<p>You are given an array of binary strings <code>strs</code> and two integers <code>m</code> and <code>n</code>.</p>

<p>Return <em>the size of the largest subset of <code>strs</code> such that there are <strong>at most</strong> </em><code>m</code><em> </em><code>0</code><em>&#39;s and </em><code>n</code><em> </em><code>1</code><em>&#39;s in the subset</em>.</p>

<p>A set <code>x</code> is a <strong>subset</strong> of a set <code>y</code> if all elements of <code>x</code> are also elements of <code>y</code>.</p>

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

<pre>
<strong>Input:</strong> strs = [&quot;10&quot;,&quot;0001&quot;,&quot;111001&quot;,&quot;1&quot;,&quot;0&quot;], m = 5, n = 3
<strong>Output:</strong> 4
<strong>Explanation:</strong> The largest subset with at most 5 0&#39;s and 3 1&#39;s is {&quot;10&quot;, &quot;0001&quot;, &quot;1&quot;, &quot;0&quot;}, so the answer is 4.
Other valid but smaller subsets include {&quot;0001&quot;, &quot;1&quot;} and {&quot;10&quot;, &quot;1&quot;, &quot;0&quot;}.
{&quot;111001&quot;} is an invalid subset because it contains 4 1&#39;s, greater than the maximum of 3.
</pre>

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

<pre>
<strong>Input:</strong> strs = [&quot;10&quot;,&quot;0&quot;,&quot;1&quot;], m = 1, n = 1
<strong>Output:</strong> 2
<b>Explanation:</b> The largest subset is {&quot;0&quot;, &quot;1&quot;}, so the answer is 2.
</pre>

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

<ul>
	<li><code>1 &lt;= strs.length &lt;= 600</code></li>
	<li><code>1 &lt;= strs[i].length &lt;= 100</code></li>
	<li><code>strs[i]</code> consists only of digits <code>&#39;0&#39;</code> and <code>&#39;1&#39;</code>.</li>
	<li><code>1 &lt;= m, n &lt;= 100</code></li>
</ul>

class Solution {
private:
  struct Counts {
    int zeros = 0;
    int ones = 0;
  };
public:
  int findMaxForm(const vector<string>& strs, int m, int n) {
    vector<int> cache(strs.size() * (m + 1) * (n + 1), -1);

    const auto get_cached = [&cache, m, n](int i, int km, int kn) -> int& {
      return cache[i * (m+1) * (n+1) + km * (n+1) + kn];
    };

    return largest_subset(strs, 0, m, n, get_cached);
  }

private:
  int largest_subset(
    const vector<string>& strs,
    int i,
    int m,
    int n,
    function<int&(int i, int m, int n)> get_cached
  ) {
    if (i >= strs.size()) return 0;

    auto& cached = get_cached(i, m, n);
    if (cached != -1) return cached;

    const string& s = strs[i];

    const int max_exclusive = largest_subset(strs, i+1, m, n, get_cached);

    const int n_zeros = count(s.begin(), s.end(), '0');
    const int n_ones = count(s.begin(), s.end(), '1');

    if (n_zeros <= m && n_ones <= n) {
      const int max_inclusive = 1 + largest_subset(strs, i+1, m - n_zeros, n - n_ones, get_cached);

      return cached = max(max_inclusive, max_exclusive);
    }

    return cached = max_exclusive;
  }
};

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