Accepting Hybrid Networks of Evolutionary Processors with Special Topologies and Small Communication

Starting from the fact that complete Accepting Hybrid Networks of Evolutionary Processors allow much communication between the nodes and are far from network structures used in practice, we propose in this paper three network topologies that restrict the communication: star networks, ring networks, and grid networks. We show that ring-AHNEPs can simulate 2-tag systems, thus we deduce the existence of a universal ring-AHNEP. For star networks or grid networks, we show a more general result; that is, each recursively enumerable language can be accepted efficiently by a star- or grid-AHNEP. We also present bounds for the size of these star and grid networks. As a consequence we get that each recursively enumerable can be accepted by networks with at most 13 communication channels and by networks where each node communicates with at most three other nodes.Comment: In Proceedings DCFS 2010, arXiv:1008.127

On the Pseudoperiodic Extension of u^l = v^m w^n

We investigate the solution set of the pseudoperiodic extension of the classical Lyndon and Sch"utzenberger word equations. Consider u_1 ... u_l = v_1 ... v_m w_1 ... w_n, where u_i is in {u, theta(u)} for all 1 = 12 or m,n >= 5 and either m and n are not both even or not all u_i\u27s are equal, all solutions are pseudoperiodic

Testing Generalised Freeness of Words

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 efficiently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variable x and its image under f, 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 efficiently a factor of the form xx^Rxxx^R, or any other pattern of such type. We also address the problem of testing efficiently, in the same setting, whether the word w contains an arbitrary pseudo-repetition of a given exponent

Matching Patterns with Variables Under Edit Distance

A pattern $\alpha$ is a string of variables and terminal letters. We say that $\alpha$ matches a word $w$, consisting only of terminal letters, if $w$ can be obtained by replacing the variables of $\alpha$ by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. If we are interested in what is the minimum Hamming distance between $w$ and any word $u$ obtained by replacing the variables of $\alpha$ by terminal words (so matching under Hamming distance), one can devise efficient algorithms and matching conditional lower bounds for the class of regular patterns (in which no variable occurs twice), as well as for classes of patterns where we allow unbounded repetitions of variables, but restrict the structure of the pattern, i.e., the way the occurrences of different variables can be interleaved. Moreover, under Hamming distance, if a variable occurs more than once and its occurrences can be interleaved arbitrarily with those of other variables, even if each of these occurs just once, the matching problem is intractable. In this paper, we consider the problem of matching patterns with variables under edit distance. We still obtain efficient algorithms and matching conditional lower bounds for the class of regular patterns, but show that the problem becomes, in this case, intractable already for unary patterns, consisting of repeated occurrences of a single variable interleaved with terminals

Fast and Longest Rollercoasters

For k >= 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least ceil[n/2] for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(n log n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,...,n}). They have also shown that every sequence of n >=slant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least n/(2(k-1)) - 3k/2, and gave an O(nk log n)-time (respectively, O(n k log log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,...,n}). In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log^2 n)-time, that is, subquadratic even for large values of k <= n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega(n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster

Matching Patterns with Variables Under Hamming Distance

A pattern ? is a string of variables and terminal letters. We say that ? matches a word w, consisting only of terminal letters, if w can be obtained by replacing the variables of ? by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. In this paper, we approach this problem in a generalized setting, by considering approximate pattern matching under Hamming distance. More precisely, we are interested in what is the minimum Hamming distance between w and any word u obtained by replacing the variables of ? by terminal words. Firstly, we address the class of regular patterns (in which no variable occurs twice) and propose efficient algorithms for this problem, as well as matching conditional lower bounds. We show that the problem can still be solved efficiently if we allow repeated variables, but restrict the way the different variables can be interleaved according to a locality parameter. However, as soon as we allow a variable to occur more than once and its occurrences can be interleaved arbitrarily with those of other variables, even if none of them occurs more than once, the problem becomes intractable

Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations

It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results