# 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): N-1 Links -> O(N) complexity Node Degree: 1-2 Diameter = N -1 Average Distance = (N+1)/3 ~ N/3 Bisection Width = 1 Ring (1-D Torus): N bi-directional Links -> O(N) complexity Node Degree: 2 Diameter N is even: N/2 N is odd: (N-1)/2 Average Distance: N is even: N 2 /4/(N-1) 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 d-1 -a d-1,..., 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 d-1 for mesh 2k d-1 for torus Examples Which is the order from low to high bisection width for 1024 nodes of the topologies a) bi-directional 2-D torus, b) bi-directional ring, c) bi-directional 2-D 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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