58 research outputs found
Isomorphic routing on a toroidal mesh
We study a routing problem that arises on SIMD parallel architectures whose communication network forms a toroidal mesh. We assume there exists a set of k message descriptors (xi, yi), where (xi, yi) indicates that the ith message's recipient is offset from its sender by xi hops in one mesh dimension, and yi hops in the other. Every processor has k messages to send, and all processors use the same set of message routing descriptors. The SIMD constraint implies that at any routing step, every processor is actively routing messages with the same descriptors as any other processor. We call this isomorphic routing. Our objective is to find the isomorphic routing schedule with least makespan. We consider a number of variations on the problem, yielding complexity results from O(k) to NP-complete. Most of our results follow after we transform the problem into a scheduling problem, where it is related to other well-known scheduling problems
The multidimensional Manhattan networks
The -dimensional Manhattan network ---a special case of
-regular digraph---is formally defined and some of its structural
properties are studied. In particular, it is shown that is a
Cayley digraph, which can be seen as a subgroup of the -dim
version of the wallpaper group . These results induce a useful
new presentation of , which can be applied to design a
(shortest-path) local routing algorithm and to study some other
metric properties. Also it is shown that the -dim Manhattan
networks are Hamiltonian and, in the standard case (that is,
dimension two), they can be decomposed in two arc-disjoint
Hamiltonian cycles. Finally, some results on the connectivity and
distance-related parameters of , such as the distribution of
the node distances and the diameter are presented
Symmetric Interconnection Networks from Cubic Crystal Lattices
Torus networks of moderate degree have been widely used in the supercomputer
industry. Tori are superb when used for executing applications that require
near-neighbor communications. Nevertheless, they are not so good when dealing
with global communications. Hence, typical 3D implementations have evolved to
5D networks, among other reasons, to reduce network distances. Most of these
big systems are mixed-radix tori which are not the best option for minimizing
distances and efficiently using network resources. This paper is focused on
improving the topological properties of these networks.
By using integral matrices to deal with Cayley graphs over Abelian groups, we
have been able to propose and analyze a family of high-dimensional grid-based
interconnection networks. As they are built over -dimensional grids that
induce a regular tiling of the space, these topologies have been denoted
\textsl{lattice graphs}. We will focus on cubic crystal lattices for modeling
symmetric 3D networks. Other higher dimensional networks can be composed over
these graphs, as illustrated in this research. Easy network partitioning can
also take advantage of this network composition operation. Minimal routing
algorithms are also provided for these new topologies. Finally, some practical
issues such as implementability and preliminary performance evaluations have
been addressed
Modelling and Analysis using Graph Transformation Systems
Communication protocols, a class of critical systems, play an important role in industry. These protocols are critical because the tolerance for faults in these systems is low and it is highly desirable that these systems work correctly. Therefore, an effective methodology for describing and verifying that these systems behave according to their specifications is vitally important.
Model checking is a verification technique in which a mathematically precise model of the system, either concrete or with abstraction, is built and a specification of how the system should behave is given. Then the system is considered correct if its model satisfies its specification. However, due to their size and complexity, critical systems, such as communication systems, are notoriously resistant to formal modelling and verification.
In this thesis, we propose using graph transformation systems (GTSs), a visual
semantic modelling approach, to model the behaviour of dynamically evolving communication protocols. Then, we show how a GTS model can facilitate verification of invariant properties of potentially unbounded communication systems. Finally, due to the use of similar isomorphic components in communication systems, we show how to exploit symmetries of these dynamically evolving models described by GTSs, to reduce the size of the model under verification.
We use graph transformation systems to provide an expressive and intuitive visual description of the system state as a graph and for the computations of the system as a finite set of rules that transform the state graphs. Our model is well-suited for describing the behaviour of individual components, error-free communication channels amongst the components, and dynamic component creation and elimination. Thus, the structure of the generated model closely resembles the way in which communication protocols are typically separated into three levels: the first describing local features or components, the second characterizing interactions among components, and the third showing the evolution of the component set. The graph transformation semantics follows this scheme, enabling a clean separation of concerns when describing a protocol. This separation of concerns is a necessity for formal analysis of system behaviour.
We prove that the finite set of graph transformation rules that describe behaviour of the system can be used to perform verification for invariant properties of the system. We show that if a property is preserved by the finite set of transformation rules describing the system model, and if the initial state satisfies the property, then the property is an invariant of the system model. Therefore, our verification method may avoid the explicit analysis of the potentially enormous state space that the transformation rules encode.
In this thesis, we also develop symmetry reduction techniques applicable to dynamically evolving GTS models. The necessity to extend the existing symmetry reduction techniques arises because these techniques are not applicable to dynamic models such as those described by GTSs, and, in addition, these existing techniques may offer only limited reduction to systems that are not fully symmetric. We present an algorithm for generating a symmetry-reduced quotient model directly from a set of graph transformation rules. The generated quotient model is bisimilar to the model under verification and may be exponentially smaller than that model
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