A Hierarchical Network HRN and Its routing Algorithms

In this paper, a new kind of hierarchical interconnection networks, HRN, its topological properties and its routing strategies are investigated. If the diameter of the atomic graph, G, is d(G), then the diameter of HRN(G,k 1 ,k 2 ,...,k m ) is (k m /2)+(k m-1 /2)+...+(k 1 /2)+d(G). After that, a subclass of HRN network, the RP(P,k 1 ,k 2 )network, is discussed. The properties of this network are compared with those of 2D Torus, 3D Torus, Hypercube and De Bruijn Graph. The result shows that the RP(P,k 1 ,k 2 )network has simple topologies and concise routing strategies. Then four routing algorithms, point-to-point, broadcast, all-to-all and permutation routing, are proposed on the RP(P,k 1 ,k 2 ) network. The performances of the first three algorithms are k 2 /2+k 1 /2+2, k 2 /2+k 1 /2+2, and 10×k 1 ×k 2 -4 routing time steps respectively. Permutation routing at the worst cast needs 4+min{k 2 ,k 1 }+(k 2 -1)×(k 1 -1) time steps, which shows the RP(P,k 1 ,k 2 ) network has good routing properties. Another contribution is that two parameters, the optimal node group and the optimal network partition, are proposed, which are used to evaluate the performances of interconnection networks. These two parameters describe the network performance when only one program or more than one programs run on interconnection networks. With these two parameters, the performance of the RP(P,k 1 ,k 2 ), 2D torus and Hypercube networks is analyzed. At last, a method to construct the optimal node group and the optimal network partition is given and an approximately optimal network partition is obtained on the RP(P,k 1 ,k 2 ) network.