1. ICN’s The n-D hypercube (n-cube) contains 2^n nodes (processors).
The nodes are neighbors in the n-cube iff their end bit binary addresses differ in a single bit

2. Neighbor nodes
Two nodes are neighbors in the n-cube iff their end bit binary addresses differ by a single bit

3. Hypercubes
Hypercubes are designed recursively
How ?

4. 2 - Cube
2^2 = 4 nodes 00 01 10 11

5. 3 - Cube
2^3 = 8 nodes 00 01 00 01 10 11 10 11

6. 3 - Cube
2^3 = 8 nodes
100
101
000
001
110
111
010
011

7. 4 - Cube
2^4 = 16 nodes
100
101
100
101
000
001
001
000
110
110
111
111
011
010
010
011

8. 4 - Cube
2^4 = 16 nodes
100
101
100
101
000
001
001
000
110
110
111
111
011
010
010
011

9. 100
110
010
000
101
111
110
100
001
011
101
000
010
111
001
011

10. 100
110
010
000
101
111
110
100
001
011
101
000
010
111
001
011

11. 0100
0110
0010
0000
0101
0111
1110
1100
0001
0011
1101
1000
1010
1111
1001
1011

12. 4 - Cube
0110
0100
0101
0000
0010
0111
0010
0011

13. Reduced hypercube
100
101
000
001
110
111
010
011

14. Folded hypercube (4)
100
101
000
001
110
111
010
011

15. 3-cube routing: data exchange
The nodes are
coded as XYZ
100
101
000
001
110
111
010
011

16. Routing by inverting X The nodes are coded as XYZ 100 101 000 001 110
111
010
011

17. Routing by inverting Y The nodes are coded as XYZ 100 101 000 001 110
111
010
011

18. Routing by inverting Z The nodes are coded as XYZ 100 101 000 001 110
111
010
011

19. Hemming distance The shortest path , between nodes, is determined by
the Hamming
distance.
100
101
000
001
110
111
010
011

20. Hemming distance XOR (source and destination) and count the 1’s 100
101
000
001
110
111
010
011

21. Examples
010 -> 111 =
100
101
000
001
110
111
010
011

22. Examples 010 -> 111 = 010 XOR 111 = 101 = 2 hops 100 101 000 001
110
111
010
011

23. Examples
010 -> 110 =
100
101
000
001
110
111
010
011

24. Examples 010 -> 110 = 010 XOR 110 = 100 = 1 hops 100 101 000 001
111
010
011

25. Examples 010 -> 101 = 010 XOR 010 = 111 = 3 hops 100 101 000 001
110
111
010
011

26. Examples 010 -> 101 = 010 XOR 010 = 111 = 3 hops 100 101 000 001
110
111
010
011

27. Hypercube The hypercube can emulate (simulate) other ICN’s
The hypercube is a general purpose or universal ICN

28. Embedding a linear array
How can we embed other ICN’s into a hypercube ?

29. Algorithm Bits must differ by a single bit
Use the binary Reflected Gray Code

30. Binary Reflected Code RGC(n); n-bit binary Reflected Gray Code
RGC(n) = [ 0 . RGC(n-1), 1 . RGC (n-1) ] -1

31. 1-cube/2-node array or 1 - Cube 0 1
RGC(n); n-bit binary Reflected Gray Code
RGC(n) = [ 0 . RGC(n-1), 1 . RGC^{-1}(n-1) ]
= [ 0 . RGC(1-1), 1 . RGC^{-1}(1-1) ]
= [ 0. RGC(0) , 1 . RGC^{-1}(0) ]
= [ ] or
1 - Cube

32. 2-cube/4-node array 00 01 10 2- cube 11
RGC(n); n-bit binary Reflected Gray Code
RGC(2) = [ 0 . RGC(2-1), 1 . RGC^{-1}(2-1) ]
= [ 0 . RGC(1) , 1 . RGC^{-1}(1) ]
= [ 0 . ( 0,1) , 1 . (1,0) ]
= [ 00, , 11, 10 ] 00 10 01 11
2- cube

33. Linear array; n = 3 RGC(n); n-bit binary Reflected Gray Code
RGC(3) = [ 0 . RGC(3-1), 1 . RGC^{-1}(3-1) ]
= [ 0 . RGC(2) , 1 . RGC^{-1}(2) ]
= [ 0.(00,01,11,10 ) , 1.(10,11,01,00) ]
= [000,001,011,010,110,111,101,100
Addresses
ARRAY =
HYPERCUBE =

34. Linear array - Hypercube
100
101
000
001
110
111
010
011

35. Linear array - Hypercube
100
101
000
001
110
111
010
011

36. Linear array - Hypercube
100
101
000
001
110
111
010
011

37. Linear array - Hypercube
100
101
000
001
110
111
010
011

38. Linear array - Hypercube
100
101
000
001
110
111
010
011

39. Linear array - Hypercube
100
101
000
001
110
111
010
011

40. Linear array - Hypercube
100
101
000
001
110
111
010
011

41. 3-Cube/16-node array 100 101 000 001 3-cube 110 111 010 011
HYPERCUBE =
100
101
000
001
3-cube
110
111
010
011

42. Embed an 8-leave Binary Tree into a 3-Cube

43. Embed an 8-leave Binary Tree into a 3-Cube
000
000
000
1. Choose the address of the parent to be the
address of the left child.

44. Embed an 8-leave Binary Tree into a 3-Cube
000
000
000
010
100
110
Choose the address of the parent to be the
address of the left child.

45. Embed an 8-leave Binary Tree into a 3-Cube
000
000
100
Height = n = 4
Nodes = 2^n-1
000
010
100
110
Choose the address of the parent to be the
address of the left child.

46. Embed an 8-leave Binary Tree into a 3-Cube
000
Level- 3
000
100
Level- 2
000
010
100
110
Level- 1
Level- 0

47. Create Tables (connection) for each level
000
Level- 3
000
100
Level- 2
000
010
100
110
Level- 1
Level- 0

48. Table: Level -1 Level- 3 Level- 2 Level- 1 Level- 0 001 ->> 000
100
Level- 2
000
010
100
110
Level- 1
Level- 0
001 ->> 000
011 ->> 010

49. Table: Level -1 Level- 3 Level- 2 Level- 1 Level- 0 001 ->> 000
100
Level- 2
000
010
100
110
Level- 1
Level- 0
001 ->> 000
011 ->> 010
101 ->> 100
111 ->> 110

50. Table: Level -2 Level- 3 Level- 2 Level- 1 Level- 0 010 ->> 000
100
Level- 2
000
010
100
110
Level- 1
Level- 0
010 ->> 000
110 ->> 100

51. Table: Level -3 Level- 3 Level- 2 Level- 1 Level- 0 100 ->> 000
010
100
110
Level- 1
Level- 0
100 ->> 000

52. Draw: Level - 1 001 ->> 000 011 ->> 010 100 101
011 ->> 010
100
101
101 ->> 100
111 ->> 110
000
001
110
111
010
011

53. Draw: Level - 2 001 ->> 000 011 ->> 010 100 101
011 ->> 010
100
101
101 ->> 100
111 ->> 110
000
001
010 ->> 000
110 ->> 100
110
111
010
011

54. Draw: Level - 3 001 ->> 000 011 ->> 010 100 101
011 ->> 010
100
101
101 ->> 100
111 ->> 110
000
001
010 ->> 000
110 ->> 100
110
111
100 ->> 000
010
011

55. 8-leave Tree/3-cube
100
101
000
001
110
111
010
011

56. 8-leave Tree/3-cube
000
001
011
010
100
101
111
110