1. Interconnection Networks
Direct
Indirect
Shared Memory
Distributed Memory (Message passing)

2. Topology Diameter: Longest path length between two processors
d = max {dmin(vi, vj) | for all vi, vj  V}
Node connectivity
min. # of nodes to be removed for the network to be disconnected
node connectivity r  There are r disjoint paths between every pair of nodes.

3. Topology Fault diameter Bisection width Cost
Suppose the node connectivity = r, diameter of the network with at most r-1 faulty nodes
Bisection width
The # of edges to be removed to separate the graph into two equal parts.
Cost
1) # of nodes & # of edges
2) Bisection width.

4. cw (Channel width): # of wires connecting two nodes.
cr (Channel rate): # of bits/sec/wire.
Channel bandwidth = cw  cr
Bisection bandwidth
= bisection width  channel bandwidth
Cost  Bisection bandwidth
bisection bandwidth is constant

5. Hypercube: N = 22n 2-D Torus: N = 22n 3-D Torus: w3d / wh = (N)1/3/4
bisection width = 22n-1
channel width = wh
bisection bandwidth = 22n-1 wh
2-D Torus: N = 22n
bisection width = 2 2n
channel width = wd
bisection bandwidth = 2n+1 wd
2n+1 wd = 22n-1 wh
wd / wh = 22n-1 / 2n+1 = 2n-2 = 2n/4 =  N /4
3-D Torus: w3d / wh = (N)1/3/4

6. Network Topology Graph Model: Processors Nodes
Wires joining processors Link

7. Linear Array # of processors = n # of links = n-1
Connection (i, i+1) for i=1,2, … , n-1
Diameter(Longest Path Length)= n-1
Communication delay
worst case: diameter
Average case: 1/n2  dij 1 2 3 n

8. Hypercube Hamming Distance Nodes are labeled as n-bit binary
DH(x, y) = # of positions in which x & y differ
x & y are binary vectors
DH(1100, 0111) = 3
Nodes are labeled as n-bit binary
Two nodes, x & y, are adjacent if DH(x, y) =1
101
111
001
011
110
100
000
010

9. Product of graphs G1 = (V1, E1) & G2 = (V2, E2)
G1  G2 = G = (V, E)= (V1  V2 , E)
Two nodes in G, ((v1, v2), (v3, v4)) are adjacent
1) if v1= v3 and (v2, v4)  E2 or
2) if v2= v4 and (v1, v3)  E1
Q2  Qn-1= Qn or Qk  Qn-k = Qk
101
111
001
011
110
100
000
010

10. Routing: A  B A=(an-1, an-2, … , a1, a0) # of steps required
B =(bn-1, bn-2, … , b1, b0)
an-1, an-2, … , a2, a1, a0
an-1, an-2, … , a2, a1, b0
an-1, an-2, … , a2, b1, b0
an-1, an-2, … , b2, b1, b0 •
an-1, bn-2, … , b2, b1, b0
bn-1, bn-2, … , b2, b1, b0
# of steps required
= DH(A, B)
Diameter of the network
= n = log2N
Average comm. delay
= 1/N (( )+2( ) … n( ))
= n2n-1/N = n2n-1/2n
= n/2 n1 n2 nn

11. Connectivity Let DH(x, y) = d Example:
n : node disjoint paths between x & y
d : paths have length d
n-d : paths have length d+2
Example:
 , d(110000,111111)=4

12. 11000
111111
Fault Diameter = n+1

13. De Bruijn Network # of nodes, N = 2n
Node A=(an-1, an-2, …, a0 ) is adjacent to
1. an-2, an-3, … , a1, a0, 0
2. an-2, an-3, … , a1, a0, 1
an-1, an-2, … , a1
an-1, an-2, … , a1

14. De Bruijn Network N=23 = 8 2n - 4 nodes have degree 4
# of links  42n / 2 = 2n+1
001
011
111
000
010
101
100
110

15. Routing: A  B A=(an-1, an-2, … , a1, a0) Example
B =(bn-1, bn-2, … , b1, b0)
an-1, an-2, … , a2, a1, a0
an-2, an-3, … , a1, a0, bn-1
an-3, an-4, … , a0, bn-1, bn-2
an-4, an-5, … , bn-1, bn-2, bn-3 •
bn-1, bn-2, … , b2, b1, b0
 log2N
Example
A = 1010
B = 1101
1010
0101
1011
0110
1101

16. bn-1, bn-2, … , b1, b0 2-disjoint paths
an-1, an-2, … , a1, a0
an-2, … , a1, a0, 1
an-3, … , a0, 1 , 1 …
1, 1, 1, … , 1, 1
b0, 1, 1, … , 1, 1
b1, b0, 1, … , 1, 1
bn-2, bn-3, … , b0, 1
an-2, … , a1, a0, 0
an-3, … , a0, 0 , 0 …
0, 0, 0, … , 0, 0
b0, 0, 0, … , 0, 0
b1, b0, 0, … , 0, 0
bn-2, bn-3, … , b0, 0
bn-1, bn-2, … , b1, b0 2-disjoint paths

17. k-ary n-cube Multidimensional torus 00 01 02 03 10 11 12 13 20 21 2223 30 31 32 33

18. k-ary n-cube
Given a node (an-1, an-2, … , a1, a0) adjacent to 2n nodes given by
(an-1, an-2, … , a1, a0±1)
(an-1, an-2, … , a1 ±1, a0) :
(an-1 ±1, an-2, … , a1, a0)
k=8, n=3 node (3 2 4) is adjacent to
(3, 2, 5), (3, 2, 3)
(3, 3, 4), (3, 1, 4)
(4, 2, 4), (2, 2, 4)
N = # of nodes in k-ary n-cube = kn

19. 00 01 02 03
101
111 10 11 12 13
001
011 20 21 22 23
110
100
000
010 30 31 32 33
k=2, n=3
Hypercube
k=4, n=2

20. k=8 Lee Distance DL ((a3, a2, a1, a0 ), (b3, b2, b1, b0 ))
=  min (ai - bi, bi - ai ) mod k
DL (1 2 3, 3 2 1) = min (1-3, 3-1) + min (2-2, 2-2) + min (3-1, 1-3) = = 4 1 7 2 6 5 3 4

21. Routing: A  B A=(an-1, an-2, … , a1, a0) Example (k=5)
B =(bn-1, bn-2, … , b1, b0)
an-1, an-2, … , a2, a1, a0 ±1
an-1, an-2, … , a2, a1, a0 ±2 :
an-1, an-2, … , a2, a1, b0
an-1, an-2, … , b2, a1 ±1, b0
an-1, an-2, … , a2, b1, b0
bn-1, bn-2, … , b2, b1, b0
Example (k=5)
(4,4,2,1) (2,1,4,4)
(4,4,2,1)
(4,4,2,0)
(4,4,2,4)
(4,4,3,4)
(4,4,4,4,)
(4,0,4,4)
(4,1,4,4)
(3,1,4,4)
(2,1,4,4)
# of steps = DL (A, B)

22. Torus of size kn-1, kn-2, … , k1, k0
Mixed radix number system
Processor (an-1, an-2, … , a1, a0 )
0  ai < ki-1 i = 0, 1, 2, … n-1
Decimal value
= an-1 (kn-2 kn-3, … , k0) + an-2 (kn-3 kn-4, … , k0) + … + a0
Two nodes, A=(an-1, an-2, … , a1, a0) &
B =(bn-1, bn-2, … , b1, b0)
are adjacent if DL (A, B) = 1

23. Torus of size kn-1, kn-2, … , k1, k0
Example: (854)
(3,2,1) = 3(54) + 24 + 1
= =
Decimal to mixed radix … 1
69 = (3,2,1) … 2
Example: (888)
(3,2,4) = 382 + 2
= 3 = … 4
212 = (3,2,4) … 2
Radix k-number
(an-1, an-2, … , a1, a0)= an-1 kn-1 + an-2 kn-2 + … + a0

24. Embedding Cycles Hypercube - Gray codes 0 00 000 0000 1 01 001 0001
1100
1101
1111 …

25. Mapping (Binary to Gray)
000  000
001  001
010  011
011  010
100  110
101  111
110  101
111  100
f(xn-1xn-2 … x0)
= (gn-1gn-2 … g0)
where
gn-1 = xn-1
gi = xi xi+1
i= n-2, n-3, … ,0 

26. k-ary n-cube f: Radix  Gray f(xn-1xn-2 … x0) = (gn-1gn-2 … g0)
gn-1 = xn-1
gi = xi - xi+1 mod k
for i= n-2, n-3, … ,0

27. 3-ary 2-cube Radix Gray 0 00  00 1 01  01 2 02  02 3 10  12
 00
 01
 02
 12
 10
 11
 21
 k=5
  2243 00 01 02 10 11 12 20 21 22

28. De Bruijn Network 23 - node sj+3 + sj+1 + sj= 0 sj+3 = sj+1 + sj mod 2
Initial conditions s0 =1, s1 = 0 and s2 = 1
s0 s1 s2 s3 s4 s5 s6 s7 s8 s9
101
011
111
110
100
000
001
010
001
011
111
000
010
101
100
110

29. How to choose this different equation?
Take an n-r order difference equation whose characteristic equation is a primitive polynomial of degree r.
Primitive Polynomial
x2+x x7+x3+1
x3+x x8+x4+ x3+x2+1
x4+x x9+x4+1
x5+x x10+x3+1
x6+x sj+5 = sj+2 + sj

30. Disjoint Cycle sj+3 = sj+1 + sj+1
Initial conditions s0 =1, s1 = 0 and s2 = 1
s0 s1 s2 s3 s4 s5 s6 s7 s8 s9
101
010
100
000
001
011
110