The solution explores all possible pairs of disjoint subsequences and calculates the product of their palindromic lengths. Using bitmasking and state transition dynamic programming allows efficient evaluation despite exponential subsequence combinations. This ensures the maximum product is identified without repeatedly checking overlapping indices.
Problem Statement
Given a string s, determine two non-overlapping subsequences that are both palindromic, such that their length product is as large as possible. Characters cannot be shared between the two subsequences.
Return the highest achievable product of the lengths of these two palindromic subsequences. A subsequence keeps character order from s, and a palindrome reads the same forward and backward.
Examples
Example 1
Input: s = "leetcodecom"
Output: 9
An optimal solution is to choose "ete" for the 1st subsequence and "cdc" for the 2nd subsequence. The product of their lengths is: 3 * 3 = 9.
Example 2
Input: s = "bb"
Output: 1
An optimal solution is to choose "b" (the first character) for the 1st subsequence and "b" (the second character) for the 2nd subsequence. The product of their lengths is: 1 * 1 = 1.
Example 3
Input: s = "accbcaxxcxx"
Output: 25
An optimal solution is to choose "accca" for the 1st subsequence and "xxcxx" for the 2nd subsequence. The product of their lengths is: 5 * 5 = 25.
Constraints
- 2 <= s.length <= 12
- s consists of lowercase English letters only.
Solution Approach
Generate all subsequences using bitmasking
Use bitmasks to represent every possible subsequence of s. Each bit indicates whether a character is included, allowing efficient enumeration of all combinations while preserving original order.
Check palindromic subsequences and disjointness
For each pair of subsequences, verify they are palindromic and have no overlapping indices. Use bitwise AND to quickly detect shared characters between two bitmasks.
Maximize product using dynamic programming
Store lengths of palindromic subsequences for each mask. Iterate over all mask pairs to compute the product of their lengths, updating a running maximum to find the final result.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(2^n * 2^n * n) in the worst case due to enumerating all subsequence pairs and checking palindromes of length n. Space complexity is O(2^n * n) to store palindrome lengths for each subsequence mask.
What Interviewers Usually Probe
- Can you generate all subsequences and efficiently check for palindromes?
- Consider bitmasking to represent subsequences and check disjoint characters quickly.
- Can you optimize by storing palindromic lengths to avoid repeated computation?
Common Pitfalls or Variants
Common pitfalls
- Failing to ensure the two subsequences are disjoint by index.
- Rechecking palindrome property repeatedly instead of caching results.
- Overlooking subsequences that maximize product due to greedy selection of longest palindromes first.
Follow-up variants
- Find the maximum sum of lengths instead of product for two disjoint palindromic subsequences.
- Allow three disjoint palindromic subsequences and maximize the product of their lengths.
- Restrict subsequences to a specific minimum length and compute the maximum product.
How GhostInterview Helps
- Automatically enumerates all possible disjoint subsequence pairs and validates palindromes efficiently.
- Caches palindromic lengths for rapid evaluation of product combinations to find the maximum.
- Provides step-by-step reasoning to avoid common pitfalls like overlapping indices or redundant checks.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the main pattern used in Maximum Product of the Length of Two Palindromic Subsequences?
The problem primarily uses state transition dynamic programming combined with bitmasking to handle all subsequence combinations efficiently.
Can this problem be solved with simple recursion?
Recursion alone is too slow due to exponential subsequence pairs; bitmasking with DP is needed for acceptable performance.
Why is bitmasking useful here?
Bitmasking allows efficient representation of subsequences and quick checks for disjoint characters using bitwise operations.
What is the maximum input size to consider?
The string length is constrained between 2 and 12 characters, so enumerating all subsequences is feasible.
How do you check if a subsequence is a palindrome efficiently?
Store previously computed palindromic lengths or use a two-pointer check on the subsequence represented by a bitmask.
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