Longest Palindromic Substring in DSA Typescript - 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 grows as the input string gets longer.

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

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function longestPalindrome(s: string): string {
  let start = 0, maxLength = 1;
  for (let i = 0; i < s.length; i++) {
    let left = i, right = i;
    while (left >= 0 && right < s.length && s[left] === s[right]) {
      if (right - left + 1 > maxLength) {
        start = left;
        maxLength = right - left + 1;
      }
      left--;
      right++;
    }
  }
  return s.substring(start, start + maxLength);
}
    

This code finds the longest odd-length palindrome by expanding around each character as the center.

Identify Repeating Operations

• Primary operation: The inner while loop expands around each center to check for palindromes.
• How many times: The outer for loop runs once per character (n times), and the inner while loop can run up to n times in the worst case.

How Execution Grows With Input

As the string length grows, the number of checks grows roughly with the square of the length.

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

Pattern observation: Doubling the input size roughly quadruples the work needed.

Final Time Complexity

Time Complexity: O(n²)

This means the time to find the longest palindrome grows roughly with the square of the string length.

Common Mistake

[X] Wrong: "The algorithm runs in linear time because it only loops through the string once."

[OK] Correct: Although the outer loop runs once per character, the inner loop can expand many times, making the total work much larger than just one pass.

Interview Connect

Understanding this time complexity helps you explain how your solution scales and shows you can analyze nested loops clearly.

Self-Check

"What if we also expand around pairs of characters to handle even-length palindromes? 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 with all unique characters

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

C. Find the longest substring with increasing characters

D. Find the longest substring that contains only vowels

Solution

1. Step 1: Understand palindrome definition

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

2. Step 2: Identify the problem goal

   The problem asks to find the longest substring inside a string that is a palindrome.

3. Final Answer:

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

4. Quick Check:

   Palindrome substring = longest palindrome [OK]

Hint: Look for substrings that mirror around a center [OK]

Common Mistakes:

• Confusing palindrome with unique characters
• Looking for longest substring without repeats
• Thinking about substrings with vowels only

2. Which of the following TypeScript function signatures correctly represents a function to find the longest palindromic substring?

easy

A. function longestPalindrome(s: string): string

B. function longestPalindrome(s: number): number

C. function longestPalindrome(s: string): number[]

D. function longestPalindrome(s: boolean): string

Solution

1. Step 1: Check input parameter type

   The input is a string, so parameter type must be string.

2. Step 2: Check return type

   The function returns the longest palindromic substring, which is a string.

3. Final Answer:

   function longestPalindrome(s: string): string -> Option A

4. Quick Check:

   Input string, output string [OK]

Hint: Input and output are both strings for this problem [OK]

Common Mistakes:

• Using wrong input type like number or boolean
• Returning array instead of string
• Mixing input and output types

3. What will be the output of the following TypeScript code?

function longestPalindrome(s: string): string {
  let start = 0, maxLength = 1;
  for (let i = 0; i < s.length; i++) {
    let low = i, high = i;
    while (low >= 0 && high < s.length && s[low] === s[high]) {
      if (high - low + 1 > maxLength) {
        start = low;
        maxLength = high - low + 1;
      }
      low--;
      high++;
    }
  }
  return s.substring(start, start + maxLength);
}

console.log(longestPalindrome('babad'))

medium

A. "bab"

B. "aba"

C. "bad"

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 one.

2. Step 2: Trace the input 'babad'

   Expanding around index 1 ('a') finds 'bab' (length 3). Around index 2 ('b') finds 'aba' (length 3). Both have length 3, but 'bab' is found first and stored.

3. Final Answer:

   "bab" -> Option A

4. Quick Check:

   First longest palindrome found = "bab" [OK]

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

Common Mistakes:

• Assuming 'aba' is always returned instead of 'bab'
• Returning entire string incorrectly
• Confusing substring indices

4. Identify the bug in the following TypeScript code for finding the longest palindromic substring:

function longestPalindrome(s: string): string {
  let start = 0, maxLength = 1;
  for (let i = 0; i < s.length; i++) {
    let low = i, high = i + 1;
    while (low >= 0 && high < s.length && s[low] === s[high]) {
      if (high - low + 1 > maxLength) {
        start = low;
        maxLength = high - low + 1;
      }
      low--;
      high++;
    }
  }
  return s.substring(start, start + maxLength);
}

medium

A. The loop should start from 1 instead of 0

B. The substring indices are incorrect in the return statement

C. The code only checks even-length palindromes, missing odd-length ones

D. The variables low and high are initialized incorrectly for odd-length palindromes

Solution

1. Step 1: Analyze initialization of low and high

   low = i and high = i + 1 means it only checks even-length palindromes centered between i and i+1.

2. Step 2: Identify missing odd-length palindrome check

   Odd-length palindromes centered at i (low = high = i) are not checked, so some palindromes are missed.

3. Final Answer:

   The code only checks even-length palindromes, missing odd-length ones -> Option C

4. Quick Check:

   Only even-length palindrome check = bug [OK]

Hint: Check both odd (low=high=i) and even (low=i, high=i+1) centers [OK]

Common Mistakes:

• Checking only even-length palindromes
• Not expanding around single character centers
• Incorrect loop start index

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

hard

A. Manacher's algorithm; longest palindrome is "forgeeksskeegfor"

B. Dynamic programming approach; longest palindrome is "forgeeksskeegfor"

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

D. Expand around center approach; longest palindrome is "geeksskeeg"

Solution

1. Step 1: Identify the longest palindrome in the string

   The substring "geeksskeeg" is the longest palindrome inside "forgeeksskeegfor".

2. Step 2: Match approach with correct palindrome

   Expand around center approach finds "geeksskeeg" correctly. The entire string is not a palindrome, so options A and B are incorrect. Brute force (C) incorrectly identifies "ss" as the longest.

3. Final Answer:

   Expand around center approach; longest palindrome is "geeksskeeg" -> Option D

4. Quick Check:

   Center expansion finds "geeksskeeg" [OK]

Hint: Use center expansion for efficient palindrome search [OK]

Common Mistakes:

• Assuming entire string is palindrome
• Confusing brute force with efficient methods
• Mixing up palindrome substrings