<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 helper(int index, int digit, string &s, vector<vector<int>>&dp)
    {
        if(index==s.length()) return 0;
        if(dp[index][digit]!=-1) return dp[index][digit];

        int res;
        if(digit==0)
        {
            if(s[index]=='0')
            {
                res= helper(index+1,digit,s,dp);
            }
            else
            {
                int flip=1+ helper(index+1,digit,s,dp);
                int not_flip =helper(index+1,1,s,dp);
                res=min(flip,not_flip);
            }
        }
        else
        {
            if(s[index]=='1')
            {
                res=helper(index+1,digit,s,dp);
            }
            else
            {
                res= 1+ helper(index+1,digit,s,dp);
            }
        }
        return dp[index][digit]=res;
    }
    int minFlipsMonoIncr(string s) {
        int n=s.length();
        // vector<vector<int>> dp(n,vector<int>(2,-1));
        // return helper(0,0,s,dp);

        vector<vector<int>> dp(n, vector<int>(2,0));
        if(s[0]=='0')
        {
            dp[0][1]=1;
        }
        else
        {
            dp[0][0]=1;
        }
        for(int i=1;i<n;i++)
        {
            if(s[i]=='0')
            {
                dp[i][0]=dp[i-1][0];
                dp[i][1]=1+min(dp[i-1][0],dp[i-1][1]);
            }
            else
            {
                dp[i][0]=1+dp[i-1][0];
                dp[i][1]=min(dp[i-1][0],dp[i-1][1]);
            }
        }
        return min(dp[n-1][0], dp[n-1][1]);
    }
};
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