Fault-tolerant meshes and hypercubes with minimal numbers of spares

Many parallel computers consist of processors connected in the form of a d-dimensional mesh or hypercube. Two- and three-dimensional meshes have been shown to be efficient in manipulating images and dense matrices, whereas hypercubes have been shown to be well suited to divide-and-conquer algorithms requiring global communication. However, even a single faulty processor or communication link can seriously affect the performance of these machines. This paper presents several techniques for tolerating faults in d-dimensional mesh and hypercube architectures. Our approach consists of adding spare processors and communication links so that the resulting architecture will contain a fault-free mesh or hypercube in the presence of faults. We optimize the cost of the fault-tolerant architecture by adding exactly k spare processors (while tolerating up to k processor and/or link faults) and minimizing the maximum number of links per processor. For example, when the desired architecture is a d-dimensional mesh and k = 1, we present a fault-tolerant architecture that has the same maximum degree as the desired architecture (namely, 2d) and has only one spare processor. We also present efficient layouts for fault-tolerant two- and three-dimensional meshes, and show how multiplexers and buses can be used to reduce the degree of fault-tolerant architectures. Finally, we give constructions for fault-tolerant tori, eight-connected meshes, and hexagonal meshes

Fault-tolerant meshes with minimal numbers of spares

This paper presents several techniques for adding fault-tolerance to distributed memory parallel computers. More formally, given a target graph with n nodes, we create a fault-tolerant graph with n + k nodes such that given any set of k or fewer faulty nodes, the remaining graph is guaranteed to contain the target graph as a fault-free subgraph. As a result, any algorithm designed for the target graph will run with no slowdown in the presence of k or fewer node faults, regardless of their distribution. We present fault-tolerant graphs for target graphs which are 2-dimensional meshes, tori, eight-connected meshes and hexagonal meshes. In all cases our fault-tolerant graphs have smaller degree than any previously known graphs with the same properties

Wildcard dimensions, coding theory and fault-tolerant meshes and hypercubes

Hypercubes, meshes and tori are well known interconnection networks for parallel computers. The sets of edges in those graphs can be partitioned to dimensions. It is well known that the hypercube can be extended by adding a wildcard dimension resulting in a folded hypercube that has better fault-tolerant and communication capabilities. First we prove that the folded hypercube is optimal in the sense that only a single wildcard dimension can be added to the hypercube. We then investigate the idea of adding wildcard dimensions to d-dimensional meshes and tori. Using techniques from error correcting codes we construct d-dimensional meshes and tori with wildcard dimensions. Finally, we show how these constructions can be used to tolerate edge and node faults in mesh and torus networks

Fault-Tolerant Cube Graphs and Coding Theory

Hypercubes, meshes, tori and Omega networks are well known interconnection networks for parallel computers. The structure of those graphs can be described in a more general framework called cube graphs. The idea is to assume that every node in a graph with q to the power of l (letter l) nodes is represented by a unique string of l (letter l) symbols over GF(q). The edges are specified by a set of offsets, those are vectors of length l (letter l) over GF(q), where the two endpoints of an edge are an offset apart. We study techniques for tolerating edge faults in cube graphs that are based on adding redundant edges. The redundant graph has the property that the structure of the original graph can be maintained in the presence of edge faults. Our main contribution is a technique for adding the redundant edges that utilizes constructions of error-correcting codes and generalizes existing ad-hoc techniques

PCODE: an efficient and reliable collective communication protocol for unreliable broadcast domain

Existing programming environments for clusters are typically built on top of a point-to-point communication layer (send and receive) over local area networks (LANs) and, as a result, suffer from poor performance in the collective communication part. For example, a broadcast that is implemented using a TCP/IP protocol (which is a point-to-point protocol) over a LAN is obviously inefficient as it is not utilizing the fact that the LAN is a broadcast medium. We have observed that the main difference between a distributed computing paradigm and a message passing parallel computing paradigm is that, in a distributed environment the activity of every processor is independent while in a parallel environment the collection of the user-communication layers in the processors can be modeled as a single global program. We have formalized the requirements by defining the notion of a correct global program. This notion provides a precise specification of the interface between the transport layer and the user-communication layer. We have developed PCODE, a new communication protocol that is driven by a global program and proved its correctness. We have implemented the PCODE protocol on a collection of IBM RS/6000 workstations and on a collection of Silicon Graphics Indigo workstations, both communicating via UDP broadcast. The experimental results we obtained indicate that the performance advantage of PCODE over the current point-to-point approach (TCP) can be as high as an order of magnitude on a cluster of 16 workstations

Partial-sum queries in OLAP data cubes using covering codes

A partial-sum query obtains the summation over a set of specified cells of a data cube. We establish a connection between the covering problem in the theory of error-correcting codes and the partial-sum problem and use this connection to devise algorithms for the partial-sum problem with efficient space-time trade-offs. For example, using our algorithms, with 44 percent additional storage, the query response time can be improved by about 12 percent; by roughly doubling the storage requirement, the query response time can be improved by about 34 percent

Tolerating faults in a mesh with a row of spare nodes

We present an efficient method for tolerating faults in a two-dimensional mesh architecture. Our approach is based on adding spare components (nodes) and extra links (edges) such that the resulting architecture can be reconfigured as a mesh in the presence of faults. We optimize the cost of the fault-tolerant mesh architecture by adding about one row of redundant nodes in addition to a set of k spare nodes (while tolerating up to k node faults) and minimizing the number of links per node. Our results are surprisingly efficient and seem to be practical for small values of k. The degree of the fault-tolerant architecture is k + 5 for odd k, and k + 6 for even k. Our results can be generalized to d-dimensional meshes such that the number of spare nodes is less than the length of the shortest axis plus k, and the degree of the fault-tolerant mesh is (d-1)k+d+3 when k is odd and (d-1)k+2d+2 when k is even

On the design and implementation of broadcast and global combine operations using the postal model

There are a number of models that were proposed in recent years for message passing parallel systems. Examples are the postal model and its generalization the LogP model. In the postal model a parameter λ is used to model the communication latency of the message-passing system. Each node during each round can send a fixed-size message and, simultaneously, receive a message of the same size. Furthermore, a message sent out during round r will incur a latency of hand will arrive at the receiving node at round r + λ - 1. Our goal in this paper is to bridge the gap between the theoretical modeling and the practical implementation. In particular, we investigate a number of practical issues related to the design and implementation of two collective communication operations, namely, the broadcast operation and the global combine operation. Those practical issues include, for example, 1) techniques for measurement of the value of λ on a given machine, 2) creating efficient broadcast algorithms that get the latency hand the number of nodes n as parameters and 3) creating efficient global combine algorithms for parallel machines with λ which is not an integer. We propose solutions that address those practical issues and present results of an experimental study of the new algorithms on the Intel Delta machine. Our main conclusion is that the postal model can help in performance prediction and tuning, for example, a properly tuned broadcast improves the known implementation by more than 20%

CCL: a portable and tunable collective communication library for scalable parallel computers

A collective communication library for parallel computers includes frequently used operations such as broadcast, reduce, scatter, gather, concatenate, synchronize, and shift. Such a library provides users with a convenient programming interface, efficient communication operations, and the advantage of portability. A library of this nature, the Collective Communication Library (CCL), intended for the line of scalable parallel computer products by IBM, has been designed. CCL is part of the parallel application programming interface of the recently announced IBM 9076 Scalable POWERparallel System 1 (SP1). In this paper, we examine several issues related to the functionality, correctness, and performance of a portable collective communication library while focusing on three novel aspects in the design and implementation of CCL: 1) the introduction of process groups, 2) the definition of semantics that ensures correctness, and 3) the design of new and tunable algorithms based on a realistic point-to-point communication model