<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 solve(int i, int prev, const string& s, vector<vector<int>>& dp) {
    if (i >= s.size())
        return 0;
    if (dp[i][prev] != -1)
        return dp[i][prev];
    
    int flip = INT_MAX;
    int noflip = INT_MAX;
    
    if (s[i] == '1') {
        if (prev == 0) {
            flip = 1 + solve(i + 1, 0, s, dp);
            noflip = solve(i + 1, 1, s, dp);
        } else if (prev == 1) {
            noflip = solve(i + 1, 1, s, dp);
        }
    } else if (s[i] == '0') {
        if (prev == 1) {
            flip = 1 + solve(i + 1, 1, s, dp);
        } else if (prev == 0) {
            flip = 1 + solve(i + 1, 1, s, dp);
            noflip = solve(i + 1, 0, s, dp);
        }
    }
    
    return dp[i][prev] = min(flip, noflip);
}

int minFlipsMonoIncr(const string& s) {
    int n = s.size();
    vector<vector<int>> dp(n, vector<int>(2, -1));
    
    // Initial call with starting character's status (0 or 1)
    return min(solve(0, 0, s, dp), solve(0, 1, s, dp));
}
};
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