<p>This might help you:
<a href="http://arxiv.org/pdf/1308.3466v1.pdf" rel="nofollow">http://arxiv.org/pdf/1308.3466v1.pdf</a> </p>

<p>If you store the last $k$ many input symbols you can easily find palindromes up to a length of $k$.<br>
If you use the algorithms of the paper you can find the midpoints of palindromes and an length estimate of its length. </p>
How to compute palindrome from a stream of characters in sub-linear space/time?
<p>I don't even know if a solution exists or not. Here is the problem in detail. You are a program that is accepting an infinitely long stream of characters (for simplicity you can assume characters are either 1 or 0). At any point, I can stop the stream (let's say after N characters were passed through) and ask you if the string received so far is a palindrome or not. How can you do this using less sub-linear space and/or time.</p>