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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