<p>You are given a binary string <code>binary</code> consisting of only <code>0</code>&#39;s or <code>1</code>&#39;s. You can apply each of the following operations any number of times:</p>

<ul>
	<li>Operation 1: If the number contains the substring <code>&quot;00&quot;</code>, you can replace it with <code>&quot;10&quot;</code>.

	<ul>
		<li>For example, <code>&quot;<u>00</u>010&quot; -&gt; &quot;<u>10</u>010</code>&quot;</li>
	</ul>
	</li>
	<li>Operation 2: If the number contains the substring <code>&quot;10&quot;</code>, you can replace it with <code>&quot;01&quot;</code>.
	<ul>
		<li>For example, <code>&quot;000<u>10</u>&quot; -&gt; &quot;000<u>01</u>&quot;</code></li>
	</ul>
	</li>
</ul>

<p><em>Return the <strong>maximum binary string</strong> you can obtain after any number of operations. Binary string <code>x</code> is greater than binary string <code>y</code> if <code>x</code>&#39;s decimal representation is greater than <code>y</code>&#39;s decimal representation.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> binary = &quot;000110&quot;
<strong>Output:</strong> &quot;111011&quot;
<strong>Explanation:</strong> A valid transformation sequence can be:
&quot;0001<u>10</u>&quot; -&gt; &quot;0001<u>01</u>&quot; 
&quot;<u>00</u>0101&quot; -&gt; &quot;<u>10</u>0101&quot; 
&quot;1<u>00</u>101&quot; -&gt; &quot;1<u>10</u>101&quot; 
&quot;110<u>10</u>1&quot; -&gt; &quot;110<u>01</u>1&quot; 
&quot;11<u>00</u>11&quot; -&gt; &quot;11<u>10</u>11&quot;
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> binary = &quot;01&quot;
<strong>Output:</strong> &quot;01&quot;
<strong>Explanation:</strong>&nbsp;&quot;01&quot; cannot be transformed any further.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= binary.length &lt;= 10<sup>5</sup></code></li>
	<li><code>binary</code> consist of <code>&#39;0&#39;</code> and <code>&#39;1&#39;</code>.</li>
</ul>



int trailing_zero_count(const string &str, size_t i)
{
    int ans = 0;
    while (i < str.size())
        if (str[i++] == '0')
            ans++;
        else
            break;
    return ans;
}

void fetch_zero(string &str, size_t i)
{
    if (i + 1 >= str.size())
        return;
    auto j = str.find_first_of('0', i + 1);
    if (j == -1)
        return;
    str[i] = '1';
    str[i + 1] = '0';
    str[j] = '1';
    fetch_zero(str, i + 1);
}

void convert(string &str, size_t i)
{
    auto zeros = trailing_zero_count(str, i);
    if (zeros == 0)
    {
        auto j = str.find_first_of('0', i + 1);
        if (j == -1)
            return;
        convert(str, j);
        return;
    }
    if (zeros == 1)
    {
        fetch_zero(str, i);
        return;
    }
    for (size_t j = 0; j < zeros; j++)
        str[i + j] = '1';
    str[i + zeros - 1] = '0';
    fetch_zero(str, i + zeros - 1);
}

class Solution
{
public:
    string maximumBinaryString(string binary)
    {
        convert(binary, 0);
        return binary;
    }
};
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