Multiswapped networks and their topological and algorithmic properties

We generalise the biswapped network Bsw(G)Bsw(G) to obtain a multiswapped network Msw(H;G)Msw(H;G), built around two graphs G and H. We show that the network Msw(H;G)Msw(H;G) lends itself to optoelectronic implementation and examine its topological and algorithmic. We derive the length of a shortest path joining any two vertices in Msw(H;G)Msw(H;G) and consequently a formula for the diameter. We show that if G has connectivity κ⩾1κ⩾1 and H has connectivity λ⩾1λ⩾1 where λ⩽κλ⩽κ then Msw(H;G)Msw(H;G) has connectivity at least κ+λκ+λ, and we derive upper bounds on the (κ+λ)(κ+λ)-diameter of Msw(H;G)Msw(H;G). Our analysis yields distributed routing algorithms for a distributed-memory multiprocessor whose underlying topology is Msw(H;G)Msw(H;G). We also prove that if G and H are Cayley graphs then Msw(H;G)Msw(H;G) need not be a Cayley graph, but when H is a bipartite Cayley graph then Msw(H;G)Msw(H;G) is necessarily a Cayley graph

Variational networks of cube-connected cycles are recursive cubes of rings

In this short note we show that the interconnection networks known as variational networks of cube-connected cycles form a sub-class of the recursive cubes of rings

A generic greedy algorithm, partially-ordered graphs and NP-completeness.

Let π be any fixed polynomial-time testable, non-trivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. We prove that the problem of deciding whether a lexicographically first maximal subgraph of G satisfying π, with respect to one of these linear orders, contains a specified vertex is NP-complete

Embedding long paths in k-ary n-cubes with faulty nodes and links

Let $k \geq 4$ be even and let $n \geq 2$. Consider a faulty k-ary n-cube $Q_n^k$ in which the number of node faults $f_n$ and the number of link faults $f_e$ are such that $f_n + f_e \leq 2n-2$. We prove that given any two healthy nodes s and e of $Q_n^k$, there is a path from s to e of length at least $k^n - 2f_n - 1$ (resp. $k^n - 2f_n - 2$) if the nodes s and e have different (resp. the same) parities (the parity of a node in $Q_n^k$ is the sum modulo 2 of the elements in the n-tuple over {0, 1, ..., k-1} representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2

An infinite hierarchy in a class of polynomial-time program schemes

We define a class of program schemes RFDPS constructed around notions of forall-loops, repeat-loops, arrays and if-then-else instructions, and which take finite structures as inputs, and we examine the class of problems, denoted RFDPS also, accepted by such program schemes. The class of program schemes RFDPS is a logic, in Gurevich's sense, in that: every program scheme accepts an isomorphism-closed class of finite structures; we can recursively check whether a given finite structure is accepted by a given program scheme; and we can recursively enumerate the program schemes of RFDPS. We show that the class of problems RFDPS properly contains the class of problems definable in inductive fixed-point logic (for example, the well-known problem Parity is in RFDPS) and that there is a strict, infinite hierarchy of classes of problems within RFDPS (the union of which is RFDPS) parameterized by the depth of nesting of forall-loops in our program schemes. This is the first strict, infinite hierarchy in any polynomial-time logic properly extending inductive fixed-point logic (with the property that the union of the classes in the hierarchy consists of all problems definable in the logic). The fact that there are problems (like Parity) in RFDPS which cannot be defined in many of the more traditional logics of finite model theory (which often have zero-one laws) essentially means that existing tools, techniques and logical hierarchy results are of limited use to us

Accelerating ant colony optimization-based edge detection on the GPU using CUDA

Ant Colony Optimization (ACO) is a nature-inspired metaheuristic that can be applied to a wide range of optimization problems. In this paper we present the first parallel implementation of an ACO-based (image processing) edge detection algorithm on the Graphics Processing Unit (GPU) using NVIDIA CUDA. We extend recent work so that we are able to implement a novel data-parallel approach that maps individual ants to thread warps. By exploiting the massively parallel nature of the GPU, we are able to execute significantly more ants per ACO-iteration allowing us to reduce the total number of iterations required to create an edge map. We hope that reducing the execution time of an ACO-based implementation of edge detection will increase its viability in image processing and computer vision

One-to-many node-disjoint paths in (n,k)-star graphs

We present an algorithm which given a source node and a set of n−1 target nodes in the (n,k)-star graph Sn,k, where all nodes are distinct, builds a collection of n−1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k−7, and the algorithm has time complexity O(k2n2)

On the mathematics of data centre network topologies.

In a recent paper, combinatorial designs were used to construct switch-centric data centre networks that compare favourably with the ubiquitous (enhanced) fat-tree data centre networks in terms of the number of servers within (given a fixed server-to-server diameter). Unfortunately there were flaws in some of the proofs in that paper. We correct these flaws here and extend the results so as to prove that the core combinatorial construction, namely the 3-step construction, results in data centre networks with optimal path diversity

Cross-check for completeness: exploring a novel use of Leximancer in a Grounded Theory study

This paper investigates the potential for Leximancer software to actively support the Grounded Theory (GT) analyst in assessing the ‘completeness’ of their study. The case study takes an existing GT study and retrospectively analyzes the data with Leximancer. The Leximancer output showed encouraging similarities to the main themes emerging from the GT analysis; but not sufficiently at the selective coding level to justifiably claim a definitive cross-check for overall theoretical saturation. Whilst Leximancer is not found to be a substitute for the ‘hard labor’ of GT coding and theory development, it can provide a very useful, efficient and relatively impartial cross-check of completeness/saturation in the open (and possibly axial) coding stage(s) of a GT study. This automated post-analysis check of GT coding is a novel use of a CAQDAS package.<br/