Longest Palindromic Subsequence After at Most K Operations
<p>You are given a string <code>s</code> and an integer <code>k</code>.</p>

<p>In one operation, you can replace the character at any position with the next or previous letter in the alphabet (wrapping around so that <code>&#39;a&#39;</code> is after <code>&#39;z&#39;</code>). For example, replacing <code>&#39;a&#39;</code> with the next letter results in <code>&#39;b&#39;</code>, and replacing <code>&#39;a&#39;</code> with the previous letter results in <code>&#39;z&#39;</code>. Similarly, replacing <code>&#39;z&#39;</code> with the next letter results in <code>&#39;a&#39;</code>, and replacing <code>&#39;z&#39;</code> with the previous letter results in <code>&#39;y&#39;</code>.</p>

<p>Return the length of the <strong>longest <span data-keyword="palindrome-string">palindromic</span> <span data-keyword="subsequence-string-nonempty">subsequence</span></strong> of <code>s</code> that can be obtained after performing <strong>at most</strong> <code>k</code> operations.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">s = &quot;abced&quot;, k = 2</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>Replace <code>s[1]</code> with the next letter, and <code>s</code> becomes <code>&quot;acced&quot;</code>.</li>
	<li>Replace <code>s[4]</code> with the previous letter, and <code>s</code> becomes <code>&quot;accec&quot;</code>.</li>
</ul>

<p>The subsequence <code>&quot;ccc&quot;</code> forms a palindrome of length 3, which is the maximum.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">s = &quot;</span>aaazzz<span class="example-io">&quot;, k = 4</span></p>

<p><strong>Output:</strong> 6</p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>Replace <code>s[0]</code> with the previous letter, and <code>s</code> becomes <code>&quot;zaazzz&quot;</code>.</li>
	<li>Replace <code>s[4]</code> with the next letter, and <code>s</code> becomes <code>&quot;zaazaz&quot;</code>.</li>
	<li>Replace <code>s[3]</code> with the next letter, and <code>s</code> becomes <code>&quot;zaaaaz&quot;</code>.</li>
</ul>

<p>The entire string forms a palindrome of length 6.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 200</code></li>
	<li><code>1 &lt;= k &lt;= 200</code></li>
	<li><code>s</code> consists of only lowercase English letters.</li>
</ul>
String; Dynamic Programming
class Solution {
    private char[] s;
    private Integer[][][] f;

    public int longestPalindromicSubsequence(String s, int k) {
        this.s = s.toCharArray();
        int n = s.length();
        f = new Integer[n][n][k + 1];
        return dfs(0, n - 1, k);
    }

    private int dfs(int i, int j, int k) {
        if (i > j) {
            return 0;
        }
        if (i == j) {
            return 1;
        }
        if (f[i][j][k] != null) {
            return f[i][j][k];
        }
        int res = Math.max(dfs(i + 1, j, k), dfs(i, j - 1, k));
        int d = Math.abs(s[i] - s[j]);
        int t = Math.min(d, 26 - d);
        if (t <= k) {
            res = Math.max(res, 2 + dfs(i + 1, j - 1, k - t));
        }
        f[i][j][k] = res;
        return res;
    }
}
