#3472

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Longest Palindromic Subsequence After at Most K Operations

Find the longest palindromic subsequence of a string after performing at most k operations to adjust letters.

Problem Statement

You are given a string s and an integer k. You are allowed to perform at most k operations, where each operation lets you replace any character in s with its adjacent letter in the alphabet, either the next or the previous one. The alphabet wraps around, so 'a' is next to 'z' and vice versa.

Return the length of the longest palindromic subsequence that can be obtained after performing at most k operations. The goal is to adjust the characters of the string so that a palindrome subsequence can be formed with the highest possible length.

Examples

Example 1

Input: s = "abced", k = 2

Output: 3

The subsequence "ccc" forms a palindrome of length 3, which is the maximum.

Example 2

Input: s = " aaazzz ", k = 4

Output: 6

The entire string forms a palindrome of length 6.

Constraints

1 <= s.length <= 200
1 <= k <= 200
s consists of only lowercase English letters.

Solution Approach

State Transition Dynamic Programming

The key to solving this problem lies in dynamic programming. Create a DP table where dp[i][j] represents the longest palindromic subsequence that can be formed from the substring s[i..j] with the given number of operations. Transition between states depends on whether you can adjust the characters at i and j within the allowed operations.

Character Adjustment Tracking

Since each operation can modify the characters, you must track how many changes are required to convert one character into another. Consider each character's distance in the alphabet and check if it's possible to make the change within the given k operations.

Maximizing Palindrome Length

Use the DP table to maximize the length of palindromic subsequences. By adjusting characters at the right positions with k operations, you aim to expand the palindrome. Carefully handle the characters that need the fewest changes and apply the operations efficiently.

Complexity Analysis

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

Time and space complexity depend on the approach used for the dynamic programming table. A straightforward implementation would have a time complexity of O(n^2 * k), where n is the string length and k is the number of operations. The space complexity is O(n^2), as the DP table requires storage for all substrings of s.

What Interviewers Usually Probe

Look for the candidate's understanding of state transition dynamic programming in string manipulation problems.
Evaluate how efficiently they handle the complexity of operations and character adjustments.
Assess the candidate's ability to optimize the solution, particularly in handling the k operations efficiently.

Common Pitfalls or Variants

Common pitfalls

Failing to consider the wrapping alphabet when calculating character adjustments.
Overcomplicating the problem by ignoring the efficiency of the dynamic programming approach.
Neglecting to optimize space usage, especially when dealing with large strings and many operations.

Follow-up variants

Consider limiting the number of changes even further (e.g., at most 1 operation).
Extend the problem to allow additional string manipulations such as insertions or deletions.
Explore the problem with a constraint that restricts certain characters from being changed.

FAQ

What is the main approach to solve the "Longest Palindromic Subsequence After at Most K Operations"?

The problem can be solved using dynamic programming with a focus on state transitions, where the state represents the longest palindromic subsequence possible with a limited number of character modifications.

How do the k operations affect the palindrome formation?

The k operations allow you to modify characters in the string, either by advancing or retreating alphabetically. These operations are crucial for creating a longer palindromic subsequence by adjusting mismatched characters.

What is the time complexity of solving the "Longest Palindromic Subsequence After at Most K Operations"?

The time complexity is O(n^2 * k), where n is the length of the string and k is the number of operations, assuming a dynamic programming solution is used.

How can we optimize the solution for larger inputs?

Optimizing space complexity and focusing on efficient state transitions can help improve the solution for larger inputs. Avoiding redundant calculations and minimizing the size of the DP table are key strategies.

What are some common mistakes in solving this problem?

Common mistakes include failing to handle the alphabet wrapping correctly, neglecting to track changes efficiently, and not optimizing space or time complexity in the dynamic programming approach.

terminal

Solution

Solution 1: Memoized Search

We design a function $\textit{dfs}(i, j, k)$, which represents the length of the longest palindromic subsequence that can be obtained in the substring $s[i..j]$ with at most $k$ operations. The answer is $\textit{dfs}(0, n - 1, k)$.

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class Solution:
    def longestPalindromicSubsequence(self, s: str, k: int) -> int:
        @cache
        def dfs(i: int, j: int, k: int) -> int:
            if i > j:
                return 0
            if i == j:
                return 1
            res = max(dfs(i + 1, j, k), dfs(i, j - 1, k))
            d = abs(s[i] - s[j])
            t = min(d, 26 - d)
            if t <= k:
                res = max(res, dfs(i + 1, j - 1, k - t) + 2)
            return res

        s = list(map(ord, s))
        n = len(s)
        ans = dfs(0, n - 1, k)
        dfs.cache_clear()
        return ans

Continue Practicing

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