Routing In Hypercubes Research Articles

A Fault-Handling Method for the Hamiltonian Cycle in the Hypercube Topology

Many routing protocols, such as distance vector and link-state protocols are used for finding the best paths in a network. To find the path between the source and destination nodes where every node is visited once with no repeats, Hamiltonian and Hypercube routing protocols are often used. Nonetheless, these algorithms are not designed to solve the problem of a node failure, where one or more nodes become faulty. This paper proposes an efficient modified Fault-free Hamiltonian Cycle based on the Hypercube Topology (FHCHT) to perform a connection between nodes when one or more nodes become faulty. FHCHT can be applied in a different environment to transmit data with a high-reliability connection by finding an alternative path between the source and destination nodes when some nodes fail. Moreover, a proposed Hamiltonian Near Cycle (HNC) scheme has been developed and implemented. HNC implementation results indicated that FHCHT produces alternative cycles relatively similar to a Hamiltonian Cycle for the Hypercube, complete, and random graphs. The implementation of the proposed algorithm in a Hypercube achieved a 31% and 76% reduction in cost compared to the complete and random graphs, respectively.

Hypercube Fault Tolerant Routing with Bit Constraint

Thanks to its simple definition, the hypercube topology is very popular as interconnection network of parallel systems. There have been several routing algorithms described for the hypercube topology, yet in this paper we focus on hypercube routing extended with an additional restriction: bit constraint. Concretely, path selection is performed on a particular subset of nodes: the nodes are required to satisfy a condition regarding their bit weights (a.k.a. Hamming weights). There are several applications to such restricted routing, including simplification of disjoint paths routing. We propose in this paper two hypercube routing algorithms enforcing such node restriction: first, a shortest path routing algorithm, second a fault tolerant point-to-point routing algorithm. Formal proof of correctness and complexity analysis for the described algorithms are conducted. We show that the shortest path routing algorithm proposed is time optimal. Finally, we perform an empirical evaluation of the proposed fault tolerant point-to-point routing algorithm so as to inspect its practical behaviour. Along with this experimentation, we analyse further the average performance of the proposed algorithm by discussing the average Hamming distance in a hypercube when satisfying a bit constraint.

Mathematical performance modelling of adaptive wormhole routing in optoelectronic hypercubes

In this paper, an analytical performance model for adaptive wormhole routing in the hypercube-based optical transpose interconnection system (OTIS-hypercube) is proposed and validated. Using this model, the performance merits of the OTIS-hypercube are discussed. We also compare the performance of the OTIS-hypercube with that of equivalent hypercube network under different implementation constraints. Compared to pure simulation results, the presented model can serve as a more appropriate and compact method to display the performance characteristics of OTIS-hypercube networks.

Fault-tolerant routing in hypercubes using partial path set-up

A new technique is proposed for fault-tolerant routing in hypercubes, which needs to set up a partial path based on local safety information. Local safety information is utilized to guide fault-tolerant routing. Local safety is a centralized metric. A feasible path from the source to the destination may not be guaranteed at the source based on local safety information when the system contains a large number of faults, although a feasible path is available. A partial path is set up for fault-tolerant routing, where the header flit is forwarded until a maximal safe subcube is found to contain the current node and the destination. Backtracking is adopted only for the header along the minimum paths or non-minimum feasible paths, if necessary, in order to set up a partial feasible path. Extensive simulation results have shown that the partial path set-up scheme is quite useful for fault-tolerant routing, while the extra cost caused by path set-up is trivial.

Failure recovery for structured p2p networks: Protocol design and performance under churn

Measurement studies indicate a high rate of node dynamics in p2p systems. In this paper, we address the question of how high a rate of node dynamics can be supported by structured p2p networks. We confine our study to the hypercube routing scheme used by several structured p2p systems. To improve system robustness and facilitate failure recovery, we introduce the property of K-consistency, K ≥ 1, which generalizes consistency defined previously. (Consistency guarantees connectivity from any node to any other node.) We design and evaluate a failure recovery protocol based upon local information for K-consistent networks. The failure recovery protocol is then integrated with a join protocol that has been proved to construct K-consistent neighbor tables for concurrent joins. The integrated protocols were evaluated by a set of simulation experiments in which nodes joined a 2000-node network and nodes (both old and new) were randomly selected to fail concurrently over 10,000 s of simulated time. In each such experiment, we took a snapshot of neighbor tables in the network once every 50 s and evaluated connectivity and consistency measures over time as a function of the churn rate, timeout value in failure recovery, and K. We found our protocols to be effective, efficient, and stable for an average node lifetime as low as 8.3 min. Experiment results also show that the average routing delay of our protocols increases only slightly even when the churn rate is greatly increased.

Failure recovery for structured P2P networks

Measurement studies indicate a high rate of node dynamics in p2p systems. In this paper, we address the question of how high a rate of node dynamics can be supported by structured p2p networks. We confine our study to the hypercube routing scheme used by several structured p2p systems. To improve system robustness and facilitate failure recovery, we introduce the property of K- consistency , K ≥ 1, which generalizes consistency defined previously. (Consistency guarantees connectivity from any node to any other node.) We design and evaluate a failure recovery protocol based upon local information for K-consistent networks. The failure recovery protocol is then integrated with a join protocol that has been proved to construct K-consistent neighbor tables for concurrent joins. The integrated protocols were evaluated by a set of simulation experiments in which nodes joined a 2000-node network and nodes (both old and new) were randomly selected to fail concurrently over 10,000 seconds of simulated time. In each such "churn" experiment, we took a "snapshot" of neighbor tables in the network once every 50 seconds and evaluated connectivity and consistency measures over time as a function of the churn rate, timeout value in failure recovery, and K . Storage and communication overheads were also evaluated. We found our protocols to be effective, efficient, and stable for an average node lifetime as low as 8.3 minutes (the median lifetime measured for Napster and Gnutella was 60 minutes [10]).

A mathematical model of deterministic wormhole routing in hypercube multicomputers using virtual channels

Although several analytical models have been proposed in the literature for binary n-cubes with deterministic routing, most of them have not included the effects of virtual channel multiplexing on network performance. The only mathematical model for deterministic wormhole routing in hypercubes with virtual channels was proposed in [Y. Boura, Design and Analysis of Routing Schemes and Routers for Wormhole-routed Mesh Architectures, Ph.D. Thesis, Department of Computer Science and Engineering, Pennsylvania State University, 1995] which uses complex combinatorial analysis with a computation time of O( N=2 n ) for an n-dimensional hypercube. This paper proposes a new and simple analytical model to compute message latency in binary n-cubes with virtual channels. It is very simple and easy to understand which uses an average case analysis and requires a computation time of O( n) for an n-dimensional hypercube. Results from simulation experiments confirm that the proposed model exhibits a good degree of accuracy for various network sizes and under different operating conditions.

Compressionless wormhole routing: an analysis for hypercube with virtual channels

Several recent studies have shown that adaptive routing algorithms based on deadlock recovery have superior performance characteristics than those based on deadlock avoidance. Most of these studies, however, have relied on software simulation due to the lack of analytical modelling tools. In an effort towards filling this gap, this paper presents a new analytical model of compressionless routing in wormhole-routed hypercubes. This routing algorithm exploits the tight coupling between wormhole routers for flow control to detect and recover from potential deadlock situations. The advantages of compressionless routing include deadlock-free adaptive routing with no extra virtual channels, simple router design, and order-preserving message transmission. The proposed analytical model computes message latency by determining the message transmission time, blocking delay at each router, multiplexing delay at each network channel, and waiting time in the source before entering the network. The validity of the model is demonstrated by comparing analytical results with those obtained through simulation experiments.

Performance analysis of adaptive wormhole routing in a two-dimensional torus

This paper presents an analytical evaluation of the performance of adaptive wormhole routing in a two-dimensional torus. Our analysis focuses on minimal and fully adaptive wormhole routing that allows a message to use any shortest path between source and destination. A validation of the analysis through simulation is presented to demonstrate the accuracy of the obtained results. Finally, we remark that no theoretical limitation prevents the extension of our analytical approach to the evaluation of the performance of adaptive wormhole routing in hypercubes or other symmetric topologies with wrap-around connections.

Communication delay in hypercubes in the presence of bit-reversal traffic

A number of studies have recently proposed analytical models of adaptive routing in wormhole-switched hypercubes under the uniform traffic pattern. However, there has been hardly any work reported yet that deals with other important non-uniform traffic patterns found in many parallel applications. This paper describes a new analytical model of adaptive routing in wormhole-switched hypercubes in the presence of highly non-uniform traffic generated by bit-reversal permutations, which is an important communication operation in parallel applications such as FFT computations. The analysis focuses Duato's fully adaptive routing algorithm [J. Duato, IEEE Trans. Parallel Distrib. Syst. 4(12) (1993) 320], which is widely accepted as the most general algorithm for achieving adaptivity in wormhole-routed networks while allowing for an efficient router implementation. Results obtained through simulation experiments confirm that the proposed model exhibits a good degree of accuracy in predicting message latency under different traffic conditions.

Fault-tolerant routing in hypercubes using probability vectors

In this paper, we propose a new fault-tolerant routing algorithm for the hypercube, which overcomes the performance limitations of the recently proposed safety vectors algorithm [IEEE Trans. Parallel Distrib. Syst. 9 (4) (1998) 321]. We present first a method for evaluating the k-level unsafety sets S k A for all 1⩽ k⩽ n in an n-dimensional hypercube. The k-level unsafety set at node A represents the set, S k A , of all nodes at Hamming distance k from A, which are faulty or unreachable from A due to faulty nodes or links. Equipped with these unsafety sets we show how each node calculates numeric probability vectors and uses them to achieve efficient fault-tolerant routing. A probability-based analysis is conducted to prove some properties of the proposed fault-tolerant algorithm. A performance comparison against the safety vectors algorithm, through extensive simulation experiments, reveals that the new algorithm exhibits superior performance in terms of routing distances and percentage of reachability.

Probability-based Fault-tolerant Routing in Hypercubes

This paper describes a new fault-tolerant routing algorithm for the hypercube, using the concept of probability vectors. To compute these vectors, a node first determines its faulty set, which represents the set of all its neighbouring nodes that are faulty or unreachable due to faulty nodes or links. Each node then calculates a probability vector, where the kth element represents the probability that a destination node at distance k cannot be reached through a minimal path due to a faulty node or link. The probability vectors are used by all the nodes to achieve an efficient fault-tolerant routing in the network. A performance comparison with the recently-proposed safety-vectors algorithm, through extensive simulation, shows that the new algorithm exhibits superior performance in terms of routing distances and percentage of reachability. Received ; revised

An analytical model of adaptive wormhole routing in hypercubes in the presence of hot spot traffic

Analytical models of fully adaptive routing for common wormhole-routed networks (e.g., hypercubes) under the uniform traffic pattern have recently been reported in the literature. However, many studies have revealed that the performance advantages of adaptive routing over deterministic routing is more noticeable when the traffic is nonuniform due to, for example, the existence of hot spots in the network. This paper proposes a new queueing model of fully adaptive routing in the hypercube in the presence of hot spot traffic. The analysis focuses on Duato's algorithm, but can easily be applied to other fully adaptive routing algorithms. Results from simulation experiments are presented to validate the model.

Modelling fully-adaptive routing in hypercubes

Analytical models for wormhole?routed hypercubes with deterministic routing have been widely reported in the literature. A model for the hypercube with fully?adaptive routing has recently been proposed in [1]. It uses M/M/1 queues, and computes a different probability of blocking at each intermediate router along the message path. As a result, the number of equations, and thus the computation steps, to evaluate latency increases with the network size. This paper proposes an alternative model that uses M/G/1 queues, and requires a constant number of computation steps irrespective of the network size. It achieves this by computing only once the mean probability of blocking across the entire path, and using it to determine the blocking time at a given router. Simulation experiments reveal that the model yields accurate latency results.

Analysis of fully adaptive wormhole routing in tori

A model of adaptive routing in the hypercube has recently been proposed (Y. Boura, C.R. Das, T.M. Jacob, A performance model for adaptive routing in hypercubes, in: Proceedings of the International Workshop on Parallel Processing, 1994, pp. 11–16). Modelling adaptive routing in a high-radix k-ary n-cube, e.g., the torus, is more complicated than in the hypercube since a message in the former may cross more than one channel along a given dimension. This paper proposes a queuing model of adaptive routing in the torus. The validity of the model is demonstrated by comparing analytical results with those obtained through simulations.

Message latency in the 2-dimensional mesh with wormhole routing

Analytical models of deterministic routing in hypercubes and tori with wormhole routing have been widely reported in the literature. Greenberg and Guan (in R. Greenberg, L. Guan, Modelling and comparison of wormhole routed mesh and torus networks, Proc. 9th IASTED Int. Conf. Parallel and Distributed Computing and Systems, October 1997) have recently described a model of deterministic routing in the 2-dimensional mesh, which has been one of the most popular multicomputer networks. Although there have been many adaptive routing algorithms proposed for this network, there has been hardly any study that proposes an analytical model for these algorithms. This paper proposes an approximate analytical model to compute message latency in the mesh with fully-adaptive routing. Results from simulation experiments are then presented to validate the model.

Node-to-set and set-to-set cluster fault tolerant routing in hypercubes

We study node-to-set and set-to-set fault tolerant routing problems in n-dimensional hypercubes H n . Node-to-set routing problem is that given a node s and a set of nodes T={ t 1,…, t k }, finds k node-disjoint paths s→ t i ,1⩽ i⩽ k. Set-to-set routing problem is that given two sets of nodes S={ s 1,…, s k } and T={ t 1,…, t k }, finds k node-disjoint paths, each path connects a node of S and a node of T. From Menger's theorem, it is known that these two problems in H n can tolerate at most n− k arbitrary faulty nodes. In this paper, we prove that both routing problems can tolerate n− k arbitrary faulty subgraphs (called cluster) of diameter 1. For 2⩽ k⩽ n, we show that, in the presence of at most n− k faulty clusters of diameter at most 1, the k paths of length at most n+2 for node-to-set routing in H n can be found in O( kn) optimal time and the k paths of length at most n+ k+2 for set-to-set routing in H n can be found in O (kn log k) time. The upper bound n+2 on the length of the paths for node-to-set routing in H n is optimal.

Comments on the Oblivious Routing Algorithm of Kaklamanis, Krizanc, and Tsantilas in the Hypercube

Kaklamanis, Krizanc, and Tsantilas (1991) gave an asymptotically optimal oblivious algorithm for many--one routing in hypercubes. It is shown that their argument needs to be modified in order for the algorithm to attain the asymptotic lower bound. They also applied the algorithm to permutation routing via the many--one and one--many routing phases. They claimed to save a factor of two by proposing to divide the packets into halves, routing the first half forward and the second half backward in the bit sequence. It is shown that this idea cannot reduce the total number of steps by a factor of two.

Optimally adaptive, minimum-distance, circuit-switched routing in hypercubes

In circuit-switched routing, the path between a source and its destination is established by incrementally reserving all required links before the data transmission can begin. If the routing algorithm is not carefully designed, deadlocks can occur in reserving these links. Deadlock-free algorithms based on dimension-ordered routing, such as the E-cube , exist. However, E-cube does not provide any flexibility in choosing a path from a source to its destination and can thus result in long latencies under heavy or uneven traffic. Adaptive, minimum-distance routing algorithms, such as the Turn Model and the UP Preference algorithms, have previously been reported. In this article, we present a new class of adaptive, provably deadlock-free, minimum-distance routing algorithms. We prove that the algorithms developed here are optimally adaptive in the sense that any further flexibility in communication will result in deadlock. We show that the Turn Model is actually a member of our new class of algorithms that does not perform as well as other algorithms within the new class. It creates artificial hotspots in routing the traffic and allows fewer total paths. We present an analytical comparison of the flexibility and balance in routing provided by various algorithms and a comparison based on uniform and nonuniform traffic simulations. The Extended UP Preference algorithm developed in this article is shown to have improved performance with respect to existing algorithms. The methodology and the algorithms developed here can be used to develop routing for other schemes such as wormhole routing, and for other recursively defined networks such as k -ary n -cubes.

Use of routing capability for fault-tolerant routing in hypercube multicomputers

The concept of routing capability is proposed to assist fault-tolerant routing in hypercubes. Routing capability is defined with respect to the entire spectrum of distance. As a result, the amount of information that is useful for message routing is increased. An algorithm is presented to facilitate efficient fault-tolerant routing of messages. The algorithm routes a message in an attempt to minimize derouting. Furthermore, the concept of directed routing capability which contains more useful information for fault-tolerant routing is introduced. Simulation results demonstrate the usefulness of our approach.