Longest Palindromic Substring in DSA Python - 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.

Specifically, how does the number of steps grow when checking all possible palindromes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def longest_palindrome(s: str) -> str:
    start, max_len = 0, 1
    for i in range(len(s)):
        # Check odd length palindrome
        l, r = i, i
        while l >= 0 and r < len(s) and s[l] == s[r]:
            if (r - l + 1) > max_len:
                start = l
                max_len = r - l + 1
            l -= 1
            r += 1
        # Check even length palindrome
        l, r = i, i + 1
        while l >= 0 and r < len(s) and s[l] == s[r]:
            if (r - l + 1) > max_len:
                start = l
                max_len = r - l + 1
            l -= 1
            r += 1
    return s[start:start + max_len]

This code checks all possible centers of palindromes and expands outwards to find the longest palindrome substring.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Expanding around each center to check palindrome substrings.
• How many times: For each character (n times), two expansions happen (odd and even length), each expanding up to n steps in worst case.

How Execution Grows With Input

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

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

Pattern observation: Doubling the input size roughly quadruples the work, showing a square growth pattern.

Final Time Complexity

Time Complexity: O(n²)

This means the time needed grows roughly with the square of the input length, so longer strings take much more time.

Common Mistake

[X] Wrong: "Since we only check around each character, the time is O(n)."

[OK] Correct: Each center can expand up to n steps, so the total checks add 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 their costs.

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. Count the total number of palindromic substrings

B. Reverse the string and compare with original

C. Check if the entire string is a palindrome

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

Solution

1. Step 1: Understand the problem statement

   The problem asks to find the longest continuous part of the string that reads the same forwards and backwards.

2. Step 2: Differentiate from related problems

   Counting palindromic substrings or checking the whole string is different from finding the longest palindromic substring.

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: Longest palindrome means longest continuous mirrored substring [OK]

Common Mistakes:

• Confusing substring with subsequence
• Counting all palindromes instead of longest
• Checking whole string palindrome only

2. Which of the following Python code snippets correctly initializes variables to find the longest palindromic substring?

easy

A. start, max_len = 0, 1

B. start = max_len = None

C. start, max_len = 1, 0

D. start = max_len = 0

Solution

1. Step 1: Understand variable roles

   We need to track the start index and maximum length of the palindrome found. Starting with 0 and 1 is common because a single character is a palindrome.

2. Step 2: Check initialization correctness

   start, max_len = 0, 1 sets start=0 and max_len=1, which is correct. start, max_len = 1, 0 reverses values incorrectly. Options A, B, and C do not set max_len properly.

3. Final Answer:

   start, max_len = 0, 1 -> Option A

4. Quick Check:

   Initialize start=0 and max_len=1 for single char palindrome [OK]

Hint: Start at 0, max length at least 1 for single char palindrome [OK]

Common Mistakes:

• Setting max_len to 0 initially
• Using None instead of integers
• Swapping start and max_len values

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

def expand_around_center(s, left, right):
    while left >= 0 and right < len(s) and s[left] == s[right]:
        left -= 1
        right += 1
    return right - left - 1

s = "babad"
length = expand_around_center(s, 1, 1)
print(length)

medium

A. 1

B. 5

C. 3

D. 0

Solution

1. Step 1: Understand the function behavior

   The function expands around the center indices (left and right) while characters match, then returns the length of the palindrome found.

2. Step 2: Trace the expansion for s='babad' at center (1,1)

   Center at index 1 ('a'). Expand left=1, right=1: 'a'=='a' true, expand to left=0, right=2: 'b'=='b' true, expand to left=-1 (stop). Length = right - left - 1 = 3 - (-1) - 1 = 3.

3. Final Answer:

   3 -> Option C

4. Quick Check:

   Expand around center returns palindrome length 3 [OK]

Hint: Expand while chars match, length = right - left - 1 [OK]

Common Mistakes:

• Not subtracting 1 in length calculation
• Confusing indices after expansion
• Assuming length equals right - left

4. Identify the error in the following code snippet for finding the longest palindromic substring:

def longest_palindrome(s):
    start, max_len = 0, 1
    for i in range(len(s)):
        len1 = expand_around_center(s, i, i)
        len2 = expand_around_center(s, i, i+1)
        length = max(len1, len2)
        if length > max_len:
            start = i - length // 2
            max_len = length
    return s[start:start+max_len]

medium

A. The loop should run till len(s)-1 to avoid index error

B. The start index calculation is incorrect for even length palindromes

C. max_len should be initialized to 0 instead of 1

D. expand_around_center function is missing

Solution

1. Step 1: Analyze start index update logic

   For even length palindromes, start index calculation differs because center is between two characters. Using i - length // 2 works for odd length but not always for even length.

2. Step 2: Identify correct start calculation

   For even length, start should be i - (length // 2) + 1 to correctly locate the palindrome start.

3. Final Answer:

   The start index calculation is incorrect for even length palindromes -> Option B

4. Quick Check:

   Start index must adjust differently for even palindromes [OK]

Hint: Adjust start index differently for even-length palindromes [OK]

Common Mistakes:

• Using same start formula for odd and even palindromes
• Ignoring even length palindrome cases
• Not updating max_len properly

5. Given the string "forgeeksskeegfor", which substring is the longest palindromic substring?

hard

A. "geeksskeeg"

B. "forgeeksskeegfor"

C. "geeksskeegf"

D. "eksskeeg"

Solution

1. Step 1: Identify palindromic substrings in the string

   Check substrings that read the same forwards and backwards. "geeksskeeg" is a palindrome of length 10 inside the string.

2. Step 2: Compare lengths of candidate substrings

   "forgeeksskeegfor" is not a palindrome as a whole. "geeksskeegf" and "eksskeeg" are shorter palindromes.

3. Final Answer:

   "geeksskeeg" -> Option A

4. Quick Check:

   Longest palindrome in "forgeeksskeegfor" is "geeksskeeg" [OK]

Hint: Look for longest mirrored substring inside the string [OK]

Common Mistakes:

• Assuming whole string is palindrome
• Choosing substring with extra characters
• Ignoring palindrome length