Mesh Embedding Research Articles

Cyclic-cubes: a new family of interconnection networks of even fixed-degrees

We introduce a new family of interconnection networks that are Cayley graphs with fixed degrees of any even number greater than or equal to four. We call the proposed graphs cyclic-cubes because contracting some cycles in such a graph results in a generalized hypercube. These Cayley graphs have optimal fault tolerance and logarithmic diameters. For comparable number of nodes, a cyclic-cube can have a diameter smaller than previously known fixed-degree networks. The proposed graphs can adopt an optimum routing algorithm known for one of its subfamilies of Cayley graphs. We also show that a graph in the new family has a Hamiltonian cycle and, hence, there is an embedding of a ring. Embedding of meshes and hypercubes are also discussed.

The offset cube: a three-dimensional multicomputer network topology using through-wafer optics

Three-dimensional packaging technologies are critical for enabling ultra-compact, massively parallel processors (MPPs) for embedded applications. Through-water optical interconnect has been proposed as a useful technology for building ultra-compact MPPs since it provides a simplified mechanism for interconnecting stacked multichip substrates. This paper presents the offset cube, a new network topology designed to exploit the packaging benefits of through-wafer optical interconnect in ultra-compact MPP systems. We validate the offset cube's topological efficiency by developing deadlock-free adaptive routing protocols with modest virtual channel requirements (only two virtual channels per link needed for full adaptivity). A preliminary analysis of router complexity suggests these protocols can be efficiently implemented in hardware. We also present a 3D mesh embedding for the offset cube. Network simulations show the offset cube performs comparably to a bidirectional 3D mesh of equal size under uniform, hot-spot, and trace-driven traffic loads. While the offset cube is not proposed as a general replacement for the mesh topology it leverages the benefits of through-wafer optical interconnect more effectively than a mesh by completely eliminating chip-to-chip wires for data signals. Hence, the offset cube is an effective topology for interconnecting ultra-compact MCM-level MPP systems.

Parallel matrix product algorithm in the de Bruijn network using emulation of meshes of trees

We present an algorithm computing the product of two square matrices in the de Bruijn network by using emulation: the algorithm is based on a [ (d − 1) 2 ]-embedding of meshes of d-ary trees into de Bruijn graphs and also on a mapping of particular trees.

96/05348 Calculation of flow and heat transfer near the entrance to film-cooling holes using three-dimensional mesh embedding with solution adaptation

96/05348 Calculation of flow and heat transfer near the entrance to film-cooling holes using three-dimensional mesh embedding with solution adaptation

Calculation of flow and heat transfer near the entrance to film-cooling holes using three-dimensional mesh embedding with solution adaptation

A three-dimensional (3-D) mesh embedding algorithm for the Navier-Stokes equations is presented. An upwind control volume discretization is used to maintain accuracy and stability at the embedding interfaces. With minor restrictions on cell shape, fluxes are also conserved across the interfaces. The hanging nodes on the interfaces are treated implicitly to avoid internal boundaries in the flow that would degrade the rate of convergence. The embedded mesh solver is used to model 90 and 30° cylindrical film-cooling holes ingesting flow from a parallel-sided duct at low-, medium- and high-suction ratios. Adaptation of the mesh to the flow solution is used to improve the resolution of high-gradient regions. The influences of the inclination angle and suction ratio on the flow and heat transfer in the vicinity of the hole and on the hole discharge coefficient are modelled and compared with reported experimental measurements.

Transposition networks as a class of fault-tolerant robust networks

The paper proposes designs of interconnection networks (graphs) which can tolerate link failures. The networks under study belong to a subclass of Cayley graphs whose generators are subsets of all possible transpositions. We specifically focus on star and bubble sort networks. Our approach is to augment existing dimensions (or generators) with one or more dimensions. If the added dimension is capable of replacing any arbitrary failed dimension, it is called a wildcard dimension. It is shown that, up to isomorphism among digits used in labeling the vertices, the generators of the star graph are unique. The minimum number of extra dimensions needed to acquire i wildcard dimensions is derived for the star and bubble sort networks. Interestingly, the optimally augmented star network coincides with the Transposition network, T/sub n/. Transposition networks are studied rigorously. These networks are shown to be optimally fault tolerant. T/sub n/ is also shown to possess wide containers with short length. Fault diameter of T/sub n/ is shown to be n. While the T can efficiently embed star and bubble sort graphs, it can also lend itself to an efficient embedding of meshes and hypercubes.

Benchmarking the computation and communication performance of the CM-5

Thinking Machines' CM-5 machine is a distributed-memory, message-passing computer. In the paper we devise a performance benchmark for the base and vector units and the data communication networks of the CM-5 machine. We model the communication characteristics such as communication latency and bandwidths of point-to-point and global communication primitives. We show, on a simple Gaussian elimination code, that an accurate static performance estimation of parallel algorithms is possible by using those basic machine properties connected with computation, vectorization, communication and synchronization. Futhermore, we describe the embedding of meshes or hypercubes on the CM-5 fat-tree topology and illustrate the performance results of their basic communication primitives.

The Embedding of Meshes and Trees into Degree Four Chordal Ring Networks

In this paper a particular computer interconnection network, the symmetric chordal ring network of degree four, is presented, and the mapping of meshes and binary trees onto chordal ring networks is analysed. Expressions for the network diameter (the maximum distance a message must travel between any pair of computers) and the mean inter-computer distance are derived for a sub-set of chordal ring networks. Such networks incorporate the maximum number of computers for a given diameter, and have a communications cost, measured either as network diameter or as the mean internode distance, of O(\/N). While these networks provide attractive properties for mesh-based applications on small- and medium-sized multicomputer systems, binary trees are restricted to five levels (31 nodes).

Matrix Representation of Graph Embedding in a Hypercube

The purpose of this paper is to demonstrate the use of matrices for the representation of graph embedding in a hypercube. We denote the image of an embedding (which is a subgraph of the hypercube) as a matrix. With this representation, we are able to simplify, unify, generalize, or improve existing results regarding multigraph embedding in a hypercube. A class of trees called regular binary-reflected trees is identified, which includes linear paths, binomial trees, and many others. We show that for any regular binary-reflected tree T, n copies of T can be simultaneously embedded in an n-cube with congestion = 2. Embeddings of binary trees and meshes are also discussed.

Three‐dimensional mesh embedding for the Navier–Stokes equations using upwind control volumes

AbstractA numerical model for the compressible Navier–Stokes equations using local mesh embedding is presented. The model solves for three‐dimensional turbulent flow using an algebraic mixing length model of turbulence. The technique of control volume upwinding is used to produce a novel treatment, whereby the hanging nodes on the mesh interfaces are left with null control volumes. This yields an efficient discretization scheme which ensures second‐order accuracy, flux conservation and stability at the mesh interfaces, whilst retaining a simple interpolative treatment for the hanging nodes. The discrete flow equations are solved using the semi‐implicit pressure correction method. The accuracy of the embedded mesh solver is demonstrated by modelling the three‐dimensional flow through a cascade of turbine vanes at design and off‐design conditions. Mesh embedding gives a saving of 48% in the number of nodes. The embedded mesh solutions compare well with fine structured mesh solutions and experimental measurements. The capability of the embedded mesh solver to perform solution adaptive calculations is demonstrated using a two‐dimensional mid‐height section of the cascade at the off‐design flow conditions.

Viscous high-speed flow computations by adaptive mesh embedding techniques

In the present work, an adaptive mesh embedding technique has been developed for the solution of viscous high Mach number flows. The equations solved are the two-dimensional full Navier Stokes equations. A Runge Kutta finite volume formulation is used with symmetric discretization of both inviscid and viscous fluxes, and adaptive dissipation. The technique has been applied to a shock-wave boundary-layer laminar interaction problem and to a variety of flows over a (double ellipse) blunt body to study the effects of grid quality and topology of the adopted region on the solution accuracy. Computed results indicate that the selection of the most appropriate criterion for adaptation depends upon the physical phenomena of interest.

Embedding meshes in Boolean cubes by graph decomposition

This paper explores the embeddings of multidimensional meshes into minimal Boolean cubes by graph decomposition. The dilation and the congestion of the product graph ( G 1 × G 2) → ( H 1 × H 2) is the maximum of the dilation and congestion for the two embeddings G 1 → H 1 and G 2 → H 2. The graph decomposition technique can be used to improve the average dilation and average congestion. The graph decomposition technique combined with some particular two-dimensional embeddings allows for minimal-expansion, dilation-two, congestion-two embeddings of about 87% of all two-dimensional meshes, with a significantly lower average dilation and congestion than by modified line compression. For three-dimensional meshes we show that the graph decomposition technique, together with two three-dimensional mesh embeddings presented in this paper and modified line compression, yields dilation-two embeddings of more than 96% of all three-dimensional meshes contained in a 512 × 512 × 512 mesh. The graph decomposition technique is also used to generalize the embeddings to meshes with wrap-around. The dilation increases by at most one compared to a mesh without wraparound. The expansion is preserved for the majority of meshes, if a wraparound feature is added to the mesh.

The banyan-hypercube networks

The authors introduce a family of networks that are a synthesis of banyans and hypercubes and are called the banyan-hypercubes (BH). They combine the advantageous features of banyans and hypercubes and thus have better communication capabilities. The networks can be viewed as consisting of interconnecting hypercubes. It is shown that many hypercube features can be incorporated into BHs with regard to routing, embedding of rings and meshes, and partitioning, and that improvements over the hypercube result are made. In particular, it is shown that BHs have better diameters and average distances than hypercubes, and they embed pyramids and multiple pyramids with dilation cost 1. An optimal routing algorithm for BHs and an efficient partitioning strategy are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Minimal mesh embeddings in binary hypercubes

The authors state and prove a result about minimal graph embeddings of meshes into hypercubes. This corrects a flaw in a previously published proof by D.W. Krumme and K.N. Venkataraman (1986). >

Embeddings in hypercubes

One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their inter-communication requirements can be modeled by a graph, and the assignment of subtasks to processors viewed as an embedding of the task graph into the graph of the hypercube network. We survey the known results concerning such embeddings, including expansion/dilation tradeoffs for general graphs, embeddings of meshes and trees, packings of multiple copies of a graph, the complexity of finding good embeddings, and critical graphs which are minimal with respect to some property. In addition, we describe several open problems.