Longest Palindromic Substring in DSA Rust - Time & Space Complexity

Time Complexity: Longest Palindromic Substring

O(n²)

Understanding Time Complexity

We want to understand how the time needed to find the longest palindromic substring changes as the input string gets longer.

How does the number of steps grow when the string size increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

fn longest_palindrome(s: &str) -> String {
    let chars: Vec = s.chars().collect();
    let n = chars.len();
    let mut start = 0;
    let mut max_len = 1;

    for i in 0..n {
        // Check odd length palindrome
        let (l1, r1) = expand_around_center(&chars, i, i);
        // Check even length palindrome
        let (l2, r2) = expand_around_center(&chars, i, i + 1);

        let len1 = r1 - l1 + 1;
        let len2 = r2 - l2 + 1;

        if len1 > max_len {
            start = l1;
            max_len = len1;
        }
        if len2 > max_len {
            start = l2;
            max_len = len2;
        }
    }

    chars[start..start + max_len].iter().collect()
}

fn expand_around_center(chars: &[char], mut left: usize, mut right: usize) -> (usize, usize) {
    while left <= right && right < chars.len() && chars[left] == chars[right] {
        if left == 0 { break; }
        left -= 1;
        right += 1;
    }
    (left + 1, right - 1)
}
    

This code finds the longest palindrome by expanding around each character and its neighbor to check odd and even length palindromes.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The outer loop runs once for each character in the string.
• How many times: For each character, the expand_around_center function expands outwards checking characters until mismatch or ends.

How Execution Grows With Input

As the string length grows, the outer loop runs more times, and each expansion can check many characters.

Input Size (n)	Approx. Operations
10	About 100 checks
100	About 10,000 checks
1000	About 1,000,000 checks

Pattern observation: The number of checks grows roughly with the square of the input size.

Final Time Complexity

Time Complexity: O(n²)

This means if the input string doubles in length, the time to find the longest palindrome roughly quadruples.

Common Mistake

[X] Wrong: "Since we only expand around centers, the time is O(n)."

[OK] Correct: Each expansion can check many characters, so the total work adds up to about n times n, not just n.

Interview Connect

Understanding this time complexity helps you explain why some palindrome solutions are slower and shows your grasp of nested loops and string processing.

Self-Check

"What if we used dynamic programming to store palindrome results? How would the time complexity change?"

Practice

1. What is the main goal of the Longest Palindromic Substring problem?

easy

A. Find the longest substring that contains only vowels

B. Find the longest substring with all unique characters

C. Find the longest substring with increasing characters

D. Find the longest substring that reads the same forwards and backwards

Solution

1. Step 1: Understand palindrome definition

   A palindrome reads the same forwards and backwards, like 'madam' or 'racecar'.

2. Step 2: Identify problem goal

   The problem asks to find the longest continuous part of a string that is a palindrome.

3. Final Answer:

   Find the longest substring that reads the same forwards and backwards -> Option D

4. Quick Check:

   Longest Palindromic Substring = Find longest palindrome [OK]

Hint: Look for substring that mirrors itself [OK]

Common Mistakes:

• Confusing substring with subsequence
• Looking for unique characters instead of palindrome
• Ignoring continuous substring requirement

2. Which Rust code snippet correctly declares a function header for finding the longest palindromic substring in a string slice s?

easy

A. fn longest_palindrome(s: &str) -> &str { }

B. fn longest_palindrome(s: String) -> String { }

C. fn longest_palindrome(s: &String) -> &String { }

D. fn longest_palindrome(s: str) -> str { }

Solution

1. Step 1: Understand Rust string types

   Rust uses &str for string slices and String for owned strings.

2. Step 2: Choose correct function signature

   Function should take a string slice &str and return a string slice &str for substring references.

3. Final Answer:

   fn longest_palindrome(s: &str) -> &str { } -> Option A

4. Quick Check:

   Use &str for slices in function signature [OK]

Hint: Use &str for input and output substrings [OK]

Common Mistakes:

• Using owned String when slice is better
• Returning owned String unnecessarily
• Incorrect parameter types like str without &

3. What will be the output of the following Rust code snippet that finds the longest palindromic substring?

fn main() {
    let s = "babad";
    let result = longest_palindrome(s);
    println!("{}", result);
}

fn longest_palindrome(s: &str) -> &str {
    let chars: Vec<char> = s.chars().collect();
    let mut start = 0;
    let mut max_len = 1;

    for i in 0..chars.len() {
        let mut left = i as i32;
        let mut right = i as i32;

        while left >= 0 && right < chars.len() as i32 && chars[left as usize] == chars[right as usize] {
            if (right - left + 1) > max_len {
                start = left as usize;
                max_len = (right - left + 1) as usize;
            }
            left -= 1;
            right += 1;
        }
    }

    &s[start..start + max_len]
}

medium

A. aba

B. bad

C. bab

D. babad

Solution

1. Step 1: Understand the code logic

   The code expands around each character to find odd-length palindromes and tracks the longest.

2. Step 2: Trace the input "babad"

   At index 0: palindrome "b" (length 1), index 1: "bab" (length 3), index 2: "aba" (length 3). The first longest palindrome found is "bab".

3. Final Answer:

   bab -> Option C

4. Quick Check:

   Longest palindrome substring = bab [OK]

Hint: Expand around center, pick first longest palindrome [OK]

Common Mistakes:

• Assuming palindrome must be unique
• Returning entire string instead of substring
• Confusing substring indices

4. Identify the bug in this Rust function that tries to find the longest palindromic substring:

fn longest_palindrome(s: &str) -> &str {
    let chars: Vec<char> = s.chars().collect();
    let mut start = 0;
    let mut max_len = 1;

    for i in 0..chars.len() {
        let mut left = i as i32;
        let mut right = i as i32 + 1;

        while left >= 0 && right < chars.len() as i32 && chars[left as usize] == chars[right as usize] {
            if (right - left + 1) > max_len {
                start = left as usize;
                max_len = (right - left + 1) as usize;
            }
            left -= 1;
            right += 1;
        }
    }

    &s[start..start + max_len]
}

medium

A. The max_len should start at 0, not 1

B. The initial right pointer should be equal to i, not i + 1

C. The loop should iterate from 1 to length, not 0 to length

D. The function should return a String, not &str

Solution

1. Step 1: Analyze pointer initialization

   Right pointer starts at i + 1, which only checks even-length palindromes, missing odd-length ones.

2. Step 2: Correct pointer for odd-length palindromes

   To check odd-length palindromes, right pointer should start at i, same as left pointer.

3. Final Answer:

   The initial right pointer should be equal to i, not i + 1 -> Option B

4. Quick Check:

   Right pointer = i for odd-length palindrome check [OK]

Hint: Start right pointer same as left for odd palindromes [OK]

Common Mistakes:

• Only checking even-length palindromes
• Wrong loop range causing index errors
• Returning wrong type causing borrow issues

5. Given the string "forgeeksskeegfor", which approach correctly finds the longest palindromic substring and what is that substring?

hard

A. Expand around center method; longest palindrome is "geeksskeeg"

B. Dynamic programming with substring table; longest palindrome is "forgeeksskeegfor"

C. Brute force checking all substrings; longest palindrome is "ss"

D. Using hash map for character counts; longest palindrome is "ks"

Solution

1. Step 1: Understand approaches to find longest palindrome

   Expand around center and dynamic programming are common methods; brute force is slow but correct.

2. Step 2: Identify longest palindrome in given string

   "forgeeksskeegfor" contains "geeksskeeg" as the longest palindrome substring, not the entire string.

3. Step 3: Match approach and answer

   Expand around center method efficiently finds "geeksskeeg"; dynamic programming would too but not the full string.

4. Final Answer:

   Expand around center method; longest palindrome is "geeksskeeg" -> Option A

5. Quick Check:

   Longest palindrome = geeksskeeg by center expansion [OK]

Hint: Expand around centers to find longest palindrome fast [OK]

Common Mistakes:

• Assuming entire string is palindrome
• Using character counts instead of substring checks
• Confusing brute force with efficient methods