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

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

Hypercubes Hypercubes are designed recursively How ?

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

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

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

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

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

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

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

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

4 - Cube 0110 0100 0101 0000 0010 0111 0010 0011

Reduced hypercube 100 101 000 001 110 111 010 011

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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) ] = [ 0 1 ] or 1 - Cube 0 1

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, 01 , 11, 10 ] 00 10 01 11 2- cube

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 = 0 1 2 3 4 5 6 7 HYPERCUBE = 0 1 3 2 6 7 5 4

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

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

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

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

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

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

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

3-Cube/16-node array 100 101 000 001 3-cube 110 111 010 011 HYPERCUBE = 0 1 3 2 6 7 5 4 100 101 000 001 3-cube 110 111 010 011

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

Embed an 8-leave Binary Tree into a 3-Cube 000 000 000 000 001 010 011 100 101 110 111 1. Choose the address of the parent to be the address of the left child.

Embed an 8-leave Binary Tree into a 3-Cube 000 000 000 010 100 110 000 001 010 011 100 101 110 111 Choose the address of the parent to be the address of the left child.

Embed an 8-leave Binary Tree into a 3-Cube 000 000 100 Height = n = 4 Nodes = 2^n-1 000 010 100 110 000 001 010 011 100 101 110 111 Choose the address of the parent to be the address of the left child.

Embed an 8-leave Binary Tree into a 3-Cube 000 Level- 3 000 100 Level- 2 000 010 100 110 Level- 1 Level- 0 000 001 010 011 100 101 110 111

Create Tables (connection) for each level 000 Level- 3 000 100 Level- 2 000 010 100 110 Level- 1 Level- 0 000 001 010 011 100 101 110 111

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

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

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

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

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

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

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

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

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