<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:
    def minFlipsMonoIncr(self, s: str) -> int:
        
        dp = {}

        def dfs(i,mono):
            if i == len(s):
                return 0 
            
            if (i,mono) in dp: 
                return dp[(i,mono)]
            # until now is 0 
            if s[i] == "0" and mono: 
                dp[(i,mono)] = min(1 + dfs(i+1,mono=False), dfs(i+1,mono))
            # until now is 1 changed to 0 
            elif s[i] == '1' and mono:  
                dp[(i,mono)] = min(1+ dfs(i+1,mono), dfs(i+1,mono=False))
            else:
                if s[i] == '1': 
                    dp[(i,mono)] = dfs(i+1,mono)
                else:
                    dp[(i,mono)] = 1 + dfs(i+1,mono)
            
            return dp[(i,mono)]

            
                




        return dfs(0,True)
97168