Honeycomb Networks Project by Jessica Wolf Hypercube Network - PowerPoint PPT Presentation

Honeycomb Networks Project by Jessica Wolf

Hypercube Network • Cayley graph of x 0 a the reflections of a square y y • Creates a 2-D c b square lattice x ywx + + + xzxywx wx w wywx + + wxzxywx + x x xwywx + + + xwxzxywx 0 xywx + + + zxywx xwx y z

Diameter of square lattice • Use mod n a • Creates the torus mod n • The diameter of b c the square torus Mod n  n , n even    n 1 , n odd a • Example mod 5 d • Example mod 6 b c • Name points using (  n x  n ) e

Routing algorithm • Many different possible paths to travel • Find the distance from A 1 2 1 3 to B: B-A A 4 • Rewrite these 5 2 coordinates in all 6 possible ways 7 B • Calculate 3 4 5 distance and take shortest path

Honeycomb Mesh y y x y x x x y z • Reflection of a triangle in the x, y axis give S 3 • Reflections of a triangle in x,y,z axis give the honeycomb mesh

Distance • Use of translations s, t • Nodes adressed as (  x  ):S 3 • Distance between (m,n,0) and     4 m 4 n 2 (0,0,0) is times (the number of m, n in common) (for m,n>0 or m,n<0) y z y x x + z z y + x y z y y x z x z y x z y y z + x y x y z 1 2 x x z y z y x x z + x y + z y s t y z z y x x y z x z y y + x y z x

Diameter and Bisection Width • N+(n-1) hexagons along the center • N hexagons along each side + • Diameter of the honeycomb torus +  2 n is + • The bisection + width is    3 n 2 n

Routing in Honeycomb Torus •Travelling from A to B similar to square torus •First calc A -1 B = (s,t,g), •Rewrite (s,t,g) in all possible combinations mod n, •Find the distances for each •Deciding on shortest route. + + + + + + + + +

Higher Dimensions • Use S 4 instead of S 3 • Coxeter groups

Applications • Hypercube used in IBM’s blue gene supercomputer • Hexagonal networks used in mobile phone networks

Conclusion Square Torus • n 2 nodes – – Bisection width is 2/n – diameter is 1/n • Honeycomb  2 6 n – nodes bisection – width is 5/6n diameter is – 1/3n.