Block-cyclic matrix triangulation on the Cartesian product of star graphs

Abstract

The star graph has drawn a lot of attention recently as an attractive alternative to hypercube. Due to its reduced diameter, the star graph theoretically supports more efficient communication than popular networks such as the hypercube and the mesh. However, practically only limited results have been obtained related to the design of parallel algorithms on the star graph. By its own nature, the star graph seems to be inadequate for certain types of algorithms especially those based on matrix computations. Furthermore, no efficient embeddings of hypercubes or meshes in the star graph are known, which would (had they existed) allow us to simulate the corresponding hypercube or mesh algorithms. In this paper, we show how to overcome these difficulties with the star graph while allowing to take advantage of its communication capabilities. We propose to consider the Cartesian product of star graphs as interconnection networks and we design and evaluate block-cyclic matrix triangulation on these networks. We demonstrate how such star graph based Cartesian product networks are more suitable for real applications than traditional star graphs, while inheriting the theoretically established communication efficiency of star graphs. We present a framework for practical implementation of block-cyclic matrix triangulation on the Cartesian product of star graphs along with a performance evaluation and comparison with other topologies. The proposed techniques for matrix decomposition and mapping are of general use and can be applied to design other matrixbased algorithms on the Cartesian product of star graphs.