SLIDE 1
Hypercube Network
- Cayley graph of
the reflections of a square
- Creates a 2-D
square lattice
x y y x a b c
+ + + + + + + + + + +
x w z y x wx ywx wywx xwywx xywx zxywx xzxywx wxzxywx xwxzxywx
+
xwx
Honeycomb Networks Project by Jessica Wolf Hypercube Network - - PowerPoint PPT Presentation
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 + + +
SLIDE 2
SLIDE 3
Diameter of square lattice
- Use mod n
- Creates the
torus mod n
- The diameter of
the square torus Mod n
- Example mod 5
- Example mod 6
- Name points
using (n x n )
a b c
a b c d e
- dd
n n even n n , 1 ,
SLIDE 4
Routing algorithm
- Many different
possible paths to travel
- Find the
distance from A to B: B-A
- Rewrite these
coordinates in all possible ways
- Calculate
distance and take shortest path
1 2 3 4 1 2 3 4 5 6 7
A B
5
SLIDE 5
Honeycomb Mesh
x y z x y x y x y
- Reflection of a triangle in the x, y
axis give S3
- Reflections of a triangle in x,y,z axis
give the honeycomb mesh
SLIDE 6
Distance
- Use of translations s, t
- Nodes adressed as (x):S3
- Distance between (m,n,0) and
(0,0,0) is times (the number of m, n in common) (for m,n>0 or m,n<0)
x x x x x x x x x x x x x y y y y y y y y y y y y z z z z z z z z z z z z z z z z z z y y y y y y y y y x x x x x
+ + + + + +
s t
1 2
2 4 4 n m
SLIDE 7
Diameter and Bisection Width
- N+(n-1)
hexagons along the center
- N hexagons
along each side
- Diameter of the
honeycomb torus is
- The bisection
width is
+ + + +
n 2
n n 2 3
SLIDE 8
Routing in Honeycomb Torus
- Travelling from A to B similar to square
torus
- First calc A-1B = (s,t,g),
- Rewrite (s,t,g) in all possible
combinations mod n,
- Find the distances for each
- Deciding on shortest route.
+ + + + + + + + +
SLIDE 9
Higher Dimensions
- Use S4 instead of S3
- Coxeter groups
SLIDE 10
Applications
- Hypercube used in IBM’s blue gene
supercomputer
- Hexagonal networks used in mobile
phone networks
SLIDE 11
Conclusion
- Square Torus
– n2 nodes – Bisection width is 2/n – diameter is 1/n
- Honeycomb
– nodes – bisection width is 5/6n – diameter is 1/3n.
2
6 n