Palindrome Partitioning DP Minimum Cuts in DSA Rust

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 smallest number of palindromic segments so each segment looks the same from both ends.

s: a b a c d c b a
cuts: [0] -> [1] -> [2] -> ... -> [n]
palindrome check matrix:
  T F T F F F F T
  F T F F F F F F
  T F T F F F F T
  F F F T F T F F
  F F F F T F F F
  F F F T F T F F
  F F F F F F T F
  T F T F F F F T

Dry Run Walkthrough

Input: string: "aab"

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

Step 1: Initialize cuts array with max cuts: cuts = [0,1,2]

cuts: [0,1,2]
string: a a b

Why: Start assuming worst case: cut before every character

Step 2: Check palindrome substrings starting at index 0

palindrome substrings: "a" (0-0), "aa" (0-1)
Update cuts for "a" at 0: cuts[0] = 0
Update cuts for "aa" at 1: cuts[1] = 0

Why: "a" and "aa" are palindromes, so no cuts needed before index 1

Step 3: Check palindrome substrings starting at index 1

palindrome substring: "a" (1-1)
cuts remain: cuts = [0,0,2]

Why: "a" is palindrome, cuts at 1 stays 0

Step 4: Check palindrome substrings starting at index 2

palindrome substring: "b" (2-2)
Update cuts for "b" at 2: cuts[2] = cuts[1] + 1 = 1

Why: "b" is palindrome, so cuts at 2 is cuts at 1 plus one cut

Result:

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

Annotated Code

DSA Rust

fn min_cut_palindrome_partition(s: &str) -> i32 {
    let n = s.len();
    if n == 0 {
        return 0;
    }
    let chars: Vec<char> = s.chars().collect();
    let mut cuts = vec![0; n];
    let mut palindrome = vec![vec![false; n]; n];

    for i in 0..n {
        cuts[i] = i as i32; // max cuts
    }

    for end in 0..n {
        for start in 0..=end {
            if chars[start] == chars[end] && (end - start < 2 || palindrome[start + 1][end - 1]) {
                palindrome[start][end] = true;
                if start == 0 {
                    cuts[end] = 0; // no cut needed if palindrome from start
                } else {
                    cuts[end] = cuts[end].min(cuts[start - 1] + 1); // cut before start
                }
            }
        }
    }
    cuts[n - 1]
}

fn main() {
    let s = "aab";
    let result = min_cut_palindrome_partition(s);
    println!("Minimum cuts needed: {}", result);
}

cuts[i] = i as i32; // max cuts

initialize cuts assuming worst case: cut before every character

if chars[start] == chars[end] && (end - start < 2 || palindrome[start + 1][end - 1]) {

check if substring s[start..=end] is palindrome

palindrome[start][end] = true;

mark substring as palindrome

if start == 0 { cuts[end] = 0; } else { cuts[end] = cuts[end].min(cuts[start - 1] + 1); }

update cuts: no cut if palindrome from start, else cut before start

Output

Minimum cuts needed: 1

Complexity Analysis

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

Space: O(n^2) for 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 partitions needed

DSA Rust

if n == 0 { return 0; }

string with all identical characters, e.g. "aaaa"

returns 0 cuts since whole string is palindrome

DSA Rust

if start == 0 { cuts[end] = 0; }

string with no palindromic substring longer than 1, e.g. "abc"

returns n-1 cuts, cutting before every character

DSA Rust

cuts[i] = i as i32; // max cuts initialization

When to Use This Pattern

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

Common Mistakes

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

Mistake: Not initializing cuts array with maximum cuts before DP
Fix: Initialize cuts[i] = i to represent worst case cuts

Summary

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

Use when you want to partition a string into palindromes with fewest cuts.

The key is to use DP to check palindromes and update cuts efficiently.

Practice

1. What does the "Palindrome Partitioning DP Minimum Cuts" problem aim to find for a given string s?

easy

A. The minimum number of cuts needed to partition s into palindromic substrings

B. The longest palindromic substring in s

C. The total number of palindromic substrings in s

D. The maximum number of cuts possible in s

Solution

1. Step 1: Understand the problem goal

   The problem asks for the fewest cuts needed so that every piece after cutting is a palindrome.

2. Step 2: Clarify what minimum cuts mean

   Minimum cuts means the smallest number of splits to make all parts palindromes.

3. Final Answer:

   The minimum number of cuts needed to partition s into palindromic substrings -> Option A

4. Quick Check:

   Minimum cuts = minimum splits for palindrome parts [OK]

Hint: Minimum cuts split string into palindrome parts [OK]

Common Mistakes:

• Confusing minimum cuts with longest palindrome
• Thinking about total palindrome count instead of cuts
• Assuming maximum cuts is the goal

2. Which of the following Rust code snippets correctly initializes a 2D vector dp of size n x n with false values for palindrome checks?

easy

A. let dp = vec![vec![true; n]; n];

B. let dp = vec![false; n * n];

C. let dp = vec![vec![false; n * n]; 1];

D. let dp = vec![vec![false; n]; n];

Solution

1. Step 1: Understand 2D vector initialization in Rust

   To create an n by n 2D vector filled with false, use vec![vec![false; n]; n].

2. Step 2: Check each option

   let dp = vec![vec![false; n]; n]; correctly creates a 2D vector with n rows and n columns filled with false.

3. Final Answer:

   let dp = vec![vec![false; n]; n]; -> Option D

4. Quick Check:

   2D vector with false = vec![vec![false; n]; n] [OK]

Hint: Use vec![vec![false; n]; n] for 2D boolean vector [OK]

Common Mistakes:

• Using 1D vector instead of 2D
• Filling with true instead of false
• Incorrect dimensions in inner or outer vec

3. Given the Rust function below for minimum cuts, what is the output for input "aab"?

fn min_cut(s: &str) -> i32 {
    let n = s.len();
    let s_bytes = s.as_bytes();
    let mut dp = vec![0; n + 1];
    let mut palindrome = vec![vec![false; n]; n];
    for i in 0..=n {
        dp[i] = i as i32 - 1;
    }
    for i in 0..n {
        for j in i..n {
            if s_bytes[i] == s_bytes[j] && (j - i < 2 || palindrome[i + 1][j - 1]) {
                palindrome[i][j] = true;
                dp[j + 1] = dp[j + 1].min(dp[i] + 1);
            }
        }
    }
    dp[n]
}

println!("{}", min_cut("aab"));

medium

A. 3

B. 2

C. 1

D. 0

Solution

1. Step 1: Trace palindrome table for "aab"

   Substrings: "a" (0,0), "aa" (0,1), "aab" (0,2), "a" (1,1), "ab" (1,2), "b" (2,2). "aa" and single letters are palindrome.

2. Step 2: Calculate dp array values

   dp[0] = -1, dp[1] = 0, dp[2] = 1, dp[3] = 2 initially. Updates: For "aa", dp[2] = min(dp[2], dp[0] + 1) = 0. For "b", dp[3] = min(dp[3], dp[2] + 1) = 1.

3. Final Answer:

   1 -> Option C

4. Quick Check:

   Minimum cuts for "aab" = 1 [OK]

Hint: Check palindrome substrings and update dp minimum cuts [OK]

Common Mistakes:

• Not initializing dp correctly
• Ignoring palindrome length 1 or 2 cases
• Off-by-one errors in dp indexing

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

fn min_cut(s: &str) -> i32 {
    let n = s.len();
    let s_bytes = s.as_bytes();
    let mut dp = vec![0; n + 1];
    let mut palindrome = vec![vec![false; n]; n];
    for i in 0..n {
        dp[i] = i as i32;
    }
    for i in 0..n {
        for j in i..n {
            if s_bytes[i] == s_bytes[j] && (j - i < 2 || palindrome[i + 1][j - 1]) {
                palindrome[i][j] = true;
                dp[j + 1] = dp[j + 1].min(dp[i] + 1);
            }
        }
    }
    dp[n]
}

medium

A. Palindrome check condition is incorrect

B. dp array initialization should start from -1 for dp[0]

C. Loop ranges for i and j are reversed

D. s_bytes should be converted to chars instead of bytes

Solution

1. Step 1: Check dp initialization

   dp[i] should be initialized as i - 1 to handle zero cuts correctly, but here dp[i] = i which is off by one.

2. Step 2: Understand impact of dp initialization

   Incorrect initialization causes wrong minimum cuts calculation because dp[0] should be -1 to represent no cuts before start.

3. Final Answer:

   dp array initialization should start from -1 for dp[0] -> Option B

4. Quick Check:

   dp[0] = -1 needed for correct minimum cuts [OK]

Hint: Initialize dp[0] = -1 for correct minimum cuts [OK]

Common Mistakes:

• Initializing dp with zeros or i instead of i-1
• Misunderstanding palindrome base cases
• Incorrect loop boundaries

5. You want to find the minimum cuts for the string "abccbaabccba" using the palindrome partitioning DP approach. Which of the following statements is TRUE about the expected minimum cuts and the DP process?

hard

A. The minimum cuts will be 0 because the entire string is a palindrome

B. The minimum cuts will be 1 because the string can be split into two palindromes

C. The minimum cuts will be 2 because the string contains two palindromic halves

D. The minimum cuts will be 3 because multiple small palindromes are needed

Solution

1. Step 1: Check if the entire string is palindrome

   "abccbaabccba" reads the same forwards and backwards, so it is a palindrome.

2. Step 2: Understand minimum cuts for a full palindrome

   If the whole string is palindrome, no cuts are needed, so minimum cuts = 0.

3. Final Answer:

   The minimum cuts will be 0 because the entire string is a palindrome -> Option A

4. Quick Check:

   Full palindrome means zero cuts [OK]

Hint: If whole string palindrome, minimum cuts = 0 [OK]

Common Mistakes:

• Assuming cuts needed even if full palindrome
• Splitting unnecessarily into halves
• Ignoring palindrome check for entire string