<p>A binary string is monotone increasing if it consists of some number of <code>0</code>&#39;s (possibly none), followed by some number of <code>1</code>&#39;s (also possibly none).</p>

<p>You are given a binary string <code>s</code>. You can flip <code>s[i]</code> changing it from <code>0</code> to <code>1</code> or from <code>1</code> to <code>0</code>.</p>

<p>Return <em>the minimum number of flips to make </em><code>s</code><em> monotone increasing</em>.</p>

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

<pre>
<strong>Input:</strong> s = &quot;00110&quot;
<strong>Output:</strong> 1
<strong>Explanation:</strong> We flip the last digit to get 00111.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;010110&quot;
<strong>Output:</strong> 2
<strong>Explanation:</strong> We flip to get 011111, or alternatively 000111.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;00011000&quot;
<strong>Output:</strong> 2
<strong>Explanation:</strong> We flip to get 00000000.
</pre>

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

<ul>
	<li><code>1 &lt;= s.length &lt;= 10<sup>5</sup></code></li>
	<li><code>s[i]</code> is either <code>&#39;0&#39;</code> or <code>&#39;1&#39;</code>.</li>
</ul>

class Solution {
public:
    int minFlipsMonoIncr(string s) {
        vector<vector<int>> memo(2, vector<int>(s.size(), -1));
        return dfs(0, false, s, memo);
    }

    int dfs(int i, bool oneFound, string& s, vector<vector<int>>& memo) {
        if (i == s.size()) {
            return 0;
        }
        if (memo[oneFound][i] != -1) {
            return memo[oneFound][i];
        }

        if (oneFound) {
            memo[oneFound][i] = (s[i] == '0') + dfs(i + 1, true, s, memo);
        }
        else {
            if (s[i] == '0') {
                memo[oneFound][i] = dfs(i + 1, false, s, memo);
            }
            else {
                memo[oneFound][i] = min(dfs(i + 1, true, s, memo), 1 + dfs(i + 1, false, s, memo));
            }
        }
        return memo[oneFound][i];
    }
};
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