Palindrome Partitioning DP Minimum Cuts in DSA Java - Execution Trace

Concept Flow - Palindrome Partitioning DP Minimum Cuts

Start with full string

↓

Check all substrings palindrome?

↓

Build palindrome table

↓

Initialize cuts array

↓

For each position i in string

↓

Find min cuts using palindrome info

↓

Return cuts for full string

We first find all palindrome substrings, then use that info to find minimum cuts needed to partition the string into palindromes.

Execution Sample

DSA Java

String s = "aab";
boolean[][] isPal = new boolean[s.length()][s.length()];
int[] cuts = new int[s.length()];
// Fill isPal and cuts arrays
return cuts[s.length() - 1];

This code finds minimum cuts to split 'aab' into palindrome substrings.

Execution Table

Step	Operation	Palindrome Table State	Cuts Array State	Visual State
1	Initialize palindrome table for single chars	[[T, F, F], [F, T, F], [F, F, T]]	[0, 0, 0]	"a" | "a" | "b"
2	Check substrings length 2: s[0..1] = "aa" palindrome?	[[T, T, F], [F, T, F], [F, F, T]]	[0, 0, 0]	"aa" | "b"
3	Check substrings length 2: s[1..2] = "ab" palindrome?	[[T, T, F], [F, T, F], [F, F, F]]	[0, 0, 0]	"aa" | "b"
4	Check substrings length 3: s[0..2] = "aab" palindrome?	[[T, T, F], [F, T, F], [F, F, F]]	[0, 0, 0]	"aab"
5	Calculate cuts for i=0 ("a")	[[T, T, F], [F, T, F], [F, F, F]]	[0, 0, 0]	"a"
6	Calculate cuts for i=1 ("aa")	[[T, T, F], [F, T, F], [F, F, F]]	[0, 0, 0]	"aa"
7	Calculate cuts for i=2 ("aab")	[[T, T, F], [F, T, F], [F, F, F]]	[0, 0, 1]	"aab"
8	Return minimum cuts for full string	[[T, T, F], [F, T, F], [F, F, F]]	[0, 0, 1]	"aab"

💡 All substrings checked and minimum cuts calculated for full string.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	Final
isPal	[[F,F,F],[F,F,F],[F,F,F]]	[[T,F,F],[F,T,F],[F,F,T]]	[[T,T,F],[F,T,F],[F,F,T]]	[[T,T,F],[F,T,F],[F,F,F]]	[[T,T,F],[F,T,F],[F,F,F]]	[[T,T,F],[F,T,F],[F,F,F]]	[[T,T,F],[F,T,F],[F,F,F]]	[[T,T,F],[F,T,F],[F,F,F]]	[[T,T,F],[F,T,F],[F,F,F]]
cuts	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,0]	[0,0,1]	[0,0,1]

Key Moments - 3 Insights

Why do we mark single characters as palindrome first?

Because every single character is a palindrome by itself. This is shown in step 1 where isPal[i][i] is set to true for all i.

Why does the cuts array start with zeros?

Initially, we assume zero cuts. We update cuts only when we find a substring that is not palindrome or needs partitioning. See steps 5-7 where cuts get updated.

Why is the substring "aab" not palindrome but "aa" is?

Because "aa" reads the same forwards and backwards, but "aab" does not. This is reflected in step 4 where isPal[0][2] remains false.

Visual Quiz - 3 Questions

Look at the execution table at step 2, what is the palindrome state of substring "aa"?

A True

B False

C Unknown

D Not checked

💡 Hint

Check the 'Palindrome Table State' column at step 2 in the execution table.

At which step does the cuts array first change from all zeros?

A Step 5

B Step 6

C Step 7

D Step 8

💡 Hint

Look at the 'Cuts Array State' column in the execution table and find when it changes from [0,0,0].

If the input string was "aaa", how would the palindrome table change at step 2?

A isPal[0][1] would be true

B isPal[1][2] would be false

C isPal[0][2] would be false

D No changes from original

💡 Hint

Check how palindrome substrings of length 2 are marked in the palindrome table at step 2.

Concept Snapshot

Palindrome Partitioning DP Minimum Cuts:
- Use a 2D boolean table isPal to mark palindrome substrings.
- Initialize all single chars as palindrome.
- Check substrings of increasing length.
- Use cuts array to store min cuts needed up to each index.
- If substring s[j..i] is palindrome, update cuts[i] = min(cuts[i], cuts[j-1]+1).
- Result is cuts[n-1], min cuts for full string.

Full Transcript

This visualization shows how to find the minimum cuts needed to partition a string into palindrome substrings using dynamic programming. We start by marking all single characters as palindrome. Then we check longer substrings and update a palindrome table. Using this table, we calculate the minimum cuts needed at each position. The cuts array is updated by checking if the substring ending at current position is palindrome and choosing the minimum cuts accordingly. Finally, the minimum cuts for the full string is returned.

Practice

1. What does the term minimum cuts mean in the context of palindrome partitioning of a string?

easy

A. The smallest number of cuts needed to split the string into palindromic substrings

B. The maximum number of palindromic substrings in the string

C. The length of the longest palindromic substring

D. The total number of substrings in the string

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 so that every substring after cutting is a palindrome.

3. Final Answer:

   The smallest number of cuts needed to split the string into palindromic substrings -> Option A

4. Quick Check:

   Minimum cuts = smallest cuts to get palindromes [OK]

Hint: Minimum cuts = least splits for palindrome parts [OK]

Common Mistakes:

• Confusing minimum cuts with maximum palindromes
• Thinking minimum cuts means longest palindrome length
• Counting total substrings instead of cuts

2. Which of the following Java code snippets correctly initializes the DP array for minimum cuts in palindrome partitioning of a string s of length n?

easy

A. int[] cuts = new int[n]; Arrays.fill(cuts, -1);

B. int[] cuts = new int[n]; for (int i = 0; i <= n; i++) cuts[i] = 0;

C. int[] cuts = new int[n]; for (int i = 0; i < n; i++) cuts[i] = i;

D. int[] cuts = new int[n+1]; Arrays.fill(cuts, Integer.MAX_VALUE);

Solution

1. Step 1: Understand DP array initialization

   We initialize cuts array where cuts[i] = i means max cuts needed for substring s[0..i].

2. Step 2: Check code correctness

   int[] cuts = new int[n]; for (int i = 0; i < n; i++) cuts[i] = i; initializes cuts array of size n and sets cuts[i] = i correctly. Other options have errors like wrong size or wrong initialization.

3. Final Answer:

   int[] cuts = new int[n]; for (int i = 0; i < n; i++) cuts[i] = i; -> Option C

4. Quick Check:

   Initialize cuts with max cuts = index [OK]

Hint: Initialize cuts[i] = i for max cuts before optimization [OK]

Common Mistakes:

• Using wrong array size causing IndexOutOfBounds
• Filling cuts with zero instead of max cuts
• Using negative values which are invalid

3. What is the output of the following Java code snippet for string s = "aab"?

public class Solution {
  public static int minCut(String s) {
    int n = s.length();
    int[] cuts = new int[n];
    boolean[][] palindrome = new boolean[n][n];
    for (int i = 0; i < n; i++) cuts[i] = i;
    for (int end = 0; end < n; end++) {
      for (int start = 0; start <= end; start++) {
        if (s.charAt(start) == s.charAt(end) && (end - start < 2 || palindrome[start + 1][end - 1])) {
          palindrome[start][end] = true;
          cuts[end] = start == 0 ? 0 : Math.min(cuts[end], cuts[start - 1] + 1);
        }
      }
    }
    return cuts[n - 1];
  }
  public static void main(String[] args) {
    System.out.println(minCut("aab"));
  }
}

medium

A. 0

B. 1

C. 2

D. 3

Solution

1. Step 1: Analyze palindrome substrings in "aab"

   "a", "a", "b" are palindromes. "aa" is palindrome. Minimum cuts needed is 1 (cut between "aa" and "b").

2. Step 2: Trace DP updates

   cuts array updates to [0,0,1]. Final cuts[n-1] = 1.

3. Final Answer:

   1 -> Option B

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindromes and update cuts with min previous cuts +1 [OK]

Common Mistakes:

• Assuming no cuts needed for "aab"
• Counting cuts as number of palindromes
• Ignoring palindrome "aa" substring

4. Identify the bug in the following Java code snippet for palindrome partitioning minimum cuts:

public int minCut(String s) {
  int n = s.length();
  int[] cuts = new int[n];
  boolean[][] palindrome = new boolean[n][n];
  for (int i = 0; i <= n; i++) cuts[i] = i;
  for (int end = 0; end < n; end++) {
    for (int start = 0; start <= end; start++) {
      if (s.charAt(start) == s.charAt(end) && (end - start < 2 || palindrome[start + 1][end - 1])) {
        palindrome[start][end] = true;
        cuts[end] = start == 0 ? 0 : Math.min(cuts[end], cuts[start - 1] + 1);
      }
    }
  }
  return cuts[n - 1];
}

medium

A. The return statement should be 'cuts[n]' instead of 'cuts[n - 1]'

B. The palindrome check condition is incorrect

C. The cuts array is not initialized properly

D. The loop 'for (int i = 0; i <= n; i++)' causes IndexOutOfBoundsException

Solution

1. Step 1: Check array initialization loop

   cuts array size is n, but loop runs i from 0 to n inclusive, causing IndexOutOfBoundsException at i = n.

2. Step 2: Confirm other parts are correct

   Palindrome check and return statement are correct. Initialization loop must be 'i < n'.

3. Final Answer:

   The loop 'for (int i = 0; i <= n; i++)' causes IndexOutOfBoundsException -> Option D

4. Quick Check:

   Array size n but loop runs to n inclusive causes error [OK]

Hint: Loop to n-1 for array size n to avoid out-of-bounds [OK]

Common Mistakes:

• Using <= n instead of < n in loops
• Misunderstanding palindrome condition
• Returning wrong index from cuts array

5. Given the string s = "ababbbabbababa", what is the minimum number of cuts needed to partition it into palindromic substrings using the DP minimum cuts approach?

hard

A. 4

B. 5

C. 3

D. 6

Solution

1. Step 1: Understand the problem and DP approach

   We find minimum cuts so that each substring is palindrome. The string "ababbbabbababa" has palindromic partitions with 4 cuts.

2. Step 2: Trace or recall known result

   Known minimum cuts for this string is 4, splitting into palindromes like "a|babbbab|b|ababa" or similar.

3. Final Answer:

   4 -> Option A

4. Quick Check:

   Minimum cuts for "ababbbabbababa" = 4 [OK]

Hint: Use DP to find min cuts; known result for this string is 4 [OK]

Common Mistakes:

• Guessing cuts without checking palindrome substrings
• Counting number of palindromes instead of cuts
• Confusing cuts with substring length