<p>Given two strings <code>word1</code> and <code>word2</code>, return <em>the minimum number of operations required to convert <code>word1</code> to <code>word2</code></em>.</p>

<p>You have the following three operations permitted on a word:</p>

<ul>
	<li>Insert a character</li>
	<li>Delete a character</li>
	<li>Replace a character</li>
</ul>

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

<pre>
<strong>Input:</strong> word1 = &quot;horse&quot;, word2 = &quot;ros&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> 
horse -&gt; rorse (replace &#39;h&#39; with &#39;r&#39;)
rorse -&gt; rose (remove &#39;r&#39;)
rose -&gt; ros (remove &#39;e&#39;)
</pre>

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

<pre>
<strong>Input:</strong> word1 = &quot;intention&quot;, word2 = &quot;execution&quot;
<strong>Output:</strong> 5
<strong>Explanation:</strong> 
intention -&gt; inention (remove &#39;t&#39;)
inention -&gt; enention (replace &#39;i&#39; with &#39;e&#39;)
enention -&gt; exention (replace &#39;n&#39; with &#39;x&#39;)
exention -&gt; exection (replace &#39;n&#39; with &#39;c&#39;)
exection -&gt; execution (insert &#39;u&#39;)
</pre>

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

<ul>
	<li><code>0 &lt;= word1.length, word2.length &lt;= 500</code></li>
	<li><code>word1</code> and <code>word2</code> consist of lowercase English letters.</li>
</ul>

class Solution {
public:
    int minDistance(const std::string& word1, const std::string& word2) {
        std::vector<std::vector<std::optional<int>>> memo(
            word1.size() + 1,
            std::vector<std::optional<int>>(word2.size() + 1)
        );
        std::function<int(int, int)> backtrack{
            [&memo, &word1, &word2, &backtrack] (int i, int j) -> int {
                if (memo[i][j]) return memo[i][j].value();
                while (
                    (i < word1.size()) && 
                    (j < word2.size()) &&
                    (word1[i] == word2[j])
                ) {
                    ++i;
                    ++j;
                }
                if (i >= word1.size()) return word2.size() - j;
                if (j >= word2.size()) return word1.size() - i;
                std::array<int, 3> options{
                    backtrack(i + 1, j),
                    backtrack(i, j + 1),
                    backtrack(i + 1, j + 1)
                };
                int min{*std::min_element(options.begin(), options.end()) + 1};
                memo[i][j] = min;
                return min;
            }
        };
        return backtrack({}, {});
    }
};
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