Dragonfly networks have been recently proposed for the interconnection network of forthcoming exascale supercomputers. Relying on large-radix routers, they build a topology with low diameter and high throughput, divided into multiple groups of routers. While minimal routing is appropriate for uniform traffic patterns, adversarial traffic patterns can saturate intergroup links and degrade the obtained performance. Such traffic patterns occur in typical communication patterns used by many HPC applications, such as neighbor data exchanges in multidimensional space decompositions. Non-minimal traffic routing is employed to handle such cases. Adaptive policies have been designed to select between minimal and nonminimal routing to handle variable traffic patterns.However, previous papers have not taken into account the effect of saturation of intra-group (local) links. This paper studies how local link saturation can be common in these networks, and shows that it can largely reduce the performance. The solution to this problem is to use nonminimal paths that avoid those saturated local links. However, this extends the maximum path length, and since all previous routing proposals prevent deadlock by relying on an ascending order of virtual channels, it would imply unaffordable cost and complexity in the network routers.In this paper we introduce a novel routing/flow-control scheme that decouples the routing and the deadlock avoidance mechanisms. Our model does not impose any dependencies between virtual channels, allowing for on-the-fly (in-transit) adaptive routing of packets. To prevent deadlock we employ a deadlock-free escape subnetwork based on injection restriction. Simulations show that our model obtains lower latency, higher throughput, and faster adaptation to transient traffic, because it dynamically exploits a higher path diversity to avoid saturated links. Notably, our proposal consumes traffic bursts 43% faster than previous ones.

A construction of 2-quasi-perfect Lee codes is given over the space Z n p for p prime, p ≡ ±5 (mod 12) and n = 2[ p 4 ]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee-metric do not exist for dimension n ≥ 3 and radius r ≥ 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between Coding and Graph Theories.

Current High-Performance Computing (HPC) and data center networks rely on large-radix routers. Hamming graphs (Cartesian products of complete graphs) and dragonflies (two-level direct networks with nodes organized in groups) are some direct topologies proposed for such networks. The original definition of the dragonfly topology is very loose, with several degrees of freedom, such as the inter-and intragroup topology, the specific global connectivity, and the number of parallel links between groups (or trunking level). This work provides a comprehensive analysis of the topological properties of the dragonfly network, providing balancing conditions for network dimensioning, as well as introducing and classifying several alternatives for the global connectivity and trunking level. From a topological study of the network, it is noted that a Hamming graph can be seen as a canonical dragonfly topology with a high level of trunking. Based on this observation and by carefully selecting the global connectivity, the Dimension Order Routing (DOR) mechanism safely used in Hamming graphs is adapted to dragonfly networks with trunking. The resulting routing algorithms approximate the performance of minimal, nonminimal, and adaptive routings typically used in dragonflies but without requiring virtual channels to avoid packet deadlock, thus allowing for lower cost router implementations. This is obtained by properly selecting the link to route between groups based on a graph coloring of network routers. Evaluations show that the proposed mechanisms are competitive with traditional solutions when using the same number of virtual channels and enable for simpler implementations with lower cost. Finally, multilevel dragonflies are discussed, considering how the proposed mechanisms could be adapted to them. ACM Reference Format:Cristóbal Camarero, Enrique Vallejo, and Ramón Beivide. 2014. Topological characterization of Hamming and dragonfly networks and its implications on routing. ACM Trans.

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