Maximum Product Of The Length Of Two Palindromic Subsequences [Python]

If you've stumbled upon this article, you're probably tackling the LeetCode problem titled Maximum Product Of The Length Of Two Palindromic Subsequences It's a bit of a mouthful, but don't worry; we'll break it down step by step.

Problem Overview

The problem can be summarized as follows: given a string s, your task is to find two disjoint palindromic subsequences of s such that the product of their lengths is maximized.

The key point here is that the two subsequences must be disjoint, meaning they cannot share a character at the same index.

Your goal is to return the maximum possible product of the lengths of the two palindromic subsequences.

Before diving into the solution, let's dissect this problem further:

Example 1:

Suppose we have the string s = "leetcodecom".

An optimal solution here is to choose "ete" as the first subsequence and "cdc" as the second subsequence.

The product of their lengths is 3 * 3 = 9.

Example 2:

For the string s = "bb", an optimal solution is to choose "b" (the first character) for the first subsequence and "b" (the second character) for the second subsequence.

The product of their lengths is 1 * 1 = 1.

Example 3:

Let's consider s = "accbcaxxcxx".

An optimal solution involves selecting "accca" for the first subsequence and "xxcxx" for the second subsequence.

The product of their lengths is 5 * 5 = 25.

Constraints:

The length of s is between 2 and 12 characters.
The string s consists of lowercase English letters only.

Now that we have a clear understanding of the problem, let's proceed to explore the solution.

Understanding the Constraints

The key to solving this problem efficiently lies in understanding how to generate all possible subsequences of a given string and check if they are palindromes.

The constraint here is not on the input string but on the number of possible subsequences.

Since there can be up to 2^12 possible subsequences for a 12-character string, a brute force approach would require checking a large number of combinations.

To tackle this efficiently, we'll employ a technique using bit manipulation, which will help us generate and validate subsequences.

Here's how it works:

We'll represent each subsequence as a bitmask, where each bit corresponds to a character in the input string.

If a bit is set (1), it indicates that the character at that position is part of the subsequence.

If a bit is not set (0), it means the character is not included.

To generate all possible subsequences, we'll iterate through integers from 1 to 2^n, where n is the length of the input string.

For each integer, we'll convert it to a bitmask and create the corresponding subsequence.

To check if a subsequence is a palindrome, we'll reverse it and compare it to the original.

If they match, it's a palindrome.

This approach allows us to efficiently generate and validate subsequences, significantly reducing the time complexity.

Now, let's dive into the Python code solution.

Maximum Product of The Length of Two Palindromic Subsequences Python Code Solution

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from functools import lru_cache

class Solution:
    def maxProduct(self, s):
        n = len(s)

        # Arrays to store the first and last occurrence of each character in s.

first, last = [0]*(1&lt;&lt;n), [0]*(1&lt;&lt;n)

        # Calculate the first occurrence of each character in s.

for i in range(n):
            for j in range(1&lt;&lt;i, 1&lt;&lt;(i+1)):
                first[j] = i

        # Calculate the last occurrence of each character in s.

for i in range(n):
            for j in range(1&lt;&lt;i, 1&lt;&lt;n, 1&lt;&lt;(i+1)):
                last[j] = i

        @lru_cache(None)
        def dp(m):
            if m &amp; (m-1) == 0: return m != 0
            l, f = last[m], first[m]
            lb, fb = 1&lt;&lt;l, 1&lt;&lt;f
            return max(dp(m-lb), dp(m-fb), dp(m-lb-fb) + (s[l] == s[f]) * 2)

        ans = 0
        for m in range(1, 1&lt;&lt;n):
            ans = max(ans, dp(m)*dp((1&lt;&lt;n) - 1 - m))

        return ans

The provided Python code implements the solution to the problem efficiently.

Let's break down the code step by step:

We use the lru_cache decorator to optimize recursive function calls.

It caches the results of function calls, making the solution faster for larger inputs.

Inside the maxProduct function, we initialize some variables, and then we perform two important preprocessing steps to calculate the first and last occurrences of each character in the string s.

This information will be crucial in the subsequent steps.

We define a recursive function dp(m) that takes a bitmask m as its argument.

This function determines if the given bitmask corresponds to a palindrome and returns True if it is a palindrome and False if it's not.

The function uses dynamic programming and memoization to optimize calculations.

The core of the solution is in the nested loop that iterates through all possible bitmasks (m) from 1 to 2^n, where n is the length of the input string.

For each m, we compute the product of lengths of two disjoint palindromic subsequences.

The final answer is updated by taking the maximum of the current answer and the product of lengths calculated for the current m.

By the end of the loop, we have the maximum product of the lengths of two disjoint palindromic subsequences.

This solution efficiently handles the constraints and provides a correct answer.

Time and Space Complexity

Now, let's analyze the time and space complexity of this solution:

Time Complexity: The time complexity is O(2^n) in the worst case, where n is the length of the input string.

This is because we generate all possible subsequences using a bitmask approach.

However, the use of memoization and efficient bitwise operations speeds up the process for practical input sizes.

Space Complexity: The space complexity is also O(2^n) because of the space required for memoization (caching results of subproblems).

This is acceptable because the input string length is constrained to a maximum of 12 characters.

Reasoning Behind Our Approach

The approach used in this solution is based on efficient bitmask manipulation and dynamic programming.

Here's the reasoning behind our approach:

Bitmask Representation: We represent each subsequence as a bitmask, where each bit corresponds to a character in the input string.

This allows us to efficiently generate and validate subsequences.

Bit manipulation is a powerful technique for handling combinations.

First and Last Occurrences

: We precompute the first and last occurrences of each character in the input string.

This information is vital for identifying palindromes and disjoint subsequences efficiently.

Dynamic Programming: The dp(m) function employs dynamic programming to determine if a given bitmask represents a palindrome.

This function uses memoization to avoid redundant calculations and optimize the process.

Efficient Subsequence Generation: We use a nested loop to iterate through all possible bitmasks, creating and validating subsequences.

This process is efficient, thanks to the bitmask representation.

Maximum Product: We keep track of the maximum product of lengths of disjoint palindromic subsequences during the iteration, ensuring that we find the optimal solution.

This approach is both correct and efficient, providing a solution that meets the constraints of the problem.

Conclusion

In this article, we tackled the LeetCode problem Maximum Product Of The Length Of Two Palindromic Subsequences We broke down the problem, discussed the constraints, and explored an efficient Python solution.

We employed a bitmask representation to generate and validate subsequences, precomputed first and last occurrences of characters, used dynamic programming with memoization, and efficiently iterated through all possible bitmasks to find the maximum product of lengths of disjoint palindromic subsequences.

By understanding the problem and following the provided solution, you can efficiently solve this medium-level problem on LeetCode.

Feel free to comment, ask questions, make suggestions, and share this content with others.

Happy coding!

Question Link: Maximum Product of The Length of Two Palindromic Subsequences on LeetCode

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