Computer Interconnection Networks Research Articles

Fault-Tolerant Routing With Load Balancing in LeTQ Networks

With the increasing scale of parallel computer interconnection network, the possibility of processor failure or link failure between processors in the network is also increasing. In the design of supercomputers, not only link overhead and communication delay should be taken into account, but also fault-tolerant performance of networks should be emphasized. Locally exchanged twisted cube ( <inline-formula><tex-math notation="LaTeX">$LeTQ$</tex-math></inline-formula> ) is a newly proposed interconnection network with lower link overhead and shorter diameter. With the increasing scale of supercomputers, fault-tolerant routing is indispensable. In this article, we propose a new load balancing fault-tolerant routing algorithm based on node contraction for <inline-formula><tex-math notation="LaTeX">$LeTQ$</tex-math></inline-formula> networks. The proposed algorithm uses the node shrinkage method to evaluate the priority of nodes. The sending node adaptively adjusts the probability of forwarding packets to the neighbor node according to the priority of the neighbor node and the state of the network. The path can be adapted to the load state of the network. The simulation results show that the fault-tolerant routing algorithm has good performance in throughput and delay.

Optimal Algorithms for Sorting Permutations with Brooms

Sorting permutations with various operations has applications in genetics and computer interconnection networks where an operation is specified by its generator set. A transposition tree T=(V,E) is a spanning tree over n vertices v1,v2,…vn. T denotes an operation in which each edge is a generator. A value assigned to a vertex is called a token or a marker. The markers on vertices u and v can be swapped only if the pair (u,v)∈E. The initial configuration consists of a bijection from the set of vertices v1,v2,…,vn to the set of markers (1,2,⋯,n−1,n). The goal is to sort the initial configuration of T, i.e., an input permutation, by applying the minimum number of swaps or moves in T. Computationally tractable optimal algorithms to sort permutations are known only for a few classes of transposition trees. We study a class of transposition trees called a broom and its variation a double broom. A single broom is a tree obtained by joining the centre vertex of a star with one of the two leaf vertices of a path graph. A double broom is an extension of a single broom where the centre vertex of a second star is connected to the terminal vertex of the path in a single broom. We propose a simple and efficient algorithm to obtain an optimal swap sequence to sort permutations with the transposition tree broom and a novel optimal algorithm to sort permutations with a double broom. We also introduce a new class of trees named millipede tree and prove that D* yields a tighter upper bound for sorting permutations with a balanced millipede tree compared to D′. Algorithms D* and D′ are designed previously.

Multidimensional Computer Interconnection Networks with the Virtual Cut–Through Routing

A theoretical model of an n-dimensional toroidal interconnection network with the virtual cut-through routing is developed. The network performance is characterized as a function of the message load for various values of the network and communication parameters. An exact analytical expression for the saturation point has been obtained. The method of statistical physics known as the mean field theory approximation has been used in order to derive analytical expressions of latency as a function of message generation rate for the full range of the network load. The saturation point has been found proportional to the number of the torus dimensions and inversely proportional to the message length and to the length of the path from the source to the destination. It was found that the transition to the saturation state is a second-order (continuous) phase transition with a critical exponent equal to 1.

Approximation algorithms for sorting permutations by extreme block-interchanges

Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two substrings that are called as blocks in literature, in a given permutation. When the blocks are restricted to be adjacent then one obtains a well studied operation: transposition. We call either a prefix or a suffix as an extreme. Restricting one of the swapped blocks to be an extreme in block-interchange operation yields a prefix or a suffix block-interchange respectively, the two types of extreme block-interchanges. For prefix block-interchange operation over permutations we design: (i) an optimum algorithm to sort reverse permutation, Rn, in n/2 moves, (ii) a simple 2-approximation algorithm, and (iii) for permutations with O(1) cycles, a 4/3 approximation algorithm. Due to symmetry, these results apply to suffix block-interchange operation also.

The Graph of Our Mind.

Graph theory in the last two decades penetrated sociology, molecular biology, genetics, chemistry, computer engineering, and numerous other fields of science. One of the more recent areas of its applications is the study of the connections of the human brain. By the development of diffusion magnetic resonance imaging (diffusion MRI), it is possible today to map the connections between the 1–1.5 cm regions of the gray matter of the human brain. These connections can be viewed as a graph. We have computed 1015-vertex graphs with thousands of edges for hundreds of human brains from one of the highest quality data sources: the Human Connectome Project. Here we analyze the male and female braingraphs graph-theoretically and show statistically significant differences in numerous parameters between the sexes: the female braingraphs are better expanders, have more edges, larger bipartition widths, and larger vertex cover than the braingraphs of the male subjects. These parameters are closely related to the quality measures of highly parallel computer interconnection networks: the better expanding property, the large bipartition width, and the large vertex cover characterize high-quality interconnection networks. We apply the data of 426 subjects and demonstrate the statistically significant (corrected) differences in 116 graph parameters between the sexes.

Fault-tolerant hamiltonian cycles and paths embedding into locally exchanged twisted cubes

The foundation of information society is computer interconnection network, and the key of information exchange is communication algorithm. Finding interconnection networks with simple routing algorithm and high fault-tolerant performance is the premise of realizing various communication algorithms and protocols. Nowadays, people can build complex interconnection networks by using very large scale integration (VLSI) technology. Locally exchanged twisted cubes, denoted by (s + t + 1)-dimensional LeTQs,t, which combines the merits of the exchanged hypercube and the locally twisted cube. It has been proved that the LeTQs,t has many excellent properties for interconnection networks, such as fewer edges, lower overhead and smaller diameter. Embeddability is an important indicator to measure the performance of interconnection networks. We mainly study the fault tolerant Hamiltonian properties of a faulty locally exchanged twisted cube, LeTQs,t − (fv + fe), with faulty vertices fv and faulty edges fe. Firstly, we prove that an LeTQs,t can tolerate up to s − 1 faulty vertices and edges when embedding a Hamiltonian cycle, for s ⩾ 2, t ⩾ 3, and s ⩽ t. Furthermore, we also prove another result that there is a Hamiltonian path between any two distinct fault-free vertices in a faulty LeTQs,t with up to (s − 2) faulty vertices and edges. That is, we show that LeTQs,t is (s − 1)-Hamiltonian and (s − 2)-Hamiltonian-connected. The results are proved to be optimal in this paper with at most (s − 1)-fault-tolerant Hamiltonicity and (s − 2) fault-tolerant Hamiltonian connectivity of LeTQs,t.

An Upper Bound For Sorting Permutations With A Transposition Tree

Abstract Cayley graphs are constructed over n! vertices that denote n! permutations of the symmetric group Sn. The edge set of a particular Cayley graph Γ is determined by the corresponding operation that consists of a well defined generator set. One such operation is a transposition tree which is a spanning tree over n vertices. Cayley graphs have been extensively studied and they have applications in genomic studies and computer interconnection networks. In computer interconnection networks each vertex of the corresponding Γ denotes a computer and an existence of an edge between two vertices indicates that a direct two way communication exists between the corresponding computers. In 2013, upper bounds for Cayley graphs generated by transposition trees were computed in polynomial time. One such upper bound δ´ was known to be the tightest until recently. In 2019, δ* was shown to be better than δ’ in a cumulative sense. We improve upon δ’* to obtain δ’* i is better than δ’ in a cumulative sense, as indicated by the experimental results. Likewise, δ’* is comparable to δ*. For corresponding classes of trees, δ’* is shown to be tighter than δ’ and δ* respectively.

Computer interconnection networks with virtual cut-through routing

This paper considers a model of a toroidal computer interconnection network with the virtual cut-through routing. The interrelationships between network parameters, load and performance are analyzed. An exact analytical expression for the saturation point and expressions for the latency as a function of the message generation rate under the mean field theory approximation have been obtained. The theoretical results have been corroborated with the results of simulation experiments for various values of network parameters. The network behavior has been found not depending on the torus linear dimensions provided that they are at least twice as large as the message path length. The saturation point has been found to be inversely proportional to the message length in good agreement with the analytical results. A good agreement with Little’s theorem has been found if the network remains in the steady state during the experiment.

Sorting permutations with transpositions in O(n3) amortized time

The problem of sorting a permutation π over alphabet ∑={1,2,3…n} with the minimum number of transpositions is known to be intractable. That is, the computation of transposition distance i.e. dt(π), is hard. Such sorting is equivalent to determining the distance between any α, β of the symmetric group Sn over ∑ under the transposition operation. It has applications in computer interconnection networks where a node is denoted by a permutation, and genetics where a genome is modeled by a permutation. The maximum distance between any pair of permutations in Sn is the diameter i.e. diam(Γ(Sn)), of the corresponding Cayley graph Γ(Sn). It corresponds to the maximum latency in the computer interconnection network modeled by Γ(Sn) and the maximum genetic dissimilarity between any pair α, β in Sn. We call transpositions, prefix transpositions, suffix transpositions and prefix/suffix transpositions as block-moves. We design an algorithm to compute dt(π) ∀π in Sn and compute diam(Γ(Sn)) with transpositions in O(n!n3) time and O(n!n2) space; at an amortized time of O(n3). The same can be computed for the remaining block moves in O(n2log⁡n) amortized time and O(n!n) space.

XOR-based HoL-blocking reduction routing mechanisms for direct networks

Abstract Routing is a key design parameter in the interconnection network of large parallel computers . Routing algorithms are classified into two different categories depending on the number of routing options available for each source–destination pair: deterministic (there is one path available) and adaptive (there are several ones). Adaptive routing has two opposed effects on network performance. On one hand, it provides routing flexibility that may help on avoiding a congested network area, thus improving network performance. On the other hand, it also may increase the Head-of-Line blocking effect due to more destination nodes sharing the port queues. Usually, adaptive routing uses virtual channels to provide routing flexibility and to guarantee deadlock freedom. Deterministic routing is simpler, which implies lower routing delay and it introduces less Head-of-Line blocking effect. In this paper, we propose an adaptive and HoL-blocking reduction routing algorithm for direct topologies that tries to combine the good properties of both worlds: It provides routing flexibility but also reduces the Head-of-Line blocking effect. To do that, this paper proposes several functions which use the XOR operation to efficiently distribute the packets among virtual channels based on their destination node. The resulting routing mechanisms have different properties depending on whether they enforce routing flexibility or Head-of-Line blocking reduction.

Tighter upper bound for sorting permutations with prefix transpositions

Permutations are sequences where each symbol in the given alphabet Σ appears exactly once. A transposition is an operation that exchanges two adjacent sublists in a permutation; if one of these sublists is restricted to be a prefix then one obtains a prefix transposition. Transpositions over permutations have applications in genome rearrangements and computer interconnection networks. The set of permutations with n symbols derived from the alphabet Σ={0,1,…,n−1} form a symmetric group, that we denote by Pn. The prefix transposition distance between two permutations π⁎∈Pn and π∼∈Pn is the minimum number of prefix transpositions, also called moves, needed to transform π⁎ into π∼. A prefix transposition can be modeled by a permutation operation. A permutation is type of sequence and it is also an operation. The generators for prefix transposition operation are a subset of permutation operation. As a result, transforming an arbitrary π⁎∈Pn to an arbitrary π∼∈Pn is equivalent to sorting some πˆ∈Pn. Thus, upper bound for transforming any π⁎∈Pn into any π∼∈Pn with prefix transpositions is simply the upper bound to sort any permutation π∈Pn with moves. In the article that establishes the best known upper bound for prefix transposition distance over Pn as n−log9/2⁡n, explicit proofs were not given for some cases. In this article, we provide the missing proofs, validating the stated upper bound. Furthermore, we establish an upper bound of n−log7/2⁡n for prefix transposition distance over Pn.

The Folded Crossed Cube : A New Interconnection Network For Parallel Systems

of parallel computer interconnection network topology has been made along with the various interconnection networks emphasizing the cube based topologies in particular. This paper proposes a new cube based topology called the Folded crossed cube with better features such as reduced diameter, cost and improved broadcast time, better fault tolerance and better message traffic density in comparison to its parent topologies: viz: hypercube and crossed cube. The one-to-one routing algorithm is also designed for the proposed network. The topological properties along with routing time are compared with the parent topologies and found to be better. Performance analysis in terms of cost, reliability is also done for the current network.

Reducing complexity in tree-like computer interconnection networks

Interconnection networks based on the k-ary n-tree topology are widely used in high-performance parallel computers. However, this topology is expensive and complex to build. In this paper we evaluate an alternative tree-like topology that is cheaper in terms of cost and complexity because it uses fewer switches and links. This alternative topology leaves unused upward ports on switches, which can be rearranged to be used as downward ports. The increase of locality might be efficiently exploited by applications. We test the performance of these thin-trees, and compare it with that of regular trees. Evaluation is carried out using a collection of synthetic traffic patterns that emulate the behavior of scientific applications and functions within message passing libraries, not only in terms of sources and destinations of messages, but also considering the causal relationships among them. We also propose a methodology to perform cost and performance analysis of different networks. Our main conclusion is that, for the set of studied workloads, the performance drop in thin-trees is less noticeable than the cost savings.

Design and performance of speculative flow control for high-radix datacenter interconnect switches

High-radix switches are desirable building blocks for large computer interconnection networks, because they are more suitable to convert chip I/O bandwidth into low latency and low cost than low-radix switches [J. Kim, W.J. Dally, B. Towles, A.K. Gupta, Microarchitecture of a high-radix router, in: Proc. ISCA 2005, Madison, WI, 2005]. Unfortunately, most existing switch architectures do not scale well to a large number of ports, for example, the complexity of the buffered crossbar architecture scales quadratically with the number of ports. Compounded with support for long round-trip times and many virtual channels, the overall buffer requirements limit the feasibility of such switches to modest port counts. Compromising on the buffer sizing leads to a drastic increase in latency and reduction in throughput, as long as traditional credit flow control is employed at the link level. We propose a novel link-level flow control protocol that enables high-performance scalable switches that are based on the increasingly popular buffered crossbar architecture, to scale to higher port counts without sacrificing performance. By combining credited and speculative transmission, this scheme achieves reliable delivery, low latency, and high throughput, even with crosspoint buffers that are significantly smaller than the round-trip time. The proposed scheme substantially reduces message latency and improves throughput of partially buffered crossbar switches loaded with synthetic uniform and non-uniform bursty traffic. Moreover, simulations replaying traces of several typical MPI applications demonstrate communication speedup factors of 2 to 10 times.

Performance of a Speculative Transmission Scheme for Scheduling-Latency Reduction

Low latency is a critical requirement in some switching applications, specifically in parallel computer interconnection networks. The minimum latency in switches with centralized scheduling comprises two components, namely, the control-path latency and the data-path latency, which in a practical high-capacity, distributed switch implementation can be far greater than the cell duration. We introduce a speculative transmission scheme to significantly reduce the average control-path latency by allowing cells to proceed without waiting for a grant, under certain conditions. It operates in conjunction with any centralized matching algorithm to achieve a high maximum utilization and incorporates a reliable delivery mechanism to deal with failed speculations. An analytical model is presented to investigate the efficiency of the speculative transmission scheme employed in a non-blocking N times NR input-queued crossbar switch with R receivers per output. Using this model, performance measures such as the mean delay and the rate of successful speculative transmissions are derived. The results demonstrate that the control-path latency can be almost entirely eliminated for loads up to 50%. Our simulations confirm the analytical results.

Design Issues in Next-Generation Merchant Switch Fabrics

Packet-switch fabrics with widely varying characteristics are currently deployed in the domains of both communications and computer interconnection networks. For economical reasons, it would be highly desirable that a single switch fabric could accommodate the needs of a variety of heterogeneous services and applications from both domains. In this paper, we consider the current requirements, technological trends, and their implications on the design of an ASIC chipset for a merchant switch fabric. We then identify the architecture upon which such a suitable and generic switch fabric could be based, and we present the general characteristics of an implementation of this switching fabric within the bounds of current state-of-the-art technology. To our knowledge, this is the first attempt to design a chipset that can be used for both communications and computer interconnection networks.

A speculative transmission scheme for scheduling‐latency reduction

AbstractLow latency is a critical requirement in some switching applications, specifically in parallel computer interconnection networks. The minimum latency in switches with centralized scheduling comprises two components, namely, the control‐path latency and the data‐path latency, which in a practical high‐capacity, distributed switch implementation can be far greater than the cell duration. We introduce a speculative transmission scheme to significantly reduce the average control‐path latency by allowing cells to proceed without waiting for a grant, under certain conditions. It operates in conjunction with any centralized matching algorithm to achieve a high maximum utilization and incorporates a reliable delivery mechanism to deal with failed speculations. The speculative transmission scheme is employed in a non‐blocking N ×N R input‐queued crossbar switch with R receivers per output. The control‐path latency can be almost entirely eliminated for loads up to 50%. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The construction of infinite families of any k-tight optimal and singular k-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is: for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained: (1) for any k ⩾ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k, e, c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ⩾ 0, an infinite family of singular k-tight optimal DLN can be constructed.

Efficient domination in circulant graphs with two chord lengths

Domination is an important property in the design of efficient computer interconnection networks. We provide a complete characterization of circulant graphs with two chord lengths that admit an efficient dominating set. In particular, for 3-regular and 4-regular circulant graphs, we give necessary and sufficient conditions for the existence of efficient dominating sets and we describe their exact structure according to the relationship between chords.

Some results on R 2-edge-connectivity of even regular graphs

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an R2-edge-cut if G-S is disconnected and contains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λn(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λn(G)=g(2k−2) for any 2k-regular connected graph G(≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.