Generalized Shuffle Permutations on Boolean Cubes

In a generalized shuffle permutation an address (a[q-1]a[1-2]...a[0]) receives its content from an address obtained through a cyclic shift on a subset of the q dimensions used for the encoding of the addresses. Big-complementation may be combined with the shift. We give an algorithm that requires (K/2)+2 exchanges for K elements per processor, when storage dimensions are part of the permutation, and concurrent communication on all ports of every processor possible. The number of element exchanges in sequence is independent of the number of processor dimensions sigma(r) in the permutation. With no storage dimensions in the permutation our best algorithm requires (sigma[r]+1)(K/2sigma[r]) element exchanges. We also give an algorithm for sigma(r)=2, or the real shuffle consists of a number of cycles of length two, that requires (K/2)+1 element exchanges in sequence when there is no bit complement. The lower bound is (K/2) for both real and mixed shuffles with no bit complementation. The minimum number of communication start-ups for sigma(r) for both cases, which is also the lower bound. The data transfer time for communication restricted to one port per processor is sigma(r)(K/2), and the minimum number of start-ups is sigma(r). The analysis is verified by experimental results on the Intel iPSC/1, and for one case also on the Connection Machine.Engineering and Applied Science

Optimal Permutation Routing for Low-dimensional Hypercubes

We consider the offline problem of routing a permutation of tokens on the nodes of a d-dimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a d-dimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability ” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large hypercubes to 2d − 4. We also find an interesting side-result, that the number of possible communication steps in a d-dimensional hypercube is the same as the number of perfect matchings in a (d + 1)-dimensional hypercube, a combinatorial quantity for which there is no closed-form expression. Finally we present some experimental observations which may lead to a proof of a more general result for arbitrarily large dimension d. 2

Investigation of the robustness of star graph networks

The star interconnection network has been known as an attractive alternative to n-cube for interconnecting a large number of processors. It possesses many nice properties, such as vertex/edge symmetry, recursiveness, sublogarithmic degree and diameter, and maximal fault tolerance, which are all desirable when building an interconnection topology for a parallel and distributed system. Investigation of the robustness of the star network architecture is essential since the star network has the potential of use in critical applications. In this study, three different reliability measures are proposed to investigate the robustness of the star network. First, a constrained two-terminal reliability measure referred to as Distance Reliability (DR) between the source node u and the destination node I with the shortest distance, in an n-dimensional star network, Sn, is introduced to assess the robustness of the star network. A combinatorial analysis on DR especially for u having a single cycle is performed under different failure models (node, link, combined node/link failure). Lower bounds on the special case of the DR: antipode reliability, are derived, compared with n-cube, and shown to be more fault-tolerant than n-cube. The degradation of a container in a Sn having at least one operational optimal path between u and I is also examined to measure the system effectiveness in the presence of failures under different failure models. The values of MTTF to each transition state are calculated and compared with similar size containers in n-cube. Meanwhile, an upper bound under the probability fault model and an approximation under the fixed partitioning approach on the ( n-1)-star reliability are derived, and proved to be similarly accurate and close to the simulations results. Conservative comparisons between similar size star networks and n-cubes show that the star network is more robust than n-cube in terms of ( n-1)-network reliability

Hypercube-Based Topologies With Incremental Link Redundancy.

Hypercube structures have received a great deal of attention due to the attractive properties inherent to their topology. Parallel algorithms targeted at this topology can be partitioned into many tasks, each of which running on one node processor. A high degree of performance is achievable by running every task individually and concurrently on each node processor available in the hypercube. Nevertheless, the performance can be greatly degraded if the node processors spend much time just communicating with one another. The goal in designing hypercubes is, therefore, to achieve a high ratio of computation time to communication time. The dissertation addresses primarily ways to enhance system performance by minimizing the communication time among processors. The need for improving the performance of hypercube networks is clearly explained. Three novel topologies related to hypercubes with improved performance are proposed and analyzed. Firstly, the Bridged Hypercube (BHC) is introduced. It is shown that this design is remarkably more efficient and cost-effective than the standard hypercube due to its low diameter. Basic routing algorithms such as one to one and broadcasting are developed for the BHC and proven optimal. Shortcomings of the BHC such as its asymmetry and limited application are clearly discussed. The Folded Hypercube (FHC), a symmetric network with low diameter and low degree of the node, is introduced. This new topology is shown to support highly efficient communications among the processors. For the FHC, optimal routing algorithms are developed and proven to be remarkably more efficient than those of the conventional hypercube. For both BHC and FHC, network parameters such as average distance, message traffic density, and communication delay are derived and comparatively analyzed. Lastly, to enhance the fault tolerance of the hypercube, a new design called Fault Tolerant Hypercube (FTH) is proposed. The FTH is shown to exhibit a graceful degradation in performance with the existence of faults. Probabilistic models based on Markov chain are employed to characterize the fault tolerance of the FTH. The results are verified by Monte Carlo simulation. The most attractive feature of all new topologies is the asymptotically zero overhead associated with them. The designs are simple and implementable. These designs can lead themselves to many parallel processing applications requiring high degree of performance

Communication efficient multi-processor FFT

Computing the Fast Fourier Transform on a distributed memory architecture by a direct pipelined radix-2 algorithm, a bi-section or multi-section algorithm, all yield the same communications requirement, if communication for all FFT stages can be performed concurrently, the input data is in normal order, and the data allocation consecutive. With a cyclic data allocation, or bit-reversed input data and a consecutive allocation, multi-sectioning offers a reduced communications requirement by approximately a factor of two. For a consecutive data allocation, normal input order, a decimation-in-time FFT requires that (P/N) + d - 2 twiddle factors be stored for P elements distributed evenly over N processors, and the axis subject to transformation distributed over 2d processors. No communication of twiddle factors is required. The same storage requirements hold for a decimation-in-frequency FFT, bit-reversed input order, and consecutive data allocation. The opposite combination of FFT type and data ordering requires a factor of log2 N more storage for N processors. The peak performance for a Connection Machine system CM-200 implementation is 12.9 Gflops/s in 32-bit precision, and 10.7 Gflops/s in 64-bit precision for unordered transforms local to each processor. The corresponding execution rates for ordered transforms are 11.1 Gflops/s and 8.5 Gflops/s, respectively. For distributed one- and two-dimensional transforms the peak performance for unordered transforms exceeds 5 Gflops/s in 32-bit precision, and 3 Gflops/s in 64-bit precision. Three-dimensional transforms executes at a slightly lower rate. Distributed ordered transforms executes at a rate of about 1/2 to 2/3 of the unordered transforms.Engineering and Applied Science

Analysis of wormhole routings in cayley graphs of permutation groups.

Over a decade, a new class of switching technology, called wormhole routing, has been investigated in the multicomputer interconnection network field. Several classes of wormhole routing algorithms have been proposed. Most of the algorithms have been centered on the traditional binary hypercube, k-ary n-cube mesh, and torus networks. In the design of a wormhole routing algorithm, deadlock avoidance scheme is the main concern. Recently, new classes of networks called Cayley graphs of permutation groups are considered very promising alternatives. Although proposed Cayley networks have superior topological properties over the traditional network topologies, the design of the deadlock-free wormhole routing algorithm in these networks is not simple. In this dissertation, we investigate deadlock free wormhole routing algorithms in the several classes of Cayley networks, such as complete-transposition and star networks. We evaluate several classes of routing algorithms on these networks, and compare the performance of each algorithm to the simulation study. Also, the performances of these networks are compared to the traditional networks. Through extensive simulation we found that adaptive algorithm outperformed deterministic algorithm in general with more virtual channels. On the network performance comparison, the complete transposition network showed the best performance among the similar sized networks, and the binary hypercube performed better compared to the star graph

Using offline routing to implement a low latenc 3D FFT in a multinode FPGA system

Thesis (M.S.)--Boston UniversityApplications that require highly parallel computing along with low latency communication due to strong scaling, such as a calculating a 3D FFT for Molecular Dynamics simulations, can be problematic for traditional high performance computing (HPC) clusters. A multinode FPGA array is a good solution for these types of problems due to the direct high speed connections and flexible internal fabric inherent in FPGAs. Offline routing uses precomputed routing information to direct packets and can avoid much of the switching and congestion communication overhead. Two architectures are explored here which show the feasibility ofusing offline routing techniques to reduce communication latencies in FPGA systems. The first architecture targets a single FPGA that was built for initial exploration and to show how the powerful and flexible a single FPGA can be. It attained a maximum clock frequency of 102MHz and latencies of 64us and 250us for 3D FFT calculations of 32^3 and 64^3 data points respectively. The second architecture targets an FPGA that is intended to be the model for each node in the array. The best multinode version is based on a multilevel switching architecture. It has a maximum clock frequency of 134MHz. When scaled to a cluster, latencies project to 2.4us and 5.5us for 3D FFT calculations of 32^3 and 64^3 data points respectively. The two designs show the potential for using a single FPGA and multi-FPGA arrays for HPC applications where communication latency is critical to the application

Cellular Automata

Modelling and simulation are disciplines of major importance for science and engineering. There is no science without models, and simulation has nowadays become a very useful tool, sometimes unavoidable, for development of both science and engineering. The main attractive feature of cellular automata is that, in spite of their conceptual simplicity which allows an easiness of implementation for computer simulation, as a detailed and complete mathematical analysis in principle, they are able to exhibit a wide variety of amazingly complex behaviour. This feature of cellular automata has attracted the researchers' attention from a wide variety of divergent fields of the exact disciplines of science and engineering, but also of the social sciences, and sometimes beyond. The collective complex behaviour of numerous systems, which emerge from the interaction of a multitude of simple individuals, is being conveniently modelled and simulated with cellular automata for very different purposes. In this book, a number of innovative applications of cellular automata models in the fields of Quantum Computing, Materials Science, Cryptography and Coding, and Robotics and Image Processing are presented