Maximum Product of the Length of Two Palindromic Subsequences

Maximum Product of the Length of Two Palindromic Subsequences

1. What is this problem about?

The Maximum Product of the Length of Two Palindromic Subsequences interview question is a challenging string manipulation problem. You are given a string s, and you need to find two disjoint (non-overlapping) subsequences that are both palindromes. You want to maximize the product of their lengths.

A subsequence is formed by deleting zero or more characters from the original string without changing the relative order of the remaining characters. "Disjoint" means each character in the original string can be used in at most one of the two subsequences.

2. Why is this asked in interviews?

This Maximum Product of the Length of Two Palindromic Subsequences coding problem is asked by companies like Ascend to test a candidate's mastery of bitmasks and exhaustive search. Since the string length is typically small (e.g., up to 12 or 15), the problem encourages using bitmasks to represent every possible subsequence. It evaluates your ability to combine bitwise operations with palindrome checks and optimization logic.

3. Algorithmic pattern used

The Bitmask, Backtracking, String, Bit Manipulation, Dynamic Programming interview pattern is perfectly suited here.

1. Use a bitmask from 1 to $2^n - 1$ to represent all possible subsequences of the string.
2. For each mask, check if the corresponding characters form a palindrome.
3. If it's a palindrome, store the mask and its length.
4. Iterate through all pairs of palindromic masks. If (mask1 & mask2) == 0 (meaning they are disjoint), calculate length1 * length2.
5. Return the maximum product.

4. Example explanation

String: "accbc" Possible palindromic subsequences:

• Mask "10001" (a...c): length 2 (not palindrome)
• Mask "01100" (cc): length 2 (palindrome)
• Mask "00010" (b): length 1 (palindrome)
• Mask "01010" (cb): length 2 (not palindrome)
• Mask "01110" (cbc): length 3 (palindrome)

Pairs:

• "cc" (01100) and "b" (00010) are disjoint. Product = 2 * 1 = 2.
• "cc" (01100) and "a" (10000) are disjoint. Product = 2 * 1 = 2. Wait, a better one: "ccc" (01110) and "a" (10000). Product = 3 * 1 = 3.

5. Common mistakes candidates make

In the Maximum Product of the Length of Two Palindromic Subsequences coding problem, a common error is confusing "substrings" with "subsequences." Subsequences do not have to be contiguous. Another mistake is using a brute-force approach that is too slow, such as nested recursion without memoization. Candidates also often struggle with the bitmasking logic, specifically checking for the disjoint condition using the bitwise AND operator.

6. Interview preparation tip

Whenever the input size is very small (N <= 15), bitmasking should be your first consideration. It allows you to represent any subset of elements as an integer, making it easy to iterate and compare. Practice checking if a bitmask represents a palindrome and using the & operator to check for overlap.