Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding eciently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variablex and its image underf, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect eciently a factor of the form xx R xxx R , or any other pattern of such type. We also address the problem of testing eciently, in the same setting, whether the wordw contains an arbitrary pseudo-repetition of a given exponent. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

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Testing Generalised Freeness of Words

Sorting networks: to the end and back again

Palindromic length of a word is defined as the smallest number n such that the word can be written as a product of n palindromes. It has been conjectured that every aperiodic infinite word has factors of arbitrarily high palindromic length. A stronger variant of this conjecture claims that every aperiodic infinite word has also prefixes of arbitrarily high palindromic length. We prove that these two conjectures are equivalent. More specifically, we prove that if every prefix of a word is a product of n palindromes, then every factor of the word is a product of 2n palindromes. Our proof quite naturally leads us to compare the properties of palindromic length in free monoids and in free groups. For example, the palindromic lengths of a word and its conjugate can be arbitrarily far apart in a free monoid, but in a free group they are almost the same.

Palindromic Length in Free Monoids and Free Groups

An important benefit of multi-objective search is that it maintains a diverse population of candidates, which helps in deceptive problems in particular. Not all diversity is useful, however: candid...

Enhanced Optimization with Composite Objectives and Novelty Selection

Optimal-depth 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].

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 less 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].

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.

The avoidability of cubes under permutations

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 less computation steps) than the previously known best networks. There- fore, we improve upon the known upper bounds for minimal depth sort- ing networks on 17;19; and 20 channels. The networks were obtained using a combination of hand-crafted first layers and a SAT encoding of sorting networks.

Mike Müller

Papers

Sorting networks: to the end and back again

New Bounds on Optimal Sorting Networks

New Bounds on Optimal Sorting Networks

The avoidability of cubes under permutations

Faster Sorting Networks for 17, 19 and 20 Inputs