Mike Müller

New Bounds on Optimal Sorting Networks

We present new parallel sorting networks for \(17\) to \(20\) inputs. For \(17, 19,\) and \(20\) inputs these new networks are faster (i.e., they require fewer computation steps) than the previously known best networks. Therefore, we improve upon the known upper bounds for minimal depth sorting networks on \(17, 19,\) and \(20\) channels. Furthermore, we show that our sorting network for \(17\) inputs is optimal in the sense that no sorting network using less layers exists. This solves the main open problem of [D. Bundala & J. Zavodný. Optimal sorting networks, Proc. LATA 2014].

The avoidability of cubes under permutations

In this paper we consider the avoidance of patterns in infinite words. Generalising the traditional problem setting, functional dependencies between pattern variables are allowed here, in particular, patterns involving permutations. One of the remarkable facts is that in this setting the notion of avoidability index (the smallest alphabet size for which a pattern is avoidable) is meaningless since a pattern with permutations that is avoidable in one alphabet can be unavoidable in a larger alphabet. We characterise the (un-)avoidability of all patterns of the form πi(x) πj(x) πk(x), called cubes under permutations here, for all alphabet sizes in both the morphic and antimorphic case.

Sorting networks: To the end and back again

Abstract New properties of the front and back ends of sorting networks are studied, illustrating their utility when searching for bounds on optimal networks. Search focuses first on the “out-sides” of the network and then on the inner part. Previous works focused on properties of the front end to break symmetries in the search. The new, out-side-in, properties shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimality. We present new, faster, parallel sorting networks for 17–20 inputs. For 17 inputs, we show that no sorting network using less layers exists.

Square-free shuffles of words

Let

Generalised Lyndon-Schutzenberger Equations

u \shuffle v

Fully bordered words

denote the set of all shuffles of the words

INNER PALINDROMIC CLOSURE

u

Cubic patterns with permutations

and

String matching with involutions

v

A note on short palindromes in square-free words

. It is shown that for each integer