Maximum Product of the Length of Two Palindromic Subsequences

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StringsDynamic ProgrammingBit Manipulation

Given a string s, find two disjoint palindromic subsequences of s such that the product of their lengths is maximized. The two subsequences are disjoint if they do not both pick a character at the same index.

Return the maximum possible product of the lengths of the two palindromic subsequences.

A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters. A string is palindromic if it reads the same forward and backward.

Example 1:

Input: s = "leetcodecom"
Output: 9
Explanation: An optimal solution is to choose "ete" for the 1^st subsequence and "cdc" for the 2^nd subsequence.
The product of their lengths is: 3 * 3 = 9.

Example 2:

Input: s = "bb"
Output: 1
Explanation: An optimal solution is to choose "b" (the first character) for the 1^st subsequence and "b" (the second character) for the 2^nd subsequence.
The product of their lengths is: 1 * 1 = 1.

Example 3:

Input: s = "accbcaxxcxx"
Output: 25
Explanation: An optimal solution is to choose "accca" for the 1^st subsequence and "xxcxx" for the 2^nd subsequence.
The product of their lengths is: 5 * 5 = 25.

Constraints:

2 <= s.length <= 12
s consists of lowercase English letters only.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum length of the input string `s`?
Does the input string `s` contain only lowercase English letters?
If no two palindromic subsequences can be found, what should I return?
Are the two palindromic subsequences required to be non-overlapping or disjoint; can they share characters?
If multiple pairs of palindromic subsequences give the same maximum product, is any of those pairs acceptable?

Brute Force Solution

Approach

The goal is to find the two palindromic subsequences within a string that, when their lengths are multiplied together, yield the largest possible product. The brute force approach involves checking every possible pair of subsequences to see if they are palindromes and then calculating their product.

Here's how the algorithm would work step-by-step:

First, consider every possible combination of characters from the original string to form the first subsequence.
Then, for each of these subsequences, check if it is a palindrome (reads the same forward and backward).
If it is a palindrome, create the second subsequence by using only the characters from the original string that were *not* used in the first subsequence.
Check if this second subsequence is also a palindrome.
If both subsequences are palindromes, multiply their lengths together.
Keep track of the largest product you find by comparing the current product to the largest one seen so far.
Repeat these steps until you have considered every possible pair of subsequences.
The final largest product is the answer.

Code Implementation

def maximum_product_of_palindromes(input_string):
    string_length = len(input_string)
    max_product = 0

    for first_subsequence_mask in range(2**string_length):
        first_subsequence = ""
        first_subsequence_indices = []

        # Construct the first subsequence based on the mask
        for index in range(string_length):
            if (first_subsequence_mask >> index) & 1:
                first_subsequence += input_string[index]
                first_subsequence_indices.append(index)

        # Check if the first subsequence is a palindrome
        if first_subsequence == first_subsequence[::-1]:

            second_subsequence = ""
            # Build the second subsequence with remaining chars
            for index in range(string_length):
                if index not in first_subsequence_indices:
                    second_subsequence += input_string[index]

            # Check if the second subsequence is also a palindrome
            if second_subsequence == second_subsequence[::-1]:

                # Update max_product if current product is larger
                product = len(first_subsequence) * len(second_subsequence)
                max_product = max(max_product, product)

    return max_product

Big(O) Analysis

Time Complexity

O(3^n) – The algorithm iterates through all possible subsequences of the string. Generating all subsequences requires considering each character, either including it or excluding it or including it in another sequence. This results in 3 possibilities for each of the n characters, leading to a time complexity of O(3^n). Checking if a subsequence is a palindrome takes O(n) time in the worst case but is dominated by the subsequence generation. Thus the overall time complexity is O(3^n).

Space Complexity

O(1) – The algorithm iterates through combinations and checks for palindromes, but it doesn't appear to use any auxiliary data structures that scale with the input string length N. It primarily involves comparisons and basic arithmetic, storing only a few variables like the maximum product found so far and lengths of subsequences. Therefore, the space required remains constant regardless of the input string's size. The auxiliary space complexity is O(1).

Optimal Solution

Approach

The goal is to find two palindromic subsequences within a given string that, when their lengths are multiplied, result in the highest possible product. The trick is to use bit manipulation to efficiently explore all possible combinations of subsequences and then check if each subsequence is a palindrome.

Here's how the algorithm would work step-by-step:

Think of each character in the string as having two possibilities: being in the first subsequence, or being in the second subsequence (or neither).
Use bits to represent these choices. Each bit corresponds to a character in the original string. If a bit is 'on', the corresponding character goes into the first subsequence. If a bit is 'off', we will later check if it can go into the second subsequence.
Go through all possible combinations of these bits. This creates every possible subsequence for the first choice.
For each first subsequence, figure out what's left of the original string.
From the remaining characters, form the second subsequence, but only including characters that don't violate having disjoint subsequences.
Check if both the first and second subsequences are palindromes.
If both are palindromes, multiply their lengths.
Keep track of the maximum product found so far.
Return the maximum product found after checking all possible combinations.

Code Implementation

def max_product(input_string):
    string_length = len(input_string)
    maximum_product = 0

    # Iterate through all possible subsequences using bit manipulation.
    for i in range(1 << string_length):
        first_subsequence = ""
        first_subsequence_indices = []
        for j in range(string_length):
            if (i >> j) & 1:
                first_subsequence += input_string[j]
                first_subsequence_indices.append(j)

        # Only process if the first subsequence is a palindrome
        if first_subsequence == first_subsequence[::-1]:
            second_subsequence = ""
            for k in range(string_length):
                # Build second subsequence with chars not in first.
                if k not in first_subsequence_indices:
                    second_subsequence += input_string[k]

            # After filtering, is the second subsequence a palindrome?
            if second_subsequence == second_subsequence[::-1]:

                # Key Step: Palindromes found, calculate product
                product = len(first_subsequence) * len(second_subsequence)
                maximum_product = max(maximum_product, product)

    return maximum_product

Big(O) Analysis

Time Complexity

O(2^n * n) – The algorithm iterates through all possible subsequences of the given string. This is done using bit manipulation, where each bit represents whether a character is included in the first subsequence. Therefore, there are 2^n possible combinations for the first subsequence. For each of these 2^n subsequences, the algorithm constructs a second subsequence from the remaining characters and checks if both subsequences are palindromes, which takes O(n) time for each palindrome check (where n is the length of the original string since the maximum subsequence length can be n). Thus, the overall time complexity is O(2^n * n).

Space Complexity

O(N) – The algorithm's space complexity is primarily determined by the creation of subsequences. In the worst-case scenario, we extract and store subsequences from the original string, potentially requiring space proportional to the input string's length, denoted as N. The 'remaining characters' implicitly create new strings of length at most N. While the palindrome check might use temporary space, the subsequence extraction and storage are the dominant factor. Therefore, the auxiliary space is O(N).

Edge Cases

Case	How to Handle
Null or empty string input	Return 0 immediately as no subsequences can be formed.
String with length 1	Return 1, as the only character is a palindrome of length 1, and the other subsequence is empty with length 0 (1 * 0 = 0, but we must account for other non-empty subsequences). We can find subsequences 'a' and ''.
String with all identical characters (e.g., 'aaaa')	The optimal solution would be to split the string as evenly as possible to maximize length (e.g. 'aa' and 'aa'), but subsequence generation must avoid overlapping chars.
Maximum length string with diverse characters (for scaling)	Ensure algorithm is efficient by using dynamic programming or bitmasking and memoization to avoid exponential time complexity.
String is already a palindrome	One palindrome subsequence can be the string itself, so find a suitable second subsequence among remaining characters, avoiding overlapping character usage, and maximize combined length.
No valid palindromic subsequences exist (e.g. a string with single unique characters)	The empty string is a valid subsequence, ensure that at least two non-overlapping subsequences of length 1, if possible are tested.
Integer overflow in length multiplication	Ensure that lengths are not excessively large and that multiplication results are handled carefully to prevent integer overflow issues by using 64 bit ints.
String contains Unicode characters or special symbols	The palindrome check and length calculation should correctly handle Unicode characters without causing errors in comparison or encoding by considering string encoding format.

Null or empty string input

How to Handle:

Return 0 immediately as no subsequences can be formed.

String with length 1

How to Handle:

Return 1, as the only character is a palindrome of length 1, and the other subsequence is empty with length 0 (1 * 0 = 0, but we must account for other non-empty subsequences). We can find subsequences 'a' and ''.

String with all identical characters (e.g., 'aaaa')

How to Handle:

The optimal solution would be to split the string as evenly as possible to maximize length (e.g. 'aa' and 'aa'), but subsequence generation must avoid overlapping chars.

Maximum length string with diverse characters (for scaling)

How to Handle:

Ensure algorithm is efficient by using dynamic programming or bitmasking and memoization to avoid exponential time complexity.

String is already a palindrome

How to Handle:

One palindrome subsequence can be the string itself, so find a suitable second subsequence among remaining characters, avoiding overlapping character usage, and maximize combined length.

No valid palindromic subsequences exist (e.g. a string with single unique characters)

How to Handle:

The empty string is a valid subsequence, ensure that at least two non-overlapping subsequences of length 1, if possible are tested.

Integer overflow in length multiplication

How to Handle:

Ensure that lengths are not excessively large and that multiplication results are handled carefully to prevent integer overflow issues by using 64 bit ints.

String contains Unicode characters or special symbols

How to Handle:

The palindrome check and length calculation should correctly handle Unicode characters without causing errors in comparison or encoding by considering string encoding format.