#1771

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximize Palindrome Length From Subsequences

Maximize Palindrome Length From Subsequences explores dynamic programming to construct the longest palindrome from two subsequences.

string dynamic programming

Problem Statement

You are given two strings, word1 and word2. Your task is to construct the longest palindrome by selecting subsequences from these strings. The subsequences must maintain the character order in both strings.

Return the length of the longest palindrome you can form from these subsequences. If no palindromes can be formed, return 0. A subsequence of a string is formed by deleting characters without changing the order of the remaining ones.

Examples

Example 1

Input: word1 = "cacb", word2 = "cbba"

Output: 5

Choose "ab" from word1 and "cba" from word2 to make "abcba", which is a palindrome.

Example 2

Input: word1 = "ab", word2 = "ab"

Output: 3

Choose "ab" from word1 and "a" from word2 to make "aba", which is a palindrome.

Example 3

Input: word1 = "aa", word2 = "bb"

Output: 0

You cannot construct a palindrome from the described method, so return 0.

Constraints

1 <= word1.length, word2.length <= 1000
word1 and word2 consist of lowercase English letters.

Solution Approach

State Transition Dynamic Programming

Use dynamic programming to explore all possible subsequences formed from both word1 and word2. This approach builds the longest palindrome by considering character matches at every position.

Optimizing with Palindromic Subsequence

The key optimization is to ignore the non-empty subsequence constraint and concatenate the two strings. Then, find the largest palindromic subsequence using dynamic programming techniques.

Complexity Reduction

Focus on optimizing the state transitions and the recurrence relation for dynamic programming. This reduces redundant calculations, enhancing the solution's efficiency for larger inputs.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity of this problem depend on the approach chosen. A direct dynamic programming approach has a time complexity of O(n^2) and space complexity of O(n^2), where n is the length of the strings. Optimizations like space compression can reduce space complexity to O(n).

What Interviewers Usually Probe

Understanding of dynamic programming and state transitions.
Ability to optimize algorithms for handling large input sizes.
Proficiency in working with palindromic subsequences in dynamic programming.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the problem by trying to form a palindrome directly from both strings, instead of finding subsequences.
Not properly optimizing the space complexity for large inputs.
Overcomplicating the dynamic programming state transitions, leading to inefficient solutions.

Follow-up variants

Different constraints on the lengths of the strings, making optimization more important.
Incorporating constraints on specific characters, affecting subsequence formation.
Adapting the problem to allow for specific subsequence combinations instead of just palindromes.

FAQ

How do I approach solving Maximize Palindrome Length From Subsequences?

Start by understanding the state transition dynamic programming approach, and then optimize for both time and space complexity when handling large input sizes.

What are the key components of the solution for this problem?

The key components are dynamic programming, state transition, and optimization for both time and space complexity.

Why is state transition dynamic programming used here?

State transition dynamic programming allows for exploring all possible subsequences while optimizing for the longest palindromic subsequence.

Can the problem be solved with a greedy algorithm?

No, a greedy approach will not work effectively due to the need to explore all subsequences and their possible combinations.

How can I optimize my solution for larger inputs?

Optimize your solution by reducing space complexity, using techniques like space compression in dynamic programming.

terminal

Solution

Solution 1: Dynamic Programming

First, we concatenate strings `word1` and `word2` to get string $s$. Then we can transform the problem into finding the length of the longest palindromic subsequence in string $s$. However, when calculating the final answer, we need to ensure that at least one character in the palindrome string comes from `word1` and another character comes from `word2`.

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class Solution:
    def longestPalindrome(self, word1: str, word2: str) -> int:
        s = word1 + word2
        n = len(s)
        f = [[0] * n for _ in range(n)]
        for i in range(n):
            f[i][i] = 1
        ans = 0
        for i in range(n - 2, -1, -1):
            for j in range(i + 1, n):
                if s[i] == s[j]:
                    f[i][j] = f[i + 1][j - 1] + 2
                    if i < len(word1) <= j:
                        ans = max(ans, f[i][j])
                else:
                    f[i][j] = max(f[i + 1][j], f[i][j - 1])
        return ans

Continue Practicing

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