The order/degree problem consists of finding the smallest diameter graph for a given order and degree. Such a graph is beneficial for designing low-latency networks with high performance for massively parallel computers. The average shortest path length (ASPL) of a graph has an influence on latency. In this paper, we propose a novel order adjustment approach. In the proposed approach, we search for Cayley graphs of the given degree that are close to the given order. We then adjust the order of the best Cayley graph to meet the given order. For some order and degree pairs, we explain how to derive the smallest known graphs from the Graph Golf 2016 and 2017 competitions. key words: order/degree problem, Cayley graph, diameter, average shortest path length (ASPL), vertex bisection, vertex injection, vertex removal * * In this paper, we do not discuss grid graphs.* * * The graph of order 1344 and degree 30 in Table 1 was created using another method. To create this graph, we concatenated two Brown graphs [11] of q = 5 2 , which have order q 2 + q + 1 = 651 and degree q + 1 = 26. We combined them to get a graph of order 1302 and then add 42 vertices. We then added edges randomly. he held an additional position as a researcher at PRESTO, Japan Science and Technology Corporation (JST). He has been a professor with the Faculty of Advanced Science and Technology at Kumamoto University since January 2016. His current research interests include high-performance, low-power computer architectures, reconfigurable computing systems, and VLSI devices and design methodologies. He is a senior member of IPSJ and IEICE, and a member of IEEE.
The order/degree problem consists of finding the smallest diameter graph for a given order and degree. Such a graph is beneficial for designing low-latency networks with high performance for massively parallel computers. The average shortest path length (ASPL) of a graph has an influence on latency. In this paper, we propose a novel order adjustment approach. In the proposed approach, we search for Cayley graphs of the given degree that are close to the given order. We then adjust the order of the best Cayley graph to meet the given order. For some order and degree pairs, we explain how to derive the smallest known graphs from the Graph Golf 2016 and 2017 competitions. key words: order/degree problem, Cayley graph, diameter, average shortest path length (ASPL), vertex bisection, vertex injection, vertex removal * * In this paper, we do not discuss grid graphs.* * * The graph of order 1344 and degree 30 in Table 1 was created using another method. To create this graph, we concatenated two Brown graphs [11] of q = 5 2 , which have order q 2 + q + 1 = 651 and degree q + 1 = 26. We combined them to get a graph of order 1302 and then add 42 vertices. We then added edges randomly. he held an additional position as a researcher at PRESTO, Japan Science and Technology Corporation (JST). He has been a professor with the Faculty of Advanced Science and Technology at Kumamoto University since January 2016. His current research interests include high-performance, low-power computer architectures, reconfigurable computing systems, and VLSI devices and design methodologies. He is a senior member of IPSJ and IEICE, and a member of IEEE.
“…More small optimal topologies with symmetries and other special properties were produced and presented in a structured table [53]. Distributed shortcut networks targeting the diameter and cable length trade-off [46] and host-switch graphs designed by minimizing diameter and/or MPL using randomized heuristics [48] were also introduced and benchmarked by simulation.…”
Communication latency has become one of the determining factors for the performance of parallel clusters. To design low-latency network topologies for high-performance computing clusters, we optimize the diameters, mean path lengths, and bisection widths of circulant topologies. We obtain a series of optimal circulant topologies of size 2 5 through 2 10 and compare them with torus and hypercube of the same sizes and degrees. We further benchmark on a broad variety of applications including effective bandwidth, FFTE, Graph 500 and NAS parallel benchmarks to compare the optimal circulant topologies and Cartesian products of optimal circulant topologies and fully connected topologies with corresponding torus and hypercube. Simulation results demonstrate superior potentials of the optimal circulant topologies for communication-intensive applications. We also find the
“…Some works analyze various properties of low-diameter topologies, for example path length, throughput, and bandwidth [212], [124], [118], [196], [123], [32], [139], [130], [97], [128], [209], [73], [129], [22], [208], [6]. Such works could be used with our multipathing analysis when developing routing protocols and architectures that take advantage of different properties of a given topology.…”
The recent line of research into topology design focuses on lowering network diameter. Many low-diameter topologies such as Slim Fly or Jellyfish that substantially reduce cost, power consumption, and latency have been proposed. A key challenge in realizing the benefits of these topologies is routing. On one hand, these networks provide shorter path lengths than established topologies such as Clos or torus, leading to performance improvements. On the other hand, the number of shortest paths between each pair of endpoints is much smaller than in Clos, but there is a large number of non-minimal paths between router pairs. This hampers or even makes it impossible to use established multipath routing schemes such as ECMP. In this work, to facilitate high-performance routing in modern networks, we analyze existing routing protocols and architectures, focusing on how well they exploit the diversity of minimal and non-minimal paths. We first develop a taxonomy of different forms of support for multipathing and overall path diversity. Then, we analyze how existing routing schemes support this diversity. Among others, we consider multipathing with both shortest and non-shortest paths, support for disjoint paths, or enabling adaptivity. To address the ongoing convergence of HPC and "Big Data" domains, we consider routing protocols developed for both traditional HPC systems and supercomputers, and for data centers and general clusters. Thus, we cover architectures and protocols based on Ethernet, InfiniBand, and other HPC networks such as Myrinet. Our review will foster developing future high-performance multipathing routing protocols in supercomputers and data centers.
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