Interconnect_Topologies

Singleton graph 50 verbces degree 7 diameter 2 a 3rd

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Unformatted text preview: g2N)/2 Node degree is n = log2N "#\$% 0-D    1-D 2-D 3-D 4-D Slide from a presentation by Prof. Dr.-Ing. Axel Hunger 16 Interconnect Topologies ! Fall 2013 Are Hypercubes opBmal? Moore Graphs    Moore bound: given degree d and diameter k, the largest number of nodes a graph can have is: Conversely, given n verBces, and degree d, the best diameter we can hope for is: k = log(d ­1)(n)    Given n = 1000, d = 10, k >= 4    A Moore graph has the maximum number of verBces possible among all graphs with maximum degree d and diameter k.    17 Interconnect Topologies Fall 2013 Known Moore graphs          Peterson Graph: 10 verBces, degree = 3, diameter = 2. Singleton graph: 50 verBces, degree = 7, diameter = 2 A 3rd graph might possibly exist, of degree 57! 18 Interconnect Topologies Fall 2013 Why did Hypercubes Die?    Disadvantages of hypercube?    Wire lengths? May span the enBre machine       Leads to reduced bandwidth per link, unless its opBcal Copper vs opBcal interconnects:         Copper is cheaper but constrained by length Number of connecBons per node increases with the size of the machine K ­ary n ­cube allows one to seek intermediate points between meshes and hypercubes.    19 N ­dimensional “hyper” cube, but number of processors along each dimension is k (instead of 2 of binary hypercube) Interconnect Topologies Fall 2013 Ring (1-D Torus): N bi-directional Links -> O(N) complexity Rings (k ­ary 1 ­cube, or wrapped chain) Node Degree: 2    iameter 2 Ddegree: N is even: N/2 N is odd: P/2 odd )/    diameter: (N-1)/2 Average Distance: N is even: N2/4/(N-1)    average distance N is odd: (N+1)/4 P/4 ~ N/4    total links/link bandwidth: P Bisection Width = 2    bisecBon bandwidth:    Why did I draw ring as a circle? What if everyone wants to send data to every one else?    20...
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