A classification of MINs

In the following we list some necessary definitions of MINs for the proposed classification.

One of the most important issues concerning an IN topology is the existence or absence of multi-paths. A uni-path network is also called a Banyan network.

Definition 8. A Banyan network [37] is a Hasse diagram of a partial ordering in which there is one and only one path from any input node to any output node.

In a uniform MIN, all switching elements of a stage are of the same degree, and a square MIN of degree r is built from SEs of size r [96]. A rectangular network is one that has the same number of inputs and outputs.

We propose in figure 2.13 a topological classification of MINs. In the following we explain each of the branches of the classification tree.

MINs

Multi-computers(distributed memory)

Multiprocessors (SMPs)

Banyan

Non-Banyan

Uniform

Non-Uniform

(Non-Square)

Square

Non-Square

Delta

Non-Delta

Square

Delta

Non-Delta

Uniform

Non-Uniform

(Non-Square)

Non-Square

Square

Delta

Non-Delta

Figure 2.13: A topological classification of MINs

As stated above, MIINs can be used as communication systems in multi- processor or multi-computer machines. In this dissertation we are interested in MINs in multiprocessor environments. Here, MINs can either be Banyan or non- Banyan.

Banyan MINs, which are the main interest of this dissertation, may or may not have thedelta property or not. Delta networks, proposed by Patel [79], are built of

be an output of index i of a crossbar in a MIN. If an input of a crossbar in stage j is connected to an output oiof another crossbar in stage j−1, then all its other inputs must be connected to outputs of the same index i of crossbars in the previous stage. We propose the following mathematical generalization of Patel’s definition of the delta property.

Definition 9. For a Banyan MIN of size N and degree r2_{, suppose that the switch’s}

inputs and outputs are presented in the base r, of the form d0, d1, . . . , dr−1. Let the inputs

and outputs of the SEs in the network have the same indexes, then digits d0 of all inputs

of a switch must be equal. If a stage has this characteristic then it has the Delta property.

Definition 10. A Banyan network is called a Delta network if all its stages have the Delta property. In this case, it is said that network is having the Delta property.

Note that a network having the Delta property possesses some kind of reg- ularity so that the network routing algorithm can be simply defined [54]. Thus, Non-Delta Banyan networks are not of interest for this dissertation.

Delta networks will be studied in more detail in the next section because they constitute one case study to be tested later.

According to the proposed classification, uniform Banyan MINs can be square or non-square. Note that considering the above definitions, a non-uniform net- work is also non-square.

A DSUB (Delta, Square (also called SW in [37]), Uniform, Banyan) network is a Delta network with all its SEs being of the same size.

The Over-Sized Delta (OS Delta) network, proposed in a later chapter, is an example of the DnSUB (Delta, non-Square, Uniform Banyan) class. In a network of this class, switching elements of the first and the last stages are multiplexers and demultiplexers as will be explained, and thus, all SEs in a same stage are of the same size, while different stages have SEs of different sizes.

A Delta network of a size which is not a power of 2 can also be built as a DnSUB MIN. Figure 2.14 demonstrates an Omega network of size 6.

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

Figure 2.14: A Delta network of size 6

Non-Banyan MINs are, in general, more expensive than Banyan networks and more complex to control, still, they often are fault tolerant and capable of using

rerouting strategies to solve some conflicts that may occur in the network. Net-

works of this class can be constructed either by the augmentation of a Banyan

network or by the construction of amultipath network such as the Clos MIN [21].

Kruskal and Snir studied in [54] two augmentation strategies: replication and dilation which are defined by the authors as follows:

Definition 11. The d-dilation of a network G is a network obtained from G by replacing each edge (link) by d distinct edges.

Definition 12. The d-replication of a network G is a network consisting of d identical distinct copies of G.

Augmented networks, when built on the base of DSUB, form an example of the SUnB MINs, while Clos network is one example of the nSUnB class.