Topological Properties


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1 Advanced Computer Architecture Topological Properties Routing Distance: Number of links on route Node degree: Number of channels per node Network diameter: Longest minimum routing distance between any two nodes, measured in hops Average distance between all pairs of nodes A network is partitioned by a set of links if their removal disconnects the graph. Bisection width: Minimum number of links whose removal disconnects the graph and cuts it in half
2 Chain and Ring Chain (Linear Array): N1 Links > O(N) complexity Node Degree: 12 Diameter = N 1 Average Distance = (N+1)/3 ~ N/3 Bisection Width = 1 Ring (1D Torus): N bidirectional Links > O(N) complexity Node Degree: 2 Diameter N is even: N/2 N is odd: (N1)/2 Average Distance: N is even: N 2 /4/(N1) N is odd: (N+1)/4 ~ N/4 Bisection Width = 2 2D Mesh and Torus Mesh: 2k(k 1) links Node degree: 2 4 Diameter: 2(k 1) Average distance? Bisection width: k Torus: 2k 2 links Node degree: 4 Diameter: k Average distance? Bisection width: 2k
3 Multidimensional Meshes 1D mesh 1D torus 2D mesh 3D mesh d dimensional mesh or torus : N = k d 1 x k d 2...x k 1 x k 0 nodes Described by d vector of coordinates (i d 1,..., i 0 ) Where 0 < i j k j for 0 j d 1 d dimensional k ary mesh: N = k d k = d N Described by d vector of radix k coordinate. Diameter = d (k 1) Multidimensional Tori 2D torus d dimensional dimensional k ary torus: Edges wrap around, every node has degree 2d and is connected to nodes that differ by one (mod k) in every dimension. d dimensional k ary cube: unidirectional links
4 Properties Distance: Relative distance: R = (b d1 a d1,..., b 0 a 0 ) Traverse r i = b i a i hops in each dimension Average Distance: d x k/3 for mesh d x k/4 for torus Degree: d to 2d for mesh 2d for torus Bisection width: k d1 for mesh 2k d1 for torus Examples Which is the order from low to high bisection width for 1024 nodes of the topologies a) bidirectional 2D torus, b) bidirectional ring, c) bidirectional 2D mesh, d) butterfly network e) binary tree?  e), b), c), d), a).  b), e), c), a), d). e), b), c), a), d) Topology: Which is the complexity in terms of switches of a butterfly network which connects 128 processors with 128 memory units?
5 Examples Interconnection networks, topology: Which is the average distance in a butterfly network with 64 nodes? a) 3 b) 4 c) 5 d) 6 = diameter log 2 N, same length for all routes Interconnection networks, topology: How many stages has a butterfly network with 16 nodes? a) 4 b) 6 c) 8 log 2 N d) 16
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