Parallel Algorithms on the Rotation-Exchange Network – A Trivalent Variant of the Star Graph Chi-Hsiang Yeh and Emmanouel A. Varvarigos Department of Electrical and Computer Engineering University of California, Santa Barbara, CA 93106-9560, USA Abstract We investigate a trivalent Cayley graph, which we call the rotation-exchange (RE) network, and present commu- nication algorithms to perform one-to-one routing, single- node broadcasting, multinode broadcasting, and total ex- change in it. The RE network can be viewed as a star- graph counterpart to the hypercubic shuffle-exchange net- work, with the important difference that the RE network is regular and symmetric. We show that RE networks can effi- ciently embed and emulate star graphs, meshes, hypercubes, cube connected cycles (CCC), pancake graphs, bubble- sort graphs, complete transposition graphs, and the shuffle- exchange permutation graphs. We also show that the per- formance of RE networks can be significantly improved for a variety of applications if the transmission rate of on-chip links is considerably higher than that of off-chip links. 1. Introduction A variety of topologies have been proposed and analyzed in the literature [2, 16, 23, 25, 29, 33] for the interconnection of processors in parallel computing systems, under several assumptions on the communication model used. Among them, the star graph [2, 3] has received a great deal of atten- tion as an attractive alternative to the hypercube for building parallel computers. Star graphs belong to the class of Cay- ley graphs [3], are symmetric and strongly hierarchical, and have diameter, average distance, and node degree that are superior to those of similar-sized hypercubes. Also, many important algorithms can be efficiently performed on the star graph [4, 6, 7, 10, 22, 24]. Even though the hypercube and the star graph have many desirable topological and algorithmic properties, their node degrees increase with the size of the network. Several constant-degree networks, such as the cube connected cy- cles (CCC) [23], the shuffle-exchange (SE) networks, the de Bruijn graphs [20], the star connected cycles (SCC) [17], the shuffle-exchange permutation (SEP) graphs [18], and the cyclic networks [30], have been proposed as alternatives to the hypercube and the star graph topologies. Since the SCC graph inherits some important properties from the star graph, and the star graph has been shown to be superior to the hypercube in several aspects, the SCC graph has some important advantages over the CCC network under certain assumptions [17]. The well-known shuffle-exchange (SE) network, which is another hypercubic network, has diameter that is somewhat smaller than that of a similar-sized CCC, and can emulate a hypercube of the same size with simpler and faster algorithms than a CCC [20]. The SE network, however, is neither symmetric nor regular. The trivalent Cayley graph to be studied in this paper can be viewed as a star-graph counterpart to the hypercubic SE network, and will be referred to as the rotation-exchange (RE) network. The RE network first appeared as an example of group graphs in [1], but its topological and algorithmic properties have not been explored in the literature before. We show that, as is the case with the SCC graph, the RE net- work inherits many desirable properties from the star graph, and is therefore in many respects superior to the CCC and SE networks under certain assumptions. Since the relation- ship between the RE network and the star graph is similar to that between the SE network and the hypercube, the RE net- work can embed and emulate a star graph of the same size as well as a variety of other network topologies with faster and considerably simpler algorithms than the corresponding embeddings and emulation for an SCC graph. In contrast to the SE network, the RE network is both regular and vertex- symmetric. We present efficient algorithms to perform one-to-one routing, single-node broadcasting, multinode broadcasting, and total exchange in RE networks. We also derive simple and efficient embeddings and emulation of star graphs [2, 3], meshes, hypercubes, CCC [23], pancake graphs [3], bubble- sort graphs [3], complete transposition graphs [19, 20], and shuffle-exchange permutation graphs [18], under a variety of assumptions on the communication model. We assume that several processors of the RE network are placed on the same module (e.g., chip, board, wafer, or multi-chip module (MCM)) and look at the case where