp[i]+1)
- Result is dp[len(s)]
- Checks all substrings for palindrome to find min cuts

Full Transcript

This concept uses dynamic programming to find the minimum number of cuts needed to partition a string into palindromes. We initialize a dp array where dp[i] represents the minimum cuts needed for the substring from start to index i. Initially, dp[i] is set to i-1, the worst case where each character is a palindrome itself. We then check every substring s[i:j] to see if it is a palindrome. If it is, we update dp[j+1] to the minimum of its current value and dp[i] + 1, representing a cut before index i plus the cuts needed for the prefix. After checking all substrings, dp[len(s)] holds the minimum cuts needed for the entire string. The example with string 'aab' shows dp updates step by step, ending with dp[3] = 1, meaning one cut is needed. This method ensures all palindrome partitions are considered to find the 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