On embeddings in cycles

We prove several exact results for the dilation of well-known interconnection networks in cycles, namely : ${\rm dil}(T_{t,r},C_{(t^{r-1}-1)/(r-1)})=\lceil t(t^{r-1}-1)/(2(r-1)(t-1))\rceil,$ for complete $r$-level $t$-ary trees, ${\rm dil}(Q_n,C_{2^n}) =\sum_{k=0}^{n-1}{k\choose \lfloor \frac{k}{2}\rfloor },$ for $n$-dimensional hypercubes, ${\rm dil}(P_n\times P_n\times P_n,C_{n^3})= \lfloor 3n^2/4+n/2\rfloor,$ for 3-dimensional meshes (where $P_n$ is an $n$-vertex path) and ${\rm dil}(P_m\times P_n,C_{mn})= {\rm dil}(C_m\times P_n,C_{mn})={\rm dil}(C_m\times C_n,C_{mn})=\min\{m,n\},$ for 2-dimensional ordinary, cylindrical and toroidal meshes, respectively. The last results solve three remaining open problems of the type $"{\rm dil}(X\times Y, Z)=?"$, where $X,\ Y$ and $Z$ are paths or cycles. The previously known dilations are: ${\rm dil}(P_m\times P_n,P_{mn})= \min \{m,n\}$, ${\rm dil}(C_m\times P_n,P_{mn})=\min \{m,2n\}$ and ${\rm dil}(C_m\times C_n,P_{mn}) =2\min \{m,n\}$, if $m\neq n$, otherwise ${\rm dil}(C_n\times C_n)=2n-1$ . The proofs of the above stated results are based on the following technique. We find a suficient condition for a graph $G$ which assures the equality ${\rm dil}(G,C_n)={\rm dil}(G,P_n)$. We prove that trees, X-trees, meshes, hypercubes, pyramides and tree of meshes satisfy the condition. Using known optimal dilations of complete trees, hypercubes and 2- and 3-dimensional meshes in path we get the above exact result

Mutation of Directed Graphs -- Corresponding Regular Expressions and Complexity of Their Generation

Directed graphs (DG), interpreted as state transition diagrams, are traditionally used to represent finite-state automata (FSA). In the context of formal languages, both FSA and regular expressions (RE) are equivalent in that they accept and generate, respectively, type-3 (regular) languages. Based on our previous work, this paper analyzes effects of graph manipulations on corresponding RE. In this present, starting stage we assume that the DG under consideration contains no cycles. Graph manipulation is performed by deleting or inserting of nodes or arcs. Combined and/or multiple application of these basic operators enable a great variety of transformations of DG (and corresponding RE) that can be seen as mutants of the original DG (and corresponding RE). DG are popular for modeling complex systems; however they easily become intractable if the system under consideration is complex and/or large. In such situations, we propose to switch to corresponding RE in order to benefit from their compact format for modeling and algebraic operations for analysis. The results of the study are of great potential interest to mutation testing

Communicability in complex networks

Many topological and dynamical properties of complex networks are defined by assuming that most of the transport on the network flows along the shortest paths. However, there are different scenarios in which non-shortest paths are used to reach the network destination. Thus the consideration of the shortest paths only does not account for the global communicability of a complex network. Here we propose a new measure of the communicability of a complex network, which is a broad generalization of the concept of the shortest path. According to the new measure, most of real-world networks display the largest communicability between the most connected (popular) nodes of the network (assortative communicability). There are also several networks with the disassortative communicability, where the most "popular" nodes communicate very poorly to each other. Using this information we classify a diverse set of real-world complex systems into a small number of universality classes based on their structure-dynamic correlation. In addition, the new communicability measure is able to distinguish finer structures of networks, such as communities into which a network is divided. A community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of nodes in the network.Comment: 20 pages, 5 figure

Implications of quantum automata for contextuality

We construct zero-error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded-error probabilistic finite automata (PFAs). Here is a summary of our results: - There is a promise problem solvable by an exact two-way QFA in exponential expected time, but not by any bounded-error sublogarithmic space probabilistic Turing machine (PTM). - There is a promise problem solvable by an exact two-way QFA in quadratic expected time, but not by any bounded-error $o(\log \log n)$-space PTMs in polynomial expected time. The same problem can be solvable by a one-way Las Vegas (or exact two-way) QFA with quantum head in linear (expected) time. - There is a promise problem solvable by a Las Vegas realtime QFA, but not by any bounded-error realtime PFA. The same problem can be solvable by an exact two-way QFA in linear expected time but not by any exact two-way PFA. - There is a family of promise problems such that each promise problem can be solvable by a two-state exact realtime QFAs, but, there is no such bound on the number of states of realtime bounded-error PFAs solving the members this family. Our results imply that there exist zero-error quantum computational devices with a \emph{single qubit} of memory that cannot be simulated by any finite memory classical computational model. This provides a computational perspective on results regarding ontological theories of quantum mechanics \cite{Hardy04}, \cite{Montina08}. As a consequence we find that classical automata based simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently powerful to simulate quantum contextuality. We conclude by highlighting the interplay between results from automata models and their application to developing a general framework for quantum contextuality.Comment: 22 page

Homomorphic Encryption for Finite Automata

We describe a somewhat homomorphic GSW-like encryption scheme, natively encrypting matrices rather than just single elements. This scheme offers much better performance than existing homomorphic encryption schemes for evaluating encrypted (nondeterministic) finite automata (NFAs). Differently from GSW, we do not know how to reduce the security of this scheme to LWE, instead we reduce it to a stronger assumption, that can be thought of as an inhomogeneous variant of the NTRU assumption. This assumption (that we term iNTRU) may be useful and interesting in its own right, and we examine a few of its properties. We also examine methods to encode regular expressions as NFAs, and in particular explore a new optimization problem, motivated by our application to encrypted NFA evaluation. In this problem, we seek to minimize the number of states in an NFA for a given expression, subject to the constraint on the ambiguity of the NFA

Passagierdampfer als Hilfskreuzer: kurze Geschichte der Entwicklung einer Schiffsgattung

To date there is no cohesive historical account of the international development of the auxiliary cruiser, and the scattered investigations of various single aspects of this ship category do not add up to a comprehensive survey. These were the circumstances discovered by the author as, upon request of the editor of this publication, he undertook to provide an introduction to the report by Captain Meyer of the only voyage of the "Kaiser Wilhelm der Grosse" as an auxiliary cruiser (see the following article). The first task was thus to gain an overall impression of this history. "Auxiliary cruisers", as defined here, were passenger steamships kept ready for battle by large shipping companies in countries with significant naval power. These ships were technically equipped for the installation of weapons, and their conversion could be carried out within a few days. The auxiliary cruiser was first to put into service by the Russians, who felt threatened by the British Royal Navy after the Russian-Turkish Balkan War. At that time a newly established shipping company, later called the "Russian Voluntary Fleet", bought three steamships from the Hamburg-America Line and had them converted for combat purposes. By the turn of the century, this originally Russian idea had been adopted by all of the world's Iarge navies . This brief study also touches on the various attempt worldwide to incorporate the legal status of the ship type "auxiliary cruiser" into international public and military law. Because of geopolitically based conflicts of interest between the naval powers, these endeavours were only partially successful. ln conclusion the author examines the auxiliary cruiser concept of the Imperial German Navy, whose goal it was to interfere with and impair the enemy's overseas commerce. The express Iiners originally designated for this purpose, however, were for various reasons inconceivably ill-suited to it. As early as 1915 the German Navy proceeded to commission cargo vessels as auxiliary cruisers, as they were inconspicuous and easier to supply. This was then the mode of operation employed by the navy during World War II

Optimal Algorithms For Dissemination Of Information In Generalized Communication Modes

Some generalized communication modes enabling the dissemination of information among processors of interconnection networks via vertex-disjoint or edge-disjoint paths in one communication step will be investigated. A thorough study of these communication modes will be presented by giving optimal algorithms for broadcasting, accumulation and gossiping in most of the well known parallel architectures. For those networks in which a Hamiltonian path exists (Hypercubes, Cube Connected Cycles, Butterflies, Shuffle Exchange, etc.) optimal algorithms can be obtained quite easily, but for complete binary trees, complete k-ary trees (k 3) and arbitrary degree bounded graphs, the optimal algorithms as well as the matching lower bound proofs are more involved. An interesting consequence of the presented algorithms is the fact that in almost all these interconnection networks the gossip problem cannot be solved in less time than the sum of time complexities of the accumulation problem and the bro..