<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:
    //return the minimum number of flips
    int dp[100001][2];
    int solve(int index, bool turn, int n, string &s)
    {
        if(index == n)return 0;

        if(dp[index][turn] != -1)return dp[index][turn];
        
        int flip = 1e9, notFlip = 1e9;

        if(index == 0)
        {
            flip = 1 + solve(index + 1, !turn, n, s);

            notFlip = 0 + solve(index + 1, turn, n , s);
        }
        else
        {
            if(s[index] == '0')
            {
                if(index - 1 >= 0 && s[index - 1] == '0' && turn == false)
                {
                    flip = 1 + solve(index + 1, !turn, n, s);

                    notFlip = 0 + solve(index + 1, turn, n , s);
                }
                else if(index - 1 >= 0 && s[index - 1] == '0' && turn == true)
                {
                    flip = 1 + solve(index + 1, turn, n, s);
                }
                else if(index - 1 >= 0 && s[index - 1] == '1' && turn == false)
                {
                    flip = 1 + solve(index + 1, !turn, n, s);
                }
                else if(index - 1 >= 0 && s[index - 1] == '1' && turn == true)
                {
                    flip = 1 + solve(index + 1, turn, n, s);

                    notFlip = 0 + solve(index + 1, !turn, n , s);
                }
            }
            else
            {
                if(index - 1 >= 0 && s[index - 1] == '1' && turn == false)
                {
                    notFlip = 0 + solve(index + 1, turn, n , s);
                }
                else if(index - 1 >= 0 && s[index - 1] == '1' && turn == true)
                {
                    flip = 1 + solve(index + 1, turn, n, s);

                    notFlip = 0 + solve(index + 1, !turn, n , s);
                }
                else if(index - 1 >= 0 && s[index - 1] == '0' && turn == true)
                {
                    notFlip = 0 + solve(index + 1, !turn, n , s);
                }
                else if(index - 1 >= 0 && s[index - 1] == '0' && turn == false)
                {
                    flip = 1 + solve(index + 1, !turn, n, s);

                    notFlip = 0 + solve(index + 1, turn, n , s);
                }
            }
        }

        return dp[index][turn] = min(flip, notFlip);
    }
    int minFlipsMonoIncr(string s) {
        int n = s.size();
        memset(dp, -1, sizeof(dp));
        return solve(0, false, n, s);
    }
};
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