<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 {
private:
    vector<vector<int>> memo;

    int solve(int i, char prev, const string &s) {
        if (i == s.size()) {
            return 0; 
        }

        
        if (memo[i][prev - '0'] != -1) {
            return memo[i][prev - '0'];
        }

        int flip = INT_MAX, noFlip = INT_MAX;

       
        if ((prev == '0' && s[i] == '0') || (prev == '1' && s[i] == '1')) {
            noFlip = solve(i + 1, s[i], s);  
        }
        else if (prev == '0' && s[i] == '1') {
            noFlip = solve(i + 1, s[i], s);  
        }
        if (prev == '0' && s[i] == '1') {
            flip = 1 + solve(i + 1, '0', s);  
        } else if (prev == '1' && s[i] == '0') {
            flip = 1 + solve(i + 1, '1', s);  
        }
        return memo[i][prev - '0'] = min(flip, noFlip);
    }

public:
    int minFlipsMonoIncr(string s) {
        int n = s.length();
        memo.assign(n, vector<int>(2, -1));
        return solve(0, '0', s);
    }
};
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