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d4 = 3<latexit sha1_base64="Vd3A3F8izvYez8zVueIwEpdh+L4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0nagl6EohePFUxbaEPZbDbt0s0m7G6EEvobvHhQxKs/yJv/xm2ag7Y+GHi8N8PMPD/hTGnb/rZKG5tb2zvl3cre/sHhUfX4pKviVBLqkpjHsu9jRTkT1NVMc9pPJMWRz2nPn94t/N4TlYrF4lHPEupFeCxYyAjWRnKDUeumOarW7LqdA60TpyA1KNAZVb+GQUzSiApNOFZq4NiJ9jIsNSOczivDVNEEkyke04GhAkdUeVl+7BxdGCVAYSxNCY1y9fdEhiOlZpFvOiOsJ2rVW4j/eYNUh9dexkSSairIclGYcqRjtPgcBUxSovnMEEwkM7ciMsESE23yqZgQnNWX10m3UXea9cZDq9a+LeIowxmcwyU4cAVtuIcOuECAwTO8wpslrBfr3fpYtpasYuYU/sD6/AHoz44X</latexit>
d4 = 2<latexit sha1_base64="iQhBlnIrfxIIvARS0pY64nfSrJw=">AAAB7HicbVBNS8NAEJ31s9avqkcvi0XwVJJa0ItQ9OKxgmkLbSibzaZdutmE3Y1QQn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF6SCa+M432htfWNza7u0U97d2z84rBwdt3WSKco8mohEdQOimeCSeYYbwbqpYiQOBOsE47uZ33liSvNEPppJyvyYDCWPOCXGSl44aNzUB5WqU3PmwKvELUgVCrQGla9+mNAsZtJQQbTuuU5q/Jwow6lg03I/0ywldEyGrGepJDHTfj4/dorPrRLiKFG2pMFz9fdETmKtJ3FgO2NiRnrZm4n/eb3MRNd+zmWaGSbpYlGUCWwSPPsch1wxasTEEkIVt7diOiKKUGPzKdsQ3OWXV0m7XnMva/WHRrV5W8RRglM4gwtw4QqacA8t8IACh2d4hTck0Qt6Rx+L1jVUzJzAH6DPH+dLjhY=</latexit>
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Path 1 Path 2 Path 3 Path 4 Path n
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…Group 1 Group 2 Group m
sj<latexit sha1_base64="VvH6gJKEE+CuhPPPbhUUbyboi4Y=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3btZhN2N0IJ/Q1ePCji1R/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHssHM0nQj+hQ8pAzaqzU1P3scdovV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzslZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TgZcITNiYgllittbCRtRRZmx+ZRsCN7yy6ukVat6F9Xa/WWlfpPHUYQTOIVz8OAK6nAHDWgCAw7P8ApvjnRenHfnY9FacPKZY/gD5/MHIJiO5A==</latexit>
S1 + S2 = 1S3 + S4 = 1
gij<latexit sha1_base64="7XzuU9o8AwiL58zrvpeL6gEx5PE=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHoxWMF+wHtUrJptk2bTZYkK5Sl/8GLB0W8+n+8+W9M2z1o64OBx3szzMwLE8GN9bxvtLa+sbm1Xdgp7u7tHxyWjo6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuH4bua3npg2XMlHO0lYEJOB5BGnxDqpOehlfDTtlcpexZsDrxI/J2XIUe+Vvrp9RdOYSUsFMabje4kNMqItp4JNi93UsITQMRmwjqOSxMwE2fzaKT53Sh9HSruSFs/V3xMZiY2ZxKHrjIkdmmVvJv7ndVIb3QQZl0lqmaSLRVEqsFV49jruc82oFRNHCNXc3YrpkGhCrQuo6ELwl19eJc1qxb+sVB+uyrXbPI4CnMIZXIAP11CDe6hDAyiM4Ble4Q0p9ILe0ceidQ3lMyfwB+jzB9Rbj0s=</latexit>
Product Networks
1. Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA 2. School of Data and Computer Science, Sun Yat-sen University, Guangzhou, Guangdong 510275, P.R. China
Optimal Routing for a Family of Scalable Interconnection NetworksZhipeng Xu1,2 and Yuefan Deng1 (Advisor)
Optimal Routing
Benchmark Re-calibration on a Beowulf Cluster
Simulation Results
• Re-calibrated the benchmark results for the research [2].
• The optimized routing strategy enhanced the performance for Grap500 on the Beowulf cluster with the optimal routing.
Graph500-BFS Graph500-SSSP
Simulation Results of MPI All-to-all
Simulation Results of Bandwidth Efficiency
Simulation Results of Benchmark NPB FT
• The optimal routing improves performance tremendously for product networks.
Future Work• The Cartesian product method is very effective
for constructing a larger graph from several specified small graphs [1].
• Starting from a series of optimal regular graphs with minimum diameter, we can create arbitrarily large graphs.
Objective Function:
• If routing of G1, G2 are optimal, then G1⊗ G2 is also optimal.
Taishan Beowulf Cluster [2]
• If the source-destination pair has only one feasible the shortest path, the fixed transit loading (hk) of the nodes on this path will add 1.
(16,3)-optimal [2]
(32,4)-optimal [2]
(14,3)-optimal (Heawood Graph)
(16,4)-optimal [2]
1.Compare our optimal routing with other routing algorithms.
2.Analyze the mixed integer nonlinear programming (MINLP) for optimal routing, to solve this problem with larger N directly.
References
(16 , 3)⊗ (16 , 3) (16 , 4)⊗ (16 , 4)
[1] Junming Xu. Topological structure and analysis of interconnection networks. Vol. 7. Springer Science & Business Media, 2013. [2] Yuefan Deng, et al. "Optimal Low-Latency Network Topologies for Cluster Performance Enhancement." arXiv preprint arXiv:1904.00513 (2019).
AcknowledgementsWe thank the Computing Center (SeaWulf Cluster) and iACS at Stony Brook University.
Contact:Zhipeng Xu ([email protected]), Yuefan Deng ([email protected])
We propose a scheme to construct a family of large and high-performance interconnection networks that are scalable, low-radix, minimum diameters. These networks, whose diameters grow linearly as their sizes grow exponentially, are generated by using the Cartesian products of smaller optimal networks of minimum diameters. For the smaller base networks, we design the vertex-balanced routing algorithm by considering the forwarding pressure at each vertex. Comparative benchmarks on a Beowulf cluster show significant improvement in performance after using the new routing algorithm. Each node of the new network generated from base graphs with low-diameter can also sustain balanced forwarding loadings if we apply optimal routing algorithms to the base network. Simulation results for larger networks show that the optimal routing algorithms achieve the gain of communication performance.
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pkj<latexit sha1_base64="4vL/M8+4Rkez3UwBAFrg7t6YPQo=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHoxWMF+wHtUrJptk2bTZYkK5Sl/8GLB0W8+n+8+W9M2z1o64OBx3szzMwLE8GN9bxvtLa+sbm1Xdgp7u7tHxyWjo6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuH4bua3npg2XMlHO0lYEJOB5BGnxDqpmfSy8WjaK5W9ijcHXiV+TsqQo94rfXX7iqYxk5YKYkzH9xIbZERbTgWbFrupYQmhYzJgHUcliZkJsvm1U3zulD6OlHYlLZ6rvycyEhsziUPXGRM7NMveTPzP66Q2ugkyLpPUMkkXi6JUYKvw7HXc55pRKyaOEKq5uxXTIdGEWhdQ0YXgL7+8SprVin9ZqT5clWu3eRwFOIUzuAAfrqEG91CHBlAYwTO8whtS6AW9o49F6xrKZ07gD9DnD+U4j1Y=</latexit>
�Node Path 1 Path 2 Path 3 Path 4 Path n
1 0 0 0 0 02 0 0 1 0 03 0 0 0 0 04 0 0 0 0 05 0 0 0 0 06 0 0 0 0 07 0 0 0 0 08 1 1 0 0 09 1 0 0 1 0
10 0 0 0 1 111 0 0 0 0 012 0 0 1 0 013 0 0 0 0 014 0 0 0 0 015 0 0 0 0 116 0 1 0 0 0
(8,3)-optimal (Cayley Graph)
minNX
k=1
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Pdk
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Subject to: 8sj 2 {0, 1}, 8pkj 2 {0, 1}ngroupX
i=1
gijsj = 1
dk = hk +nrouteX
j=1
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