Modeling hexagonal constellations with Eisenstein-Jacobi graphs

Martínez, Carmen; Stafford, Esteban; Beivide, Ramón; Gabidulin, E.M.

doi:10.1134/s0032946008010018

Cited by 29 publications

(28 citation statements)

References 8 publications

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“…An EJ number α = a + bω is used to generate an EJ interconnection network [6], [7] where the nodes of the network are the congruence classes modulo α, and two nodes x = x 1 + x 2 ω and y = y 1 + y 2 ω are adjacent if and only if y − x equals ±1, ±ω, or ±ω 2 modulo α. Therefore, α generates a homogeneous, degree six EJ network which has |α| 2 = a 2 + ab + b 2 nodes.…”

Section: The Hexagonal Torus Topologymentioning

confidence: 99%

“…The EJ network was originally developed over two decades ago [6]. It is a 2D wraparound network of degree six and its topological properties were explored in [7] and [8]. Recently [9] shows that the EJ networks are a generalization of the hexagonal wraparound mesh (hexagonal torus) networks developed earlier in [10], [11], and [12].…”

Section: Introductionmentioning

confidence: 97%

“…Recently, a two component addressing scheme for hexagonal networks has been proposed in [7] and [8]. This addressing scheme requires the minimal number of addressing bits and addresses every node utilizing its position at the angles of 0 and 60 degrees from the horizontal axis.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive routing in hexagonal torus interconnection networks

Shamaei

Bose

Flahive

2013

2013 IEEE High Performance Extreme Computing Conference (HPEC)

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The hexagonal torus network is a degree six toroidal network with rich topological properties. It was used in the design of HARTS machine at the University of Michigan, and more recently it has been proposed for cellular networks. The low diameter and less average hop distance of this network make it advantageous over other 2D toroidal network such as meshes and tori. This paper proposes a fully adaptive and deadlockfree routing algorithm for hexagonal torus networks based on virtual channel partitioning and the algorithm requires three virtual channels per physical channel to remain deadlock-free. Simulation results show that this algorithm is superior to the fully adaptive routing algorithm for 2D meshes and 2D tori of the same size.

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Section: The Hexagonal Torus Topologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Adaptive routing in hexagonal torus interconnection networks

Shamaei

Bose

Flahive

2013

2013 IEEE High Performance Extreme Computing Conference (HPEC)

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“…Among the most popular NoC topologies developed for CMPs such as n-dimensional mesh, torus, folded torus, hypercube, and octagon, low-dimensional networks, like 2D-mesh and 2D-torus, offer better performance in terms of higher throughput and lower latency than highdimensional networks (high dimensional k-aryn-cubes and meshes) [13], [14]. Recently, two other wrap-around networks have been proposed as suitable alternatives to the 2D torus network: the degree 4 Gaussian networks [15], [16], [17] and the degree 6 Eisenstein-Jacobi (EJ) networks [17], [18], [19], [20]. In [1], the authors showed that EJ networks are generalizations of the hexagonal mesh topology developed earlier in [20], [21], and [22].…”

Section: Introductionmentioning

confidence: 97%

All-To-All Broadcast in Hexagonal Torus Networks On-Chip

Touzene

2015

IEEE Trans. Parallel Distrib. Syst.

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Hexagonal torus networks are special family of Eisenstein-Jacobi (EJ) Networks which have gained popularity as good candidates network On-Chip (NoC) for interconnecting Multiprocessor System-on-Chips (MPSoCs). They showed better topological properties compared to the 2D torus networks with the same number of nodes. All-to-all broadcast is a collective communication algorithm used frequently in some parallel applications. Recently, an off-chip all-to-all broadcast algorithm has been proposed for hexagonal torus networks assuming half-duplex links and all-ports communication. The proposed all-to-all broadcast algorithm does not achieve the minimum transmission time and requires 24k extra buffers, where k is the network diameter. We first extend this work by proposing an efficient all-to-all broadcast on hexagonal torus networks under full-duplex links and all-ports communications assumptions which achieves the minimum transmission delay but requires 36k extra buffers per router. In a second stage, we develop a new all-to-all broadcast more suitable for hexagonal torus network on-chip that achieves optimal transmission delay time without requiring any extra buffers per router. By reducing the amount of buffer space, the new all-to-all broadcast reduces the routers cost which is an important issue in NoCs architectures.

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“…[10,8]. Recently, an efficient interconnection topology called EJ network has been proposed in [15] and some further properties of this network have been studied in [2,9]. Furthermore, the hexagonal network is reviewed in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Edge disjoint Hamiltonian cycles in Eisenstein–Jacobi networks

Hussain

Bose

Al-Dhelaan

2015

Journal of Parallel and Distributed Computing

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h i g h l i g h t s• Applications of Hamiltonian cycles are given in the introduction. • Rectangular representation is constructed to help finding the solution since it gives a clear visualization of the network. • The first 2 edge disjoint Hamiltonian cycles are constructed based on the rectangular representation.• The third Hamiltonian cycle is first divided into two cases, when norm is odd or even, and then it is constructed. a b s t r a c tMany communication algorithms in parallel systems can be efficiently solved by obtaining edge disjoint Hamiltonian cycles in the interconnection topology of the network. The Eisenstein-Jacobi (EJ) network generated by α = a+bρ, where ρ = (1+i √ 3)/2, is a degree six symmetric interconnection network. The hexagonal network is a special case of the EJ network that can be obtained by α = a+(a+1)ρ. Generating three edge disjoint Hamiltonian cycles in the EJ network with generator α = a + bρ for gcd(a, b) = 1 has been shown before. However, this problem has not been solved when gcd(a, b) = d > 1. In this paper, some results to this problem are given.

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Modeling hexagonal constellations with Eisenstein-Jacobi graphs

Cited by 29 publications

References 8 publications

Adaptive routing in hexagonal torus interconnection networks

Adaptive routing in hexagonal torus interconnection networks

All-To-All Broadcast in Hexagonal Torus Networks On-Chip

Edge disjoint Hamiltonian cycles in Eisenstein–Jacobi networks

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