<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: 
    vector<vector<int>> dp;

    int help(string& s, int ind, int prev) {
        if (ind >= s.size()) {
            return 0;
        }
        
        if (dp[ind][prev] != -1) {
            return dp[ind][prev];
        }

        int flip = INT_MAX; 
        int noflip = INT_MAX;

        if (s[ind] == '0') {
            if (prev == 0) {
                flip = 1 + help(s, ind + 1, 1);
                noflip = help(s, ind + 1, 0);
            } else {
                flip = 1 + help(s, ind + 1, 1);
            }
        } else if (s[ind] == '1') {
            if (prev == 1) {
                noflip = help(s, ind + 1, 1);
            } else {
                flip = 1 + help(s, ind + 1, 0);
                noflip = help(s, ind + 1, 1);
            }
        }

        return dp[ind][prev] = min(flip, noflip);
    }

    int minFlipsMonoIncr(string s) {
        int n = s.size();
        dp.resize(n, vector<int>(2, -1));
        return help(s, 0, 0);
    }
};

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