- Author:
- Osman Guzide,
- Adviser:
- Meghanad D. Wagh
Performance of distributed memory parallel architectures is heavily dependent upon the topology of the underlying interconnection network used to communicate data and intermediate results between processors. Butterfly is a popular network topology that has low diameter, symmetry, fixed node degree, a large number of processors and applicability to a large number of parallel algorithms.
This dissertation gives a new model for the butterfly network based on a direct product of a group and a finite field. This model allows the network connectivity to be expressed as an algebraic relation between the elements of the abstract structure. Using this model, all the possible cycle subgraph mappings on butterfly are obtained. The largest tree has also been mapped on a butterfly (of dimension less than 16) with unit load and dilation 2.
The binary butterfly is generalized to the p -ary butterfly and is shown to follow a similar model. Most binary butterfly results may be generalized to this network. Three new networks, the enhanced butterfly, the extended butterfly and the product butterfly, derived from the binary butterfly are also discussed. All of these networks are Cayley graphs with fixed node degree and significantly improve the mapping and combinatorial characteristics of the butterfly network.
The enhanced butterfly reduces the butterfly diameter 67% by adding one link per processor. It allows better cycle mappings than the butterfly. The extended butterfly has the same diameter as the butterfly but increases the cardinality by a factor equal to the dimension at the cost of four extra links per processor. It supports mappings similar to those on the butterfly. The product butterfly of dimension nk has a diameter equal to the sum of the diameters of butterflies of dimension n and k , and a cardinality, which is a product of the two cardinalities. Besides all the mappings supported by the butterfly, it also supports the mapping of mesh.
This dissertation has, for the first time, modeled fixed node degree symmetric interconnection networks by using groups and finite fields. Abstract algebraic tools can then be used to explore properties, mappings and designs of these networks.
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