2013/finals/archiver.html

<p>You are writing a new revolutionary archiver. The archive is essentially a
pair of non-decreasing sequences of integers of equal length <strong>K</strong>: <strong>0&le;x<sub>1</sub>&le;...&le;x<sub>k</sub></strong> and
<strong>0&le;y<sub>1</sub>&le;...&le;y<sub>k</sub></strong>.</p>

<p>
The decompression algorithm proceeds as follows:
<ol>
<li>Sequence <strong>(0,0), (x<sub>1</sub>,y<sub>1</sub>), ... (x<sub>k</sub>,y<sub>k</sub>), (x<sub>k</sub>, 0), (0, 0)</strong> defines a polygon <strong>P</strong></li>
<li>Starting from the point <strong>(0,0)</strong>, increase either <strong>x</strong> or <strong>y</strong> coordinate by 1 without
moving outside of <strong>P</strong>. If both moves are available, you should increase y.
After each step write <strong>0</strong> to output if incremented <strong>x</strong> or <strong>1</strong> otherwise.</li>
<li>Repeat step 2 until you end up in point <strong>(x<sub>k</sub>,y<sub>k</sub>)</strong>.</li>
</ol>
</p>

<p>Example: decompression of sequence <strong>(3,4), (7,6), (7,8)</strong> will produce string <strong>010101100100111</strong>.</p>

<p>Your task is to write a compression rate calculator, that is given
binary string s find the smallest value of <strong>K</strong> for which there exists archive
that decompresses to s.</p>

<h2>Input</h2>
<p>
The first line contains a single integer <strong>T</strong>, <strong>T</strong> &le; 20. <strong>T</strong> test cases follow, where each test case consists of one binary string with length <strong>&le; 1,000,000</strong>.
</p>

<h2>Output</h2>
<p>Output a single line containing the smallest possible <strong>K</strong>.</p>