<p>You are given an array of strings <code>arr</code>. A string <code>s</code> is formed by the <strong>concatenation</strong> of a <strong>subsequence</strong> of <code>arr</code> that has <strong>unique characters</strong>.</p>

<p>Return <em>the <strong>maximum</strong> possible length</em> of <code>s</code>.</p>

<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>

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

<pre>
<strong>Input:</strong> arr = [&quot;un&quot;,&quot;iq&quot;,&quot;ue&quot;]
<strong>Output:</strong> 4
<strong>Explanation:</strong> All the valid concatenations are:
- &quot;&quot;
- &quot;un&quot;
- &quot;iq&quot;
- &quot;ue&quot;
- &quot;uniq&quot; (&quot;un&quot; + &quot;iq&quot;)
- &quot;ique&quot; (&quot;iq&quot; + &quot;ue&quot;)
Maximum length is 4.
</pre>

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

<pre>
<strong>Input:</strong> arr = [&quot;cha&quot;,&quot;r&quot;,&quot;act&quot;,&quot;ers&quot;]
<strong>Output:</strong> 6
<strong>Explanation:</strong> Possible longest valid concatenations are &quot;chaers&quot; (&quot;cha&quot; + &quot;ers&quot;) and &quot;acters&quot; (&quot;act&quot; + &quot;ers&quot;).
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> arr = [&quot;abcdefghijklmnopqrstuvwxyz&quot;]
<strong>Output:</strong> 26
<strong>Explanation:</strong> The only string in arr has all 26 characters.
</pre>

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

<ul>
	<li><code>1 &lt;= arr.length &lt;= 16</code></li>
	<li><code>1 &lt;= arr[i].length &lt;= 26</code></li>
	<li><code>arr[i]</code> contains only lowercase English letters.</li>
</ul>

#include <vector>
#include <string>
#include <unordered_map>

class Solution {
public:
    // Function to check if two strings have any common characters
    bool hasCommon(const std::string &cstr, const std::string &tmp) {
        std::vector<int> a(26, 0);
        for (const auto &ch : cstr) {
            a[ch - 'a']++;
        }
        for (const auto &ch : tmp) {
            if (a[ch - 'a'] > 0) return true;
        }
        return false;
    }

    // Helper function with memoization
    int helper(int i, std::string cstr, std::vector<std::string> &arr, std::unordered_map<std::string, int> &memo) {
        if (i >= arr.size()) return 0;

        // Create a key for the current state
        std::string key = cstr + "#" + std::to_string(i);
        
        // Check if the result for this state is already computed
        if (memo.find(key) != memo.end()) return memo[key];

        int np = helper(i + 1, cstr, arr, memo);
        int f = 1;
        if (hasCommon(cstr, arr[i])) f = 0;

        std::vector<int> tmp(26, 0);
        for (const auto &ch : arr[i]) {
            if (tmp[ch - 'a'] == 1) {
                f = 0;
                break;
            } else {
                tmp[ch - 'a'] = 1;
            }
        }

        int p = 0;
        if (f) {
            cstr += arr[i];
            p = arr[i].size() + helper(i + 1, cstr, arr, memo);
        }

        // Memoize the result
        return memo[key] = std::max(p, np);
    }

    // Main function to find the maximum length of a concatenated string with unique characters
    int maxLength(std::vector<std::string> &arr) {
        std::string cstr = "";
        std::unordered_map<std::string, int> memo;
        return helper(0, cstr, arr, memo);
    }
};

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