Properties of Interconnection Networks

Static interconnection networks use fixed links between the nodes. They can be described by a connection graph G =(V,E) where V is a set of nodes to be con-nected and E is a set of direct connection links between the nodes. If there is a direct physical connection in the network between the nodes u ∈ V andv ∈V , then it is (u, v)∈E. For most parallel systems, the interconnection network is bidirectional.

This means that along a physical link messages can be transferred in both directions at the same time. Therefore, the connection graph is usually defined as an undirected graph. When a message must be transmitted from a node u to a nodev and there is no direct connection between u andv in the network, a path from u tov must be selected which consists of several intermediate nodes along which the message

is transferred. A sequence of nodes (v0, . . . , vk) is called path of length k between v0andvk, if (vi, vi+1)∈ E for 0≤i <k. For parallel systems, all interconnection networks fulfill the property that there is at least one path between any pair of nodes u, v∈V .

Static networks can be characterized by specific properties of the connection graph, including the following properties: number of nodes, diameter of the net-work, degree of the nodes, bisection bandwidth, node and edge connectivity of the network, and flexibility of embeddings into other networks as well as the embedding of other networks. In the following, a precise definition of these properties is given.

The diameterδ(G) of a network G is defined as the maximum distance between any pair of nodes:

δ(G)= max

u,v∈V min

ϕpath from u tov

{k|k is the length of the pathϕfrom u tov}.

The diameter of a network determines the length of the paths to be used for message transmission between any pair of nodes. The degree g(G) of a network G is the maximum degree of a node of the network where the degree of a node n is the number of direct neighbor nodes of n:

g(G)=max{g(v)|g(v) degree ofv ∈V}.

In the following, we assume that|A|denotes the number of elements in a set A.

The bisection bandwidth B(G) of a network G is defined as the minimum number of edges that must be removed to partition the network into two parts of equal size without any connection between the two parts. For an uneven total number of nodes, the size of the parts may differ by 1. This leads to the following definition for B(G):

B(G)= min

U₁,U₂partition of V

||U1|−|U2||≤1

|{(u, v)∈ E |u ∈U1, v∈U2}|.

B(G)+1 messages can saturate a network G, if these messages must be transferred at the same time over the corresponding edges. Thus, bisection bandwidth is a mea-sure for the capacity of a network when transmitting messages simultaneously.

The node and edge connectivity of a network measure the number of nodes or edges that must fail to disconnect the network. A high connectivity value indicates a high reliability of the network and is therefore desirable. Formally, the node con-nectivity of a network is defined as the minimum number of nodes that must be deleted to disconnect the network, i.e., to obtain two unconnected network parts (which do not necessarily need to have the same size as is required for the bisection bandwidth). For an exact definition, let GV\M be the rest graph which is obtained by deleting all nodes in M ⊂V as well as all edges adjacent to these nodes. Thus, it is GV\M=(V \M,E∩((V\M)×(V\M))). The node connectivity nc(G) of G is then defined as

nc(G)= min

M⊂V{|M| | there exist u, v∈V\M, such that there exists no path in GV\Mfrom u tov}.

Similarly, the edge connectivity of a network is defined as the minimum number of edges that must be deleted to disconnect the network. For an arbitrary subset F ⊂ E, let GE\F be the rest graph which is obtained by deleting the edges in F , i.e., it is GE\F =(V,E\F ). The edge connectivity ec(G) of G is then defined as

ec(G)=min

F⊂E{|F| | there exist u, v∈V, such that there exists no path in GE\F from u tov}.

The node and edge connectivity of a network is a measure of the number of indepen-dent paths between any pair of nodes. A high connectivity of a network is important for its availability and reliability, since many nodes or edges can fail before the network is disconnected. The minimum degree of a node in the network is an upper bound on the node or edge connectivity, since such a node can be completely sepa-rated from its neighboring nodes by deleting all incoming edges. Figure 2.11 shows that the node connectivity of a network can be smaller than its edge connectivity.

Fig. 2.11 Network with node connectivity 1, edge connectivity 2, and degree 4.

The smallest degree of a node is 3

The flexibility of a network can be captured by the notion of embedding. Let G = (V,E) and G = (V,E) be two networks. An embedding of G into G assigns each node of Gto a node of G such that different nodes of Gare mapped to different nodes of G and such that edges between two nodes in Gare also present between their associated nodes in G [19]. An embedding of Ginto G can formally be described by a mapping functionσ : V→V such that the following holds:

• if u =vfor u, v∈V, thenσ(u) =σ(v) and

• if (u, v)∈E, then (σ(u), σ(v))∈E.

If a network Gcan be embedded into a network G, this means that G is at least as flexible as G, since any algorithm that is based on the network structure of G, e.g., by using edges between nodes for communication, can be re-formulated for G with the mapping functionσ, thus using corresponding edges in G for communication.

The network of a parallel system should be designed to meet the requirements formulated for the architecture of the parallel system based on typical usage pat-terns. Generally, the following topological properties are desirable:

• a small diameter to ensure small distances for message transmission,

• a small node degree to reduce the hardware overhead for the nodes,

• a large bisection bandwidth to obtain large data throughputs,

• a large connectivity to ensure reliability of the network,

• embedding into a large number of networks to ensure flexibility, and

• easy extendability to a larger number of nodes.

Some of these properties are conflicting and there is no network that meets all demands in an optimal way. In the following, some popular direct networks are presented and analyzed. The topologies are illustrated in Fig. 2.12. The topological properties are summarized in Table 2.2.