What is Cube-Connected Networks in Computer Architecture?

Cube-connected networks are patterned after the n-cube structure. An n-cube (hypercube of order n) is defined as an undirected graph having 2n vertices labeled 0 to 2n - 1 such that there is an edge between a given pair of vertices if and only if the binary representation of their addresses differs by one and only one bit. In a cube-based multiprocessor system, processing elements are positioned at the vertices of the graph. Edges of the graph represent the point-to-point communication links between processors.

Each processor in a 4-cube is connected to four other processors. In an n-cube, each processor has communication links to n other processors. This property allows for a simple message routing mechanism. The route of a message originating at node i and destined for node j can be found by XOR-ing the binary address representation of i and j. If the XOR-ing operation results in a 1 in a given bit position, then the message has to be sent along with the link that spans the corresponding dimension.

For example, if a message is sent from source (S) node 0101 to destination (D) node 1011, then the XOR operation results in 1110. That will mean that the message will be sent only along dimensions 2, 3, and 4 (counting from right to left) to arrive at the destination. The order in which the message traverses the three dimensions is not important. Once the message traverses the three dimensions in any order it will reach its destination.

In an n-cube, each node has a degree n. The degree of a node is defined as the number of links incident on the node. The upper limit on the number of disjoint paths in an n-cube is n. The hypercube is referred to as a logarithmic architecture. This is because the maximum number of links a message has to traverse to reach its destination in an n-cube containing N - 2n nodes is log2 N = n links.

One of the desirable features of hypercube networks is the recursive nature of their constructions. An n-cube can be constructed from two subcubes each having an (n- 1) degree by connecting nodes of similar addresses in both subcubes.