<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(string word1, string word2) {
        const int N = word1.size();
        const int M = word2.size();
        std::vector<std::vector<int>> cache(N, std::vector<int>(M, 0));
        std::vector<std::vector<int>> in_cache(N, std::vector<int>(M, 0));

        std::function<int(int, int)> dp = [&](int i, int j) -> int {
            // If we reach the end, then the minimum number of ops
            // we need is how many characters we have left to process.
            if (i == N) {
                return M - j;
            }
            if (j == M) {
                return N - i;
            }
            if (in_cache[i][j]) {
                return cache[i][j];
            }

            // if chars are equal, move onto next char. Nothing to do here
            if (word1[i] == word2[j]) {
                in_cache[i][j] = true;
                return cache[i][j] = dp(i + 1, j + 1);
            }
            // 3 choices:
            // insert char
            int insert = 1 + dp(i, j + 1);
            // delete char
            int del = 1 + dp(i + 1, j);
            // replace char
            int replace = 1 + dp(i + 1, j + 1);
            in_cache[i][j] = true;
            return cache[i][j] = std::min({insert, del, replace});
        };

        return dp(0, 0);
    }
};
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