Longest Palindromic Subsequence After at Most K Operations

Longest Palindromic Subsequence After at Most K Operations

What is this problem about?

The Longest Palindromic Subsequence After at Most K Operations interview question tasks you with finding the longest palindromic subsequence in a given string. However, you are allowed to perform up to K operations. In one operation, you can change any character to its adjacent character in the alphabet (e.g., 'a' to 'b', or 'z' to 'a' if it wraps around, though typical constraints define specific shift costs). Your goal is to maximize the palindrome's length without exceeding the K operation budget.

Why is this asked in interviews?

This is an advanced problem that tests a candidate's ability to blend String manipulation with multi-dimensional Dynamic Programming (DP). It assesses whether you can add an extra constraint (the operation budget K) to a classic DP template (Longest Palindromic Subsequence). Interviewers use it to identify senior-level engineers who can confidently manage complex state transitions and cost-based optimizations.

Algorithmic pattern used

This problem relies on a 3D Dynamic Programming pattern. Let dp[left][right][k] be the maximum palindromic subsequence length within the substring from index left to right using at most k operations. To transition, you can either:

1. Ignore s[left] and find the max in [left+1][right][k].
2. Ignore s[right] and find the max in [left][right-1][k].
3. Match s[left] and s[right]. To do this, calculate the cyclical distance (cost) between the two characters. If cost <= k, you can add 2 to the length of dp[left+1][right-1][k - cost].

Example explanation

Imagine string s = "abdc" and K = 2. We evaluate the outer characters 'a' and 'c'. The distance to shift 'c' down to 'a' or 'a' up to 'c' is 2. Since our budget K = 2, we can afford this shift. If we match 'a' and 'c' (cost 2), we are left with the substring "bd" and budget 0. For "bd" with budget 0, we cannot match 'b' and 'd' (cost 2). The best we can do is pick one character, yielding a length of 1. So the total palindrome length is 2 (for the outer match) + 1 (for the inner) = 3. The mutated sequence could be "aba" or "cdc".

Common mistakes candidates make

A common error is mishandling the cost calculation between characters, especially if the problem specifies cyclical shifts (where 'a' is adjacent to 'z'). Candidates often just use abs(char1 - char2), forgetting that 26 - abs(char1 - char2) might be a cheaper path. Another mistake is allocating a massive 3D array that exceeds memory limits; you must optimize your DP table or use memoization.

Interview preparation tip

When preparing for the Longest Palindromic Subsequence After at Most K Operations coding problem, make sure you deeply understand the standard 2D Longest Palindromic Subsequence DP template first. Once you master the base case, adding the K dimension is just a matter of passing along a "budget" parameter and subtracting from it when you force a match.