# Tag Archives: Theoretical Computer Science

## Regular Encoding of Binary Trees

A couple of months ago, a question on cstheory.SE asked whether there is a regular encoding of binary trees. My immediate gut feeling was: no way! Binary trees are equivalent to Dyck words and the language of those is not regular1.

I was able to back this feeling with an approach using generating functions. I showed that the number of binary trees with $$n$$ nodes does not have a rational generating function; as all regular languages have rational structure functions, I claimed that this shows that binary trees can not be encoded with a regular language. This argument turned out to be somewhat fishy or incomplete since an encoding does not have to map all trees of size $$n$$ to encodings of size $$f(n)$$; at this point, the counting argument breaks down.

1. Easily shown with Pumping Lemma for regular languages.

## N. Misra on Polynomial Kernelization

Neeldhara Misra, PhD student from IMSc (Chennai, India), recently visited our department at Chalmers. I had the opportunity to attend her well-given talk about the occasional infeasibility of polynomial kernelization. I liked the ideas she presented so I want to share them; you can read her own summary and download her complete work on her website.

#### Idea of Kernelization

Kernelization is about formalising preprocessing for hard problems and figuring out what can and cannot be achieved. In particular, NP-complete problems are studied; for these problems no fast (i.e. with polynomial bound on runtime) algorithms have been found yet. The idea is to crunch down a given instance of size to a size that is manageable by — more or less — naive algorithms while preserving equivalence, that is the reduced instance should have a solution if and only if the original instance has one. The notion of manageability is captured by a problem and instance dependent parameter ; we say that we have a polynomial kernel if there is an equivalent problem whose size is bounded by a polynomial in (i.e. independent of ). If is small — which is often the case in practical scenarios — the kernel can be solved reasonably quickly. Of course, the reduction process should then also be fast. Remains to mention that has to be chosen wisely; only knowing at least an upper bound for enables useful kernelizations.